Abdelaati Daouia
email: daouia@cict.fr

Irène Gijbels
email: irene.gijbels@wis.kuleuven.be

Robustness and inference in nonparametric partial-frontier modeling

Keywords: JEL classification: C13, C14, D20 asymptotics, breakdown values, econometric frontiers, outliers detection

A major aim in recent nonparametric frontier modeling is to estimate a partial frontier well inside the sample of production units but near the optimal boundary. Two concepts of partial boundaries of the production set have been proposed: an expected maximum output frontier of order m = 1, 2, . . . and a conditional quantiletype frontier of order α ∈]0, 1]. In this paper, we answer the important question of how the two families are linked. For each m, we specify the order α for which both partial production frontiers can be compared. We show that even one perturbation in data is sufficient for breakdown of the nonparametric order-m frontiers, whereas the global robustness of the order-α frontiers attains a higher breakdown value. Nevertheless, once the α-frontiers break down, they become less resistant to outliers than the orderm frontiers. Moreover, the m-frontiers have the advantage to be statistically more efficient. Based on these findings, we suggest a methodology for identifying outlying data points. We establish some asymptotic results, contributing to important gaps in the literature. The theoretical findings are illustrated via simulations and real data.

Introduction

In the economics, statistics, management science and related literatures a major aim is to estimate the upper boundary of a sample {(X i , Y i ), i = 1, . . . , n} of independent copies of a random production unit (X, Y ) with support defined by {(x, y) ∈ R p+1 + |0 ≤ y ≤ ϕ(x)}. Econometric considerations lead to the natural assumption that the frontier function ϕ is monotone nondecreasing. Let (Ω, A, P) be the probability space on which the vector of inputs X ∈ R p + and the single output Y are defined. Then following [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF], the optimal value ϕ(x) can be characterized as the right-endpoint of the conditional distribution function F (y|x) = P(Y ≤ y|X ≤ x) = F (x, y)/F X (x), with F (•, •) and F X (•) being respectively the joint and marginal distribution functions of (X, Y ) and X.

The conventional estimate for ϕ is the Free Disposal Hull (FDH) estimator, i.e. the lowest nondecreasing step surface covering all sample points, that is φn (x) := sup{y ≥ 0| Fn (y|x) < 1} = max i|X i ≤x Y i , where Fn (y|x) = F (x, y)/ FX,n (x), with F (x, y) = (1/n) n i=1 1I(X i ≤ x, Y i ≤ y) and FX,n (x) = F (x, ∞). When the frontier function ϕ is also assumed to be concave, a popular estimator is the Data Envelopment Analysis (DEA) estimator, which is the lowest concave surface covering the FDH frontier. Both FDH and DEA estimators derive from the pioneering work of [START_REF] Farrell | The measurement of productive efficiency[END_REF]. The DEA frontier has been popularized by Charnes at al (1978), while the FDH has been proposed by [START_REF] Deprins | Measuring labor inefficiency in post offices[END_REF]. See [START_REF] Simar | Statistical Inference in Nonparametric Frontier Models: recent developments and perspectives[END_REF] for a survey on inference techniques using FDH and DEA estimators. By construction, these envelopment estimators are very sensitive to extremes and/or outliers in the output-direction. This dramatic lack of robustness results in poor estimation of the corresponding economic efficiencies; the efficiency score of a firm is estimated via the distance between the attained produced output and the optimal production level given by the frontier function. Of course, in production activity, outlying outputs Y i are highly desirable. But in absence of information on whether the observations are measured accurately, it is prudent to seek frontier estimators which are not determined by very few extreme observations. The underlying idea of the two existing methods in the econometric literature is to estimate a partial frontier well inside the cloud of data points but near the upper boundary.

The first concept of a partial boundary of the joint support of (X, Y ) has been introduced by [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF]. Given an integer m ≥ 1, they define a notion of expected maximum output function of order m as ξ m (x) := ϕ(x) 0

(1 -[F (y|x)] m )dy. This partial frontier function converges to the full frontier function ϕ(x) as m → ∞. It is estimated by ξm,n (x) := φn(x) 0

(1-[ Fn (y|x)] m )dy. To summarize the properties of this estimator, for fixed sample size n we have lim m→∞ ↑ ξm,n (x) = φn (x), and for fixed order m we have √ n( ξm,n (x)ξ m (x)) → N (0, σ 2 (x, m)) as n → ∞, where the asymptotic variance σ 2 (x, m) is given in (4.2). The second concept of a partial frontier function, suggested by [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF], is defined as the αth quantile function q α (x) := inf{y ≥ 0|F (y|x) ≥ α}, with α ∈]0, 1]. This orderα frontier function converges to ϕ(x), as α ↑ 1, and is estimated by qα,n (x) := inf{y ≥ 0| Fn (y|x) ≥ α}. For n fixed, this estimator satisfies lim α→1 ↑ qα,n (x) = φn (x), and for fixed order α we have √ n(q α,n (x)q α (x)) → N (0, σ 2 (α, x)) as n → ∞, where σ 2 (α, x) is given in (4.3), provided that F (•|x) is differentiable at q α (x) with derivative f (q α (x)|x) > 0.

Unlike usual (FDH, DEA) methods, both alternatives {q α,n (x)} and { ξm,n (x)} are qualitatively robust and bias-robust as shown in [START_REF] Daouia | Robust Nonparametric Frontier Estimators: Influence Function and Qualitative Robustness[END_REF]. But the order-α quantile frontiers can be more robust to extremes than the order-m frontiers when estimating the true full frontier ϕ (i.e. when α ↑ 1 and m ↑ ∞) since the influence function is no longer bounded for order-m frontiers as m tends to infinity, while it remains bounded for the conditional quantile frontiers as the order α tends to one. This advantage is proved only under the condition that the conditional density function f (•|x) is not null and continuous on its support. No attention was devoted however to the difference between the reliability of the two sequences of estimators {q α,n (x)} and { ξm,n (x)} in the general setting. Moreover, the influence function only offers a local quantification of robustness by measuring the sensitivity of estimators to infinitesimal perturbations, but it is well known that estimators can be infinitesimally robust and yet still highly sensitive to small, finite perturbations. To measure the global robustness of an estimator, the richest quantitative information is provided by the finite sample breakdown point as shown by [START_REF] Donoho | The notion of breakdown point[END_REF]. It measures the smallest fraction of contamination of an initial sample that can cause an estimator to take values arbitrarily far from its value at the initial sample.

In this paper, we deal with global robustness and some asymptotic aspects of the two sequences {q α,n } and { ξm,n } as estimators of the partial frontiers q α and ξ m respectively, for fixed orders α and m, and as estimators of the full frontier ϕ itself when α → 1 and m → ∞.

In Section 2 we first focus on the replacement breakdown values of the estimators. We

show that, as expected, both FDH and DEA frontiers may break down for any contamination (Lemma A.1). The surprising result is that even one outlying observation is sufficient for breakdown of the partial frontier ξm,n (x) for any order m, whereas the partial order-α frontier qα,n (x) has the desirable robustness in withstanding the contamination of outlying observations. While the asymptotic breakdown value is 0 for any order-m partial frontier, it is (1α)F X (x) for the sequence {q α,n (x)} n≥1 . But, once the α-frontiers break down, they become less resistant to outliers than the order-m frontiers. A natural question arising is: how to compare the reliability between the two sequences of partial frontiers once the order-α frontier also breaks down? A more general question is: given a fixed order m, which frontier function qα,n (x) can be analyzed and compared with ξm,n (x)?

The families {ξ m (•), m ≥ 1} and {q α (•), α ∈]0, 1]} have emerged in the econometric literature as two different theoretical concepts of partial production frontiers. See e.g. [START_REF] Daraio | Advanced Robust and Nonparametric Methods in Efficiency Analysis[END_REF] for statistical properties of both concepts of partial boundaries together with several appealing economic features. The estimators ξm,n and qα,n (of ξ m and q α respectively) cannot be compared since they do not estimate the same quantity except for the limiting case, when m ↑ ∞ and α ↑ 1 (when both estimate the true full frontier ϕ). We however establish in Proposition 2.2 that the two concepts of partial boundaries are closely linked in the sense that for each order m ≥ 1, there exists a well-specified order α = α(m) = (1/2) 1/m such that the theoretical order-m and order-α frontiers are respectively the mean and median of the same distribution, namely that of the maximum of m independent random variables drawn from the law of Y given X not exceeding some level of inputs. This result also confirms the advantage of qα,n over ξm,n in terms of finite sample breakdown point and gross-error sensitivity, but such a robust proposal may sacrifice statistical efficiency.

We show in Section 3 how these results can be exploited to detect outlying data points in the output-direction. In the frontier modeling context, descriptive methods for identifying outliers have been proposed by [START_REF] Wilson | Detecting outliers in deterministic nonparametric frontier models with multiple outputs[END_REF][START_REF] Wilson | Detecting influential observations in data envelopment analysis[END_REF]. Although very useful, these methods are very computer intensive as the sample size increases and are based on some tuning parameters whose choice is not justified. See further discussions in Sections 3 and 6.

Section 4 contributes to important gaps in the asymptotic theory for the estimators ξm,n (x) and qα,n (x). We establish pointwise and functional asymptotic representations for √ n( ξm,n (x)ξ m (x)) and improve its order of convergence to O( √ log log n). Similar asymptotic properties for √ n(q α,n (x)q α (x)) can be found in [START_REF] Daouia | Asymptotic Representation Theory for Nonstandard Conditional Quantiles[END_REF] and [START_REF] Daouia | Functional Convergence of Quantiletype Frontiers with Application to Parametric Approximations[END_REF]. However, unlike ξ m (x), the computation of the asymptotic confidence interval of q α (x) requires estimation of the quantile density function f (q α (x)|x), often resulting in estimates of unsatisfactory accuracy for finite samples. To avoid this problem, we derive an alternative asymptotic confidence interval for q α (x) not requiring knowledge of f (q α (x)|x).

Finally, we show under general conditions that the asymptotic normality of both ξm,n (x) and qα,n (x) is still valid when m = m n → ∞ and α = α n → 1 as n → ∞. Section 5 illustrates the theoretical findings through simulations and real data and Section 6 concludes.

Robustness

The most successful notion of global robustness of an estimator T at a sample (Z) n = (Z 1 , . . . , Z n ) is provided by the finite sample breakdown point of [START_REF] Donoho | The notion of breakdown point[END_REF]:

RB(T, (Z) n ) = min k n : k = 1, .
. . , n, and satisfies sup

(Z) n k |T {(Z) n k } -T {(Z) n }| = ∞ ,
where (Z) n k denotes the contaminated sample by replacing k points of (Z) n with arbitrary values. Given m ≥ 1, α ∈]0, 1] and x ∈ R p + , the partial boundaries ξm,n (x) and qα,n (x) are representable as functionals of the joint empirical distribution function F (•, •), or equivalently, of the data set (X, Y ) n = {(X i , Y i ), i = 1, . . . , n}:

ξ m (x) = S m,x (F ) ξm,n (x) = S m,x ( F ) = S m,x ((X, Y ) n ) , q α (x) = T α,x (F ) qα,n (x) = T α,x ( F ) = T α,x ((X, Y ) n )
where the operators S m,x and T α,x associate to a distribution function

G(•, •) on R p + × R + such that G(x, ∞) > 0, the real values S m,x (G) = ∞ 0 1 - G(x, y) G(x, ∞) m dy and T α,x (G) = inf y ≥ 0| G(x, y) G(x, ∞) ≥ α ,
with the integrand being identically zero for

y ≥ inf{y | G(x, y)/G(x, ∞) = 1}.
As expected we can easily show that even one outlying observation is sufficient for breakdown of the FDH frontier (see Lemma A.1, Appendix), and consequently for breakdown of the DEA frontier. But the surprising result is that the partial order-m boundary breaks down for the same fraction, 1/n, of contamination as the envelopment FDH and DEA estimators, for any order m.

Theorem 2.1. Let x ∈ R p + such that FX,n (x) > 0. Then, for any order m ≥ 1,

RB( ξm,n (x), (X, Y ) n ) = 1/n.
Hence the asymptotic breakdown value is 0 for any order-m partial frontier. In contrast, by an appropriate choice of the order α as a function of n and FX,n (x), we can derive a partial quantile-based frontier qα,n (x) capable of withstanding arbitrary perturbations of a significant proportion of the data points without disastrous results.

Theorem 2.2. Let x ∈ R p + such that FX,n (x) > 0.
Then, for any order α ∈]0, 1],

RB(q α,n (x), (X, Y ) n ) =    n(1 -α) FX,n (x) + 1 /n if αn FX,n (x) = 1, 2, 3, . . . n FX,n (x) -αn FX,n (x) /n otherwise,
where [αn FX,n (x)] denotes the integer part of αn FX,n (x).

Remark 2.1. The asymptotic breakdown value for the sequence {q α,n (x)} n≥1 is then (1α)F X (x). When the order α is fixed, this theorem reflects how the corresponding partial frontier qα,n (x) suffers from the left-border effect when the vector x ∈ R p + of inputs-usage is too small. Likewise, increasing the dimension p of input factors x decreases FX,n (x), and hence RB(q α,n (x), (X, Y ) n ) goes down. On the other hand, once we know that qα,n (x) = T α,x ((X, Y ) n ) does not break down for the fraction (k * -1)/n of contamination, with k * /n = RB(q α,n (x), (X, Y ) n ), it is of interest to know how large the bias |T α,x ((X, Y ) n k * -1 ) -T α,x ((X, Y ) n )| can be. For this purpose we compute the upper bound of this bias, here we only focus on contaminated samples (X, Y ) n k * -1 := (X, Y ) n,y k * -1 in the direction of Y obtained by replacing k * -1 points (X i , Y i ) with outlying extreme-values (X i , Y * i ).

Proposition 2.1. Let x ∈ R p + such that FX,n (x) > 0. Then, for any order α ∈]0, 1],

0 ≤ T α,x ((X, Y ) n,y k * -1 ) -T α,x ((X, Y ) n ) ≤ φn (x) -qα,n (x)
for any contaminated sample (X, Y ) n,y k * -1 .

Note that when the order α goes to 1, i.e. when estimating the full frontier function ϕ(x) itself, the maximal bias tends to zero since lim α↑1 qα,n (x) = φn (x). However, when α is fixed, the maximal bias φn (x)qα,n (x) may become too large as x increases. So, to estimate ϕ(x) by qα,n (x), the order α should be chosen appropriately as a function of both x and n.

We next answer the important question of how the two families of order-α and order-m boundaries are linked. We show that these concepts of partial frontiers are closely linked in the sense that {q α (x), α ∈]0, 1]} defines a "robustified" variant of the family {ξ m (x), m ≥ 1} while the latter defines an "efficient" variant of the former. We provide an explicit and exact expression of α as a function of m that allows to select which frontier qα,n can be analyzed and compared with ξm,n . Indeed, it is easy to see that ξ m (x) = E[max(Y x1 , . . . , Y xm )] for any sequence (Y x1 , . . . , Y xm ) of m independent random variables drawn from the conditional distribution of Y given X ≤ x. Since the median is known to be more robust than the mean (see e.g. Hampel 1968), we can "robustify" the expected value of the maximum, ξ m (x), by simply replacing the expectation with the median to obtain a median-type expected maximum output frontier.

Proposition 2.2. Consider the robust-variant of ξ m (x) defined as

ξm (x) = Median [max(Y x1 , . . . , Y xm )].
Then for any order m ≥ 1, there exists an order α(m) = (1/2) 1/m such that ξm (x) = q α(m) (x).

Hence for each expected-maximum output m-frontier, there exists a quantile-type frontier of a well-specified order α = α(m) such that their pointwise values ξ m (x) and q α (x) are respectively the theoretical mean and median of the same distribution, namely that of

Y ′ i s such that X i ≤ x, we first have qα,n (x) = Y x (αNx) if αN x = 1, 2, 3, . . . Y x ([αNx]+1) otherwise, (2.1)
where Y x (i) denotes the ith order statistic of the points Y x 1 , . . . , Y x Nx . Likewise, we have

φn (x) -ξm,n (x) = Nx-1 i=1 (i/N x ) m {Y x (i+1) -Y x (i) }. (2.2)
This difference being a sum of weighted spacings, ξm,n (x) is more resistant to FDH points in the sense that it converges slowly to φn (x) as m increases, whereas qα(m),n (x), as an order statistic, converges rapidly to φn (x) once it breaks down. It is easy to see that

qα(m),n (x) = Nx i=1 Y x (i) 1I i -1 N x m < 1 2 ≤ i N x m , (2.3)
and so qα(m),n (x) coincides with φn (x) for all m > log(2)/ log(N x /(N x -1)), which is not the case for

ξm,n (x) = Nx i=1 Y x (i) i N x m - i -1 N x m .
(2.4)

These sensitivity and resistance characteristics of ξm,n (x) and qα(m),n (x), as well as their statistical efficiency, are illustrated in Subsection 5.1 with simulated and real data sets.

To conclude, the α(m)-frontier ξm,n is sometimes preferred over the m-frontier ξm,n and sometimes not according to the values of m. So a sensible practice is not to restrict the frontier analysis to one procedure, but to check whether both concepts of partial boundaries point toward similar conclusions. See the practical guidelines in Subsection 5.4.

Detection of anomalous data

The word "anomalous" is used here for detecting isolated data points in the direction of Y .

From now on, we write ξm,n := qα(m),n .

Local distance : Let (x a , y a ) be an isolated outlier, that is, (x a , y a ) = (x a , φn (x a )) is an FDH observation clearly outlying the cloud of data points. We know that both partial boundaries ξm,n (x a ) and ξm,n (x a ) ր φn (x a ) as m → ∞. We distinguish between two different behaviors of ξm,n (x a ) and ξm,n (x a ) as the order m increases :

i. While ξm,n (x a ) breaks down (i.e. ξm,n (x a ) becomes attracted by the outlying value y a = φn (x a )) for any order m ≥ 1 in view of Theorem 2.1, the quantile-type value ξm,n (x a ), being determined solely by the frequency α(m), remains unaffected even when m increases (quantiles are known to be robust in this sense). In this situation, the distance between the robust value ξm,n (x a ) and the influencable value ξm,n (x a ) shall increase rapidly as m increases;

ii. However, when m achieves a sufficiently large threshold m a , the partial boundary ξm,n (x a ) also breaks down in view of Theorem 2.2 and converges rapidly, as an order statistic (see (2.1) and (2.3)), to the outlying maximum value φn (x a ). Even more strongly, it is easy to see that ξm,n coincides overall with the FDH frontier φn for any m ≥ log(1/2)/ log((n -1)/n). In contrast, ξm,n (x a ) being a linear combination of order statistics (see (2.2) and (2.4)), converges more slowly to the largest order statistic φn (x a ). Hence, although its sensitivity to the magnitude of the outlying value φn (x a ) for any m ≥ 1, ξm,n (x a ) becomes more resistant than ξm,n (x a ) as m exceeds m a . Thus, the distance between ξm,n (x a ) and ξm,n (x a ) shall decrease slowly as m > m a increases.

To summarize, if φn (x a ) is really outlying, the curve m → | ξm,n (x a ) -ξm,n (x a )| shall have roughly a "Λ" structure, that is, a sharp positive slope (indicating that ξm,n (x a ) breaks down while ξm,n (x a ) remains still unaffected as m increases) followed by a smooth decreasing slope (indicating that ξm,n (x a ) becomes non-robust for m large enough whereas ξm,n (x a ) is more resistant). Here the "Λ" effect appears at m a -1 such that the value of | ξm,n (x a ) -ξm,n (x a )| at m = (m a -1) is sufficiently large compared with its initial value at m = 1.

However, if φn (x a ) is only extreme (not really isolated), the graph of m → | ξm,n (x a )ξm,n (x a )| will have a slight "∧" curvature, that is, a non-decreasing slope followed by a nonincreasing slope such that the maximal value of the distance | ξm,n (x a ) -ξm,n (x a )| is very close to its initial value at m = 1.

So, in general, if the graph of the distance function m → | ξm,n (x i ) -ξm,n (x i )| shows clearly a sharp "Λ" curvature for a given observed value x i , this indicates a potential outlier in the data set. The suspicious outlying point can be then easily recovered: it corresponds to the FDH point (x k , y k ) for which y k = φn (x i ). This is the basic idea of our procedure.

Global distance : Consider now the maximal "distance" between the partial boundaries ξm,n and ξm,n , defined as In this case, as described above, the shape of the entire curve m → d(m) (and not only a part of this graph) should be a sharp "Λ".

d(m) = max 1≤i≤n | ξm,n (x i ) -ξm,n (x i )|. Assume that (x a ,
If, instead, the sample contains two isolated outliers (x a , y a ) and (x b , y b ) with x a < x b , it is easy to see from Theorem 2.2 that ξm,n (x a ) breaks down before ξm,n (x b ). Let m a and m b be respectively the values of m at which ξm,n (x a ) and ξm,n (x b ) break down. Then m a < m b . On the other hand, due to the conditioning on X ≤ x, both ξm,n and ξm,n are more resistant to outliers at x b than at x a (left-border effect). It follows that : In summary, in presence of two outliers far from the cloud of data points, the shape of the entire graph m → d(m) should be either one sharp "Λ" or two successive "Λ" effects showing an "M" structure. It should be also clear that if (x a , y a ) is only a suspicious extreme (not really isolated), then the strong "Λ" effect corresponding to the outlier (x b , y b ) could be preceded by a slight "∧" oscillation due to the presence of the extreme observation (x a , y a ).

In general, in presence of k outliers, the graph m → d(m) shows at least one sharp "Λ" effect and at most k "Λ" effects. However, in absence of outliers, the graph shows only slight "∧" oscillations as m increases and shall have a decreasing trend. To avoid any ambiguity of appreciation between sharp "Λ" effects and slight "∧" oscillations, we also make use of the concave envelopment of m → d(m) (i.e. the lowest concave curve enveloping the graph).

The methodology : For a given order m, let x(m) denote the observed input x j for which d(m) = | ξm,n (x j ) -ξm,n (x j )|. Then the basic tool will be a picture plotting the graph of d(m) and its concave envelopment for increasing equidistant values of m. Remember that d(m) ց 0 as m → ∞. So, if the graph of d(m) ends with an increasing slope, it should be redone by adding larger values of m until it ends with a decreasing slope. Note also that, if the graph of d(m) is plotted by using (2J + 1) or (2J + 2) values of m (with J = 1, 2, . . .), then it has at most J sharp "Λ" effects or slight "∧" oscillations. The different possible behaviors of the graph of d(m) and its concave envelopment can be summarized as follows:

(a) If the shape of the entire graph of m → d(m) is a sharp "Λ", then the order m * at which the graph is maximal should indicate that the FDH point (x k , y k ), with y k = φn (x(m * )), is an isolated outlier. The concave envelopment curve should have also a sharp "Λ" effect. Likewise, if the entire graph of d(m) shows a sharp "Λ" effect followed by a second one, that is, a structure "M", then each local maximum m * allows to detect an outlier. In this case, the concave envelopment curve should have roughly a structure "∩".

(b) If the graph of d(m) begins with a sharp positive slope as m increases, it could have a global structure of at most J successive "Λ" effects. The local maxima corresponding to these sharp effects will allow to detect isolated outliers. Here also, the concave envelopment curve should have a structure "Λ" or "∩".

(c) If, in contrast, the graph of d(m) begins with a smooth positive slope followed by a decreasing trend showing a global maximum value very close from the initial value d(1), this indicates the presence of only suspicious extreme observations (not really isolated). In this case, the concave envelopment curve should not have a clear structure "Λ" or "∩".

(d) If, in contrary, the graph of d(m) decreases overall, this indicates clearly the absence of both outliers and suspicious extremes. Here also, a structure "Λ" or "∩" of the concave envelopment curve should not appear.

(e) If, instead, the graph of d(m) begins with a decreasing slope followed by an increasing one, then we distinguish between two situations: either (e1) the (short) decreasing slope is too smooth compared with the (longer) increasing one showing roughly a curvature " " for the first values of m, or (e2) the decreasing deviation is, at least, as important as the increasing one. In situation (e1), the concave envelopment curve should have a structure "Λ" or "∩" whose maxima allow to detect isolated outliers as described in (a). In situation (e2), the concave envelopment curve should behave as the graph of d(m) in (c) or (d) leading thus to the same conclusions.

In conclusion, the above description tells us that a "Λ" or "∩" structure of the concave envelopment curve is necessary and sufficient for detecting outliers. It is also important to note that looking only at the graph of d(m) may result in some confusion between the desirable "Λ" effects (isolated outliers) and possibly contestable "∧" oscillations (suspicious extremes). To overcome such a subjectivity of appreciation, it is best to overlay in the same picture the graph of d(m) and its concave envelopment. Then only sharp "Λ" deviations of the graph of d(m) whose maximal points belong to the concave envelopment curve should be retained to identify potential outliers. This smoothing strategy however allows to detect only few outliers per picture. An outlier can "mask" other outliers situated near the first one and who are less isolated. To avoid such a masking effect pointed out earlier by [START_REF] Wilson | Detecting outliers in deterministic nonparametric frontier models with multiple outputs[END_REF][START_REF] Wilson | Detecting influential observations in data envelopment analysis[END_REF], the analysis should be redone without the identified outliers until the concave envelopment curve shows no more "Λ" or "∩" effects. Then, a careful analysis is to plot again the last graph of d(m) and its concave envelopment by using a refined sequence of "small" equidistant values of m in order to detect potential masked outliers at the left-border of the sample. Indeed, when the increasing values of m are large, our procedure cannot detect outliers having too small values of x i since, in this case, ξm,n (x i ) = ξm,n (x i ) = φn (x i ). All this results in the following simple practical algorithm (illustrated in Subsection 5.3): [2] If the concave envelopment curve shows a "Λ" or "∩" effect, then the order m * at which this curve attains its maximum indicates that the FDH point (x k , y k ), with

y k = φn (x(m * ))
, is a potential outlier. This suspicious point can be really identified as an isolated outlier only if the maximal value d(m * ) is clearly distant above from the initial value d(1). To avoid the masking effect, proceed again to step [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] without the identified outliers.

[3] If the concave envelopment curve shows neither a "Λ" nor a "∩" effect, let m 1 > 1 be the first value of m in the chosen sequence in step [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] at which the graph of d(m)

shows a decreasing deviation. Then,

[3a] if m 1 /10 ≤ 1, there are no isolated outliers in the sample of interest.

[3b] if m 1 /10 > 1, proceed to [1] by using m = 1, [ m 1 10 ], [ 2m 1 10 ], . . . , [ 9m 1 10 ], m 1 .
Multivariate extensions : Let us now extend the ideas to the full multivariate setup where a set of inputs X ∈ R p + is used to produce a set of outputs Y ∈ R q + . Let Ψ denote the joint support of the random vector (X, Y ) that we assume to be free disposal, i.e., (x, y) ∈ Ψ implies (x ′ , y ′ ) ∈ Ψ as soon as x ′ ≥ x and y ′ ≤ y (the inequalities here have to be understood componentwise). Let Y (j) , (y (j) ) denote the jth component of Y , (of y). Since a natural ordering of Euclidean spaces of dimension greater than one does not exist, we overcome the difficulty by utilizing the conditional distribution of the dimensionless transformation Y y := min j=1,...,q Y (j) /y (j) given X ≤ x instead of the multivariate distribution of Y ∈ R q + conditioned by X ≤ x. The distribution function of this univariate transformation is given by

P(Y y ≤ λ|X ≤ x) = 1 -P(Y > λy|X ≤ x) = 1 -S Y |X (λy|x) for all λ ≥ 0,
where

S Y |X (•|x) denotes the conditional survival function of Y given X ≤ x. Its endpoint λ(x, y) := sup{λ ≥ 0|S Y |X (λy|x) > 0}
coincides with the conventional Farrell efficiency score, sup{λ ≥ 0|(x, λy) ∈ Ψ}, for the unit (x, y) ∈ Ψ, and the set Y ∂ (x) := {λ(x, y)y | y : (x, y) ∈ Ψ} represents the set of maximal outputs a unit operating at the level x can produce. The point y ∂ (x) := λ(x, y)y is the radial projection of (x, y) on the support frontier Y ∂ := {(x, λ(x, y)y) | (x, y) ∈ Ψ} in the output-orientation (orthogonal to the vector x). In the particular case of q = 1, we have the equalities λ(x, y) ≡ ϕ(x)/y and Y ∂ (x) ≡ {ϕ(x)}.

Parallely to the concepts of partial frontier functions q α (x) and ξ m (x) related to the conditional distribution of Y given X ≤ x in the case of one output, we define the quantile function of order α and the expected maximum output function of order m for the dimensionless distribution of Y y given X ≤ x, respectively, as

Q α (x, y) := inf{λ ≥ 0|1 -S Y |X (λy|x) ≥ α}, X m (x, y) := λ(x,y) 0 {1 -[1 -S Y |X (λy|x)] m }dλ.
As a matter of fact, X m (x, y) coincides with the order-m output efficiency score for the unit (x, y), introduced by Cazals et al. (2002), while Q α (x, y) coincides with the αth quantile output efficiency score favored by [START_REF] Daouia | Nonparametric efficiency analysis: A multivariate conditional quantile approach[END_REF]. The sets

Y ∂ α := {(x, Q α (x, y)y) | (x, y) ∈ Ψ} and Y ∂ m := {(x, X m (x, y)y) | (x, y) ∈ Ψ} represent
, respectively, the efficient order-α and order-m partial surfaces in the output direction. In the particular case of one output, Q α (x, y) = q α (x)/y and X m (x, y) = ξ m (x)/y. In this case, the sets Y ∂ α and Y ∂ m coincide with the graphs of the frontier functions q α (•) and ξ m (•), respectively. See, e.g., [START_REF] Daraio | Advanced Robust and Nonparametric Methods in Efficiency Analysis[END_REF] for a detailed description of both partial efficiency measures and for their economic meaning.

The sample estimators Qα,n (x, y) and Xm,n (x, y) of Q α (x, y) and X m (x, y), respectively, are obtained by replacing

S Y |X (λy|x) with its empirical version ŜY |X (λy|x) = n i=1 1I(X i ≤ x, Y i > λy)/ n i=1 1I(X i ≤ x).
They can be easily computed in the same way as the quantities qα,n (x) and ξm,n (x), respectively, by simply replacing in (2.1) and (2.4) the Y i 's such that X i ≤

x with the dimensionless observations Y y i such that X i ≤ x. Moreover, it is not hard to show that all robustness and sensitivity properties established in the univariate case for the classes {q α (x), qα,n (x)} and {ξ m (x), ξm,n (x)} hold true for the transformations {Q α (x, y), Qα,n (x, y)} and {X m (x, y), Xm,n (x, y)}. In particular, the practical algorithm described above in the three steps [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF]- [START_REF] Charnes | Evaluating program and managerial efficiency: an application of data envelopment analysis to program follow through[END_REF] for detecting potential outliers remains still valid in the full multivariate case up to two natural adaptations : i. the maximal distance d(m) between the curves of qα(m),n (•) and ξm,n (•) in the case of q = 1 extends naturally to the distance

d(m) = max 1≤i≤n || Qα(m),n (X i , Y i )Y i -Xm,n (X i , Y i )Y i || between the empirical partial surfaces Ŷ ∂ α(m),n = {(X i , Qα(m),n (X i , Y i )Y i ) | i = 1, . . . , n} and Ŷ ∂ m,n = {(X i , Xm,n (X i , Y i )Y i ) | i = 1, .
. . , n} in the general case of q ≥ 1, where || • || denotes the Euclidean norm on R q ;

ii. the outlying FDH point (X k , Y k ) to be identified in step [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF], with Y k = φn (x(m * )) in the case of one output, is determined in the case of multi-outputs by

Y k = λn (x(m * ), y(m * )) y(m * ), where λn (x, y) = sup{λ ≥ 0| ŜY |X (λy|x) > 0} = max i|X i ≤x min j=1,...,q Y (j) i /y (j)
is the FDH estimator of λ(x, y), and where (x

(m * ), y(m * )) is the observation (X j , Y j ) for which d(m * ) = || Qα(m),n (X j , Y j )Y j -Xm,n (X j , Y j )Y j ||.
Subsection 5.3 illustrates the procedure with simulated and real data.

Asymptotic properties

We first derive the following pointwise and uniform asymptotic representations for ξm,n (x).

x ∈ R p + such that F X (x) > 0, we have √ n( ξm,n (x) -ξ m (x)) = √ nΦ m,n (x) + o p (1) as n → ∞ (4.1)
where

Φ m,n (x) = m F X (x) FX,n (x) ϕ(x) 0 F m-1 (y|x)[F (y|x) -Fn (y|x)]dy.
(ii) Suppose the upper boundary of the support of Y is finite. Then, for all m ≥ 1 and any

X ⊂ R p + such that inf x∈X F X (x) > 0, (4.1) holds uniformly in x ∈ X , i.e. { √ n( ξm,n (x) -ξ m (x)); x ∈ X } = { √ nΦ m,n (x); x ∈ X } + o p (1).
As an immediate consequence of Proposition 4.

1(i), √ n( ξm,n (x)-ξ m (x)
) is asymptotically normal with mean 0 and variance

σ 2 (x, m) = E m F X (x) 1I(X ≤ x) ϕ(x) 0 F m-1 (y|x) [F (y|x) -1I(Y ≤ y)] dy 2 (4.2) = 2m 2 F X (x) ϕ(x) 0 ϕ(x) 0 F m (y|x)F m-1 (u|x)[1 -F (u|x)]1I(y ≤ u)dydu.
Even more strongly, it follows from Proposition 4.1(ii) (see also the proof) that the process { √ n( ξm,n (x)ξ m (x)), x ∈ X } converges in distribution in the space L ∞ (X ) of bounded functions on X to the centered Gaussian process {G m (x); x ∈ X } as n → ∞, where

G m (x) = m F X (x) ϕ(x) 0 F m-1 (y|x) [F(x, ∞)F (y|x) -F(x, y)] dy
with F being a (p + 1) dimensional F -Brownian bridge. Similar results can be found in Cazals et al (2002, Theorem 3.1 and Appendix B). Their techniques of proof rely on the differentiability of the operator S m,x in the Fréchet sense with respect to the sup-norm. In statistical applications however, Fréchet differentiability may not hold, whereas Hadamard differentiability does, the latter being a less restrictive concept of differentiability than the former. The results in Proposition 4.1 are derived by applying the functional delta method in conjunction with the (less restrictive) Hadamard differentiability.

Note that the functional convergence of the process { √ n( ξm,n (x)ξ m (x)), x ∈ X } in Proposition 4.1(ii) provides the consistency and asymptotic distribution of parametric approximations of the order-m frontiers, as shown in [START_REF] Florens | Parametric Approximations of Nonparametric Frontier[END_REF]. Their elegant approach tries to capture the shape of the cloud points near its boundary by combining parametric and nonparametric approaches.

Next we show that √ n( ξm,n (x)ξ m (x)) also obeys a law of the iterated logarithm, which improves the order of convergence to O( √ log log n) and even gives the proportionality constant.

Theorem 4.1. For all m ≥ 1 and any x ∈ R p + such that F X (x) > 0, we have almost surely for either choice of sign

lim sup n→∞ ± √ n( ξm,n (x) -ξ m (x)) (2 log log n) 1/2 = σ(x, m).
By the asymptotic normality we have

lim n→∞ P{ξ m (x) ∈ [ ξm,n (x) ± 2σ(x, m)/ √ n]} = 2Φ(2) -1 ≈ 95%
, where Φ denotes the standard normal distribution function. An intriguing implication of the law of the iterated logarithm (see, e.g., [START_REF] Serfling | Approximation Theorems of Mathematical Statistics[END_REF]) is that we can be sure that ξ m (x) is outside the asymptotic confidence interval [ ξm,n (x) ± 2σ(x, m)/ √ n] infinitely often, but this is of little practical consequence. Monte-Carlo experiments are provided in Subsection 5.2 to illustrate the performance of the asymptotic confidence interval

Q n := [ ξm,n (x) ± zσ(x, m)/ √ n] which satisfies lim n→∞ P[ξ m (x) ∈ Q n ] = 2Φ(z) -1 for any z > 0, where σ2 (x, m) is a strongly consistent estimator of σ 2 (x, m): σ2 (x, m) = m 2 FX,n (x) φn(x) 0 φn(x) 0 [ F (y|x) F (u|x)] m-1 F (y ∧ u|x) -F (y|x) F (u|x) dydu.
Note that similar results to Proposition 4.1 and Theorem 4.1 have been proved for √ n(q α,n (x)q α (x)) in [START_REF] Daouia | Asymptotic Representation Theory for Nonstandard Conditional Quantiles[END_REF] and [START_REF] Daouia | Functional Convergence of Quantiletype Frontiers with Application to Parametric Approximations[END_REF]. Note also that, as pointed in Section 1, we have √ n (q α,n (x)q α (x))

d -→ N (0, σ 2 (α, x)) as n → ∞, where σ 2 (α, x) = α(1 -α)/f 2 (q α (x)|x)F X (x). (4.3)
Then the interval

I n = [q α,n (x)±zσ(α, x)/ √ n] satisfies lim n→∞ P[q α (x) ∈ I n ] = 2Φ(z)-1, for any z > 0. Putting z = Φ -1 (1 -a/2
) to be the (1a/2)th quantile of Φ, we obtain (1a) as the confidence coefficient. However, the computation of the asymptotic confidence interval I n requires estimation of the quantile density function f (q α (x)|x), which often results in estimates of unsatisfactory accuracy for finite samples. In the following theorem, we derive an alternative confidence interval for q α (x) which is asymptotically equivalent to I n , but does not need f (q α (x)|x) to be known or estimated.

Theorem 4.2. Let 0 < α 1 < α 2 < 1 and assume that F (•|x) is continuously differentiable on the interval [a, b] := [q α 1 (x) -ε, q α 2 (x) + ε]
for some ε > 0, with strictly positive derivative

f (•|x). For any α ∈]α 1 , α 2 [ and any z > 0, let C n =]q α n1 ,n (x), qα n2 ,n (x)[ where α n1 = α -z[α(1 -α)/n FX,n (x)] 1/2 and α n2 = α + z[α(1 -α)/n FX,n (x)] 1/2 . Then lim n→∞ P[q α (x) ∈ C n ] = 2Φ(z) -1 and √ n|length(C n ) -length(I n )| p -→ 0 as n → ∞.
In case the true partial frontiers q α (•) and ξ m (•) coincide, one can compare the performances of the asymptotic confidence intervals C n and Q n . See Subsection 5.2.

It should be clear that the estimation of the partial frontiers q α (•) and ξ m (•) instead of the full frontier ϕ(•) itself is mainly motivated by the construction of robust frontier estimators which are well inside the sample {(X i , Y i ), i = 1, . . . , n} but near its upper boundary. It is then natural to investigate whether the asymptotic normality of the estimators qα,n (x) and ξm,n (x) is still valid when α = α n → 1 and m = m n → ∞ as n → ∞. First, note that lim α↑1 qα,n (x) = lim m↑∞ ξm,n (x) = φn (x).

Note also that the necessary and sufficient condition under which the FDH estimator φn (x) converges to a non-degenerate distribution is given by

1 -F (y|x) = ℓ x {ϕ(x) -y} -1 {ϕ(x) -y} ρx as y ↑ ϕ(x) (4.4)
(Daouia et al (2010), Theorem 2.1), where ρ x > 0 is a constant and ℓ x is a slowly varying function, i.e., lim t↑∞ ℓ x (tz)/ℓ x (t) = 1 for all z > 0. In the particular case where ℓ x ({ϕ(x) -

y} -1 ) = ℓ(x) is a strictly positive function in x, it is shown in Daouia et al (2010, Corollary 2.1) that {nℓ(x)} 1/ρx ϕ(x) -φn (x) d -→ Weibull(1, ρ x ) as n → ∞.
For the estimator qαn,n (x) to keep the same limit Weibull distribution as φn (x), it suffices to choose α n → 1 rapidly so that n 1+1/ρx (1α n ) → 0 (see [START_REF] Daouia | Frontier Estimation and Extreme Value Theory[END_REF], Theorem 2.2).

This result has been also proved by [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] in the restrictive case where the joint density of (X, Y ) has a sudden jump at the frontier, which corresponds to ρ x = p + 1 in Instead of the Weibull extreme-value distribution, we provide in the next proposition sufficient conditions under which qαn,n (x) and ξmn,n (x) are rather asymptotically normal.

Proposition 4.2. (i) Suppose (4.4) holds with ℓ x ({ϕ(x) -y} -1 ) = ℓ(x) > 0 and F (•|x) is differentiable in a left neighborhood of ϕ(x) with a strictly positive derivative f (•|x). If n(1 -α n ) → ∞ as n → ∞, then √ n{σ(α n , x)} -1 (q αn,n (x) -q αn (x)) d -→ N (0, 1). (ii) If m n → ∞ and mn(mn-1) σ(x,mn) = O √ n log log n as n → ∞, then √ n{σ(x, m n )} -1 ( ξmn,n (x) - ξ mn (x)) d -→ N (0, 1).
Thus, the convergence in distribution of both

√ n σ(α,x) (q α,n (x) -q α (x)) and √ n σ(x,m) ( ξm,n (x) - ξ m (x))
to N (0, 1), for fixed orders α and m, is still valid when the partial frontiers q α (x) and ξ m (x) approach the true full frontier ϕ(x).

Numerical Illustration

We present simulation studies to illustrate the robustness and statistical efficiency of the empirical partial boundaries ξm,n and ξm,n := qα(m),n and to compare the asymptotic confidence intervals C n and Q n . We also provide illustrations with a real data set.

Comparing ξm,n and ξm,n

Simulated example: Consider the Cobb-Douglas model Y = X 1/2 exp (-U), where X is uniform on [0, 1] and U is exponential with mean 1/3. This model was studied by Gijbels et al (1999) among others. Here ϕ(x) = x 1/2 and F (y|x) = 3x -1 y 2 -2x -3/2 y 3 for 0 < x ≤ 1 and 0 ≤ y ≤ ϕ(x). As can be seen from Figure 1, in this example the theoretical partial frontiers ξ m (solid lines) and ξm = q α(m) (dotted lines) are very close. We also represent in Figure 1 a simulated sample of size 100 (green points) and we add five outliers (blue points) to this sample. For the resulting sample (X, Y ) n of size n = 105, we compute the finite sample breakdown points RB( ξm,n (x)) := RB(q α(m),n (x), (X, Y ) n ) for several values of m and x and provide in Table 1 the values n × RB( ξm,n (x)).

Since the data set contains five outlying points in the output-direction, the estimator ξm,n (x) can break down whenever RB( ξm,n (x)) ≤ 5/n. This is clearly seen from Fig- On the other hand, for too small values of x (e.g. x = 0.1), we see that both ξm,n (x) and ξm,n (x) coincide with the non-robust FDH estimator, or at least, are drastically attracted by φn (x). As pointed out in Remark 2.1, this left-border defect is due to the conditioning on X ≤ x in the construction of these two estimators. However, when the number n FX,n (x) of observations (X i , Y i ) with X i ≤ x increases, we see clearly that both ξm,n and ξm,n become more robust to the outlying points.

We also simulated 1000 samples of size n = 1000 to analyze the bias and the mean squared error (MSE) of ξm,n and ξm,n as estimators of ξ m ≃ ξm . According to the numerical results reported in Table 2 (l-h.s), we can see that ξm,n is slightly more efficient than ξm,n in terms of MSE, whereas the latter estimator is better than the former in terms of bias. When the data are contaminated by adding five outliers (indicated by " * " in Figure 1), we see in Table 2 (r-h.s) the improvement of ξm,n over ξm,n in terms of MSE. Moreover, ξm,n still outperforms ξm,n in terms of bias. Therefore, we can say that ξm,n is globally more robust to the outlying points than ξm,n in this particular example. This can be explained by the fact that, even when the α(m)-frontier ξm,n breaks down at a value x, it is influenced only locally on a right neighborhood of x, whereas the m-frontier ξm,n remains attracted overall between the 5 outliers as illustrated in Figure 2. Remember in comparing the α(m)-and m-frontiers that, in absence of outliers, ξm,n is almost overall larger than or equal to ξm,n , which is no more the case when adding the five outliers.

ξ m (•) coincides with q α (•) if and only if α = 1 2 (1 -cos[3 arccos( 1 2 -B m ) -4π]), with B m = m j=0 ( m j )3 j (-2) m-j /(3m -j + 1)
. For example, we obtain α = .8557 for m = 5 and α = .9242 for m = 10. In this case, the partial frontier ξ m ≡ q α can be estimated by ξm,n as well as qα,n , and one can compare the confidence intervals Q n and C n . The true partial frontier and its 95% confidence intervals are displayed in Figure 4 with (m = 5, α = .8557) on the l-h.s and (m = 10, α = .9242) on the r-h.s. Here we consider two simulated samples of size n = 100 (top) and n = 1000 (bottom).

By construction, the upper bound qα n2 ,n (x) of C n does not exist (see the upper blue solid lines) for small inputs-usage x and high values of α which result in orders α n2 > 1. This is the major drawback of the confidence interval C n . On the other hand, even if the confidence bands of Q n (in red lines) are overall well-defined, they do not contain ξ m (x) for small levels x and high orders m. Apart from these left-border defects, we observe that C n and Q n have very similar lower bounds, but Q 100 performs globally better than C 100 in terms of upper bounds. This is the price to be paid in order to avoid the estimation of the conditional quantile density f (q α (x)|x) involved in the asymptotic variance σ 2 (α, x)

of qα,n (x). For n = 1000, the two confidence intervals provide more similar results. Note also that Q n is computationally prohibitive when the sample size is of the order of several thousands. On the contrary, C n is very easy and very fast to implement. comparing the MSE and bias of ξm,n and qα,n as estimators of ξ m (x) = q α (x), we find here also that ξm,n is more efficient than qα,n in terms of MSE and that qα,n performs better in terms of bias. We do not reproduce the tables in order to save place.

Detection of anomalous data

A univariate simulated example: We consider the cloud of n = 105 points represented in Figure 1. The procedure based on the analysis of the curve of m → d(m) and its concave envelopment will detect only the five points '*' as isolated outliers. The first picture in Figure 5 (l-h.s) gives these two curves for m = 1, 10, 20, . . . , 90, 100: here the graph of d(m), in solid blue line, shows clearly two sharp "Λ" effects and the concave envelopment curve, in dotted red line, has roughly a structure "∩". The "Λ" effects attain their maximal values To avoid the masking effect, we redid the same work on the same data set without the detected two outliers. The second picture of Figure 5 (from left to right) provides the resulting curve of d(m) and its concave envelopment: here also looking to the sharp "Λ" effects of the graph of d(m) which appear at m * = 10 and m * = 50, we identify respectively the additional outlying points (0.3, 0.7) and (0.7, 1).

When these two outliers are also deleted from the sample, we obtain the curves in the third picture of Figure 5: clearly the graph of d(m) begins with a too smooth decreasing slope followed by a sharp "Λ" effect which attains its maximum at m * = 20, and ends with a slight "∧" oscillation. Here, the shape of the entire concave envelopment curve shows an indisputable sharp "Λ" effect which attains its maximum at m * = 20, indicating that the FDH point (x(20), φn (x(20))) ≡ (0.3, 0.65) is an outlier. The slight "∧" oscillation of the graph of d(m), attaining its maximum at m = 50, indicates only the presence of a suspicious extreme observation (not really isolated) since its maximal value d( 50) is very close to d(1). Now, when the five outliers (0.1, 0.5), (0.5, 0.9), (0.3, 0.7), (0.7, 1) and (0.3, 0.65) are deleted from the sample, we get the curves in the last picture of Figure 5 (r-h.s): the concave envelopment curve shows neither a sharp "Λ" effect nor a "∩" structure, which indicates the absence of really isolated outliers. Here, the graph of d(m) begins with a decreasing deviation followed by a slight "∧" oscillation whose maximal value d( 50) is very close to the initial value d( 1) and so cannot be used for detecting outliers. Therefore only the five points indicated by '*' in Figure 1 are detected by our semi-automatic procedure, which is quite remarkable although "no optimal procedure nor miracle procedure can be defined to detect outliers" as stated by [START_REF] Simar | Detecting Outliers in Frontiers Models: a Simple Approach[END_REF]. , n (here n = 4000), we obtain successively the pictures in Figure 7 from left to right and from top to bottom. Except for the last picture, the concave envelopment curves in dotted lines show roughly "Λ" or "∩" effects allowing to identify at most three outliers per picture indicated in Figure 6 by the number of the corresponding picture (from #1 to #14). Looking at the last picture, #15, we see a too smooth increasing slope (an approximately horizontal deviation) of the concave curve followed by a sharp decreasing slope, which makes the "Λ" effect clearly more contestable than the one appearing in the preceding picture. So we cannot proceed to step [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF]. shows a decreasing deviation being m 1 = 80, the last picture is redone by using the refined sequence m = 1, 8, 16, . . . , 80. This gives the second picture of Figure 8, which allows to identify only one potential outlier indicated by #17 in Figure 6. When this point is deleted from the sample, we obtain the third picture of Figure 8 which shows no more "Λ" or "∩" effects of the concave envelopment curve and so, there are no more outlying post offices.

In summary, our semi-automatic procedure detects 22 potential outliers. Some of these points (e.g. #1,#2,#3) are clearly outlying due to measurement errors, but other isolated observations (e.g. #4,#9,#10,#17) might contain useful information on the process under analysis and so, they deserve to be carefully examined. A multivariate simulated example: Here a multi-input and multi-output (p = q = 2) data set is simulated as in [START_REF] Park | The FDH Estimator for Productivity Efficiency Scores : Asymptotic Properties[END_REF]. In this setup, the function describing the efficient frontier is given by y (2) = 1.0845(x (1) ) 0.3 (x (2) ) 0.4y (1) , where y (j) , (x (j) ), stands for the jth component of y, (of x), for j = 1, 2. We draw X (j) i independent uniforms on (1, 2) and Ỹ (j) i independent uniforms on (0.2, 5). Then the generated random rays in the output space are characterized by the slopes

K i = Ỹ (2) i / Ỹ (1)
i . The generated random points on the frontier are defined by

Y (1) i,ef f = 1.0845(X (1) i ) 0.3 (X (2) i ) 0.4 K i + 1 , Y (2)
i,ef f = 1.0845(X

(1)

i ) 0.3 (X (2) i ) 0.4 -Y (1)
i,ef f .

The efficiencies are generated by exp(-U i ) where U i are drawn from an exponential with mean 1/3. Finally, we define Y i = Y i,ef f * exp(-U i ). We simulate 100 observations according to this scenario and we add five outliers #1, . . . , #5, as in [START_REF] Daouia | Nonparametric efficiency analysis: A multivariate conditional quantile approach[END_REF], respectively at the following values of X: (1.25,1.5), (1.25, 1.75), (1.5,1.5), (1.75, 1.25) and (1.5, 1.25); the corresponding values for the slopes in the Y space are (0.25, 0.75, 1, 3, 5).

Our working algorithm results in the successive graphs of d(m) and their concave envelopment curves displayed on Figure 9, with m = 1, 10, 20, . . . , 100. In the first picture (from left to right and from top to bottom), the graph of d(m) (solid blue line) and its concave envelopment curve (dashed red line) have a sharp structure "∩" which attains its maximal values at m * = 10 and m * = 30 and allows to identify only the outlying observation #5.

A similar "∩" structure is obtained in the second picture after removing the first detected outlier : here also the attained maximal values at m * = 10 and m * = 30 allow to identify the same outlier #1. When this additional outlier is deleted from the sample, the new graphs in the third picture show a sharp "Λ" effect which appears at m * = 10 and results in the identification of the outlier #2. The fourth picture provides the resulting graphs after removing this outlier : looking here to the structure "∩" which attains its maximal values at m * ∈ {20, 30, 60}, we identify the same outlying point #3. The new graphs obtained without this outlier are shown in the fifth picture, where a sharp "Λ" effect of the concave envelopment curve, appearing at m * = 20, allows to identify the outlier #4. For the same data set without this outlier, we get in the last picture a decreasing concave envelopment curve, indicating the absence of any suspicious observation among the remaining simulated 100 points. Thus, only the introduced five outliers are detected by our procedure. We repeated the same exercise with other simulated data sets with the same kind of results.

Application to PFT (Program Follow Through) data:

We examine here the popular data set reported by Charnes, Cooper and Rhodes (1981) on an experimental education program administrated in 70 US schools, with p = 5 inputs and q = 3 outputs. The observations #59 and #44 are detected by the procedure of [START_REF] Wilson | Detecting outliers in deterministic nonparametric frontier models with multiple outputs[END_REF] as potential outliers. The results obtained by [START_REF] Simar | Detecting Outliers in Frontiers Models: a Simple Approach[END_REF] confirm this and point out two additional suspicious observations #54 and #1 that deserve at least careful attention. Our methodology confirms that only the units #59 and #44 are really isolated from the sample in the outputorientation. Moreover, it turns out that the unit #52 is more suspicious than the extreme observations #54 and #1. Proceeding to step [START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] and then to step [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF] of our algorithm for m = 1, 7, 14, 21, . . . , 70, we find successively the eight pictures displayed on Figure 10.

In the first picture, the concave envelopment curve (dashed line) indicates a clear structure "Λ", and the graph of As a matter of fact, due to the high 8-dimensional space with a small sample (n = 70), we can identify more extreme points as potential outliers. However, we recall that before deleting any suspicious observation from the sample, our methodology requires to first check whether the suspicious point is really isolated in the output-orientation by comparing the maximal value d(m * ) of the corresponding "Λ" effect with the initial value d(1). As it can be seen from Figure 11 (l-h.s), which represents the difference d(m * )d(1) for each suspicious point, only the two units #59 and #44 can be really identified as potential outliers since, for each one of them, d(m * ) is clearly distant above from d(1). The three suspicious units #52, #54 and #1 cannot be viewed as isolated outliers, but they are certainly extreme/influential observations. In contrast, the remaining units (#21, #10, #27, #12, #50, #20, #16,...) are not even suspicious since the difference d(m * )d( 1) is clearly negligible for all of them. 1) for the suspicious observations. (r-h.s) Evolution of the % of sample points outside the partial frontiers ξm,3978 and ξm,3978 .

Practical guidelines

In view of Proposition 2.2 and Theorems 2.1-2.2, we know that a significant difference between the expected-maximum estimate ξm,n and the median-maximum estimate ξm,n indicates the presence of influential extreme observations above the order-m frontier that could be outlying. This suggests the following two steps in order to perform the frontier estimation:

Step 1: Apply the semi-automatic prescription (as illustrated above) in order to detect any potential outliers. Then consider the sample without the identified anomalous data points.

For the median-and mean-maximum estimators ξm,n and ξm,n to provide similar conclusions, an intuitive idea is to seek the order m for which the percentage of sample points above each partial frontier is approximately the same. This leads to Step 2.

Step 2: Overlay in a same picture the evolution of the percentage of observations outside each partial frontier with respect to m. Remember that the sample still contains extreme points (not really outlying) that influence ξm,n more or less than ξm,n following the values of m. Therefore the two decreasing percentage curves shall "cross" since ξm,n is less sensitive (and so envelopes less points) than ξm,n to the magnitude of extreme outputs even when m increases, but once m attains a sufficiently large threshold, ξm,n becomes more resistant (and so envelopes less points) than ξm,n . The value of m at which the two percentage curves cross corresponds to the most similar large order-m and order-α(m) frontiers that capture the shape of the cloud points near its optimal boundary.

The extreme observations left outside the resulting similar frontiers ξm,n and ξm,n might be useful to emulate: the managers of any decision-making unit (DMU) operating at (x, y) and situated below these partial frontiers could study the relevant peers (X i , Y i ) above ξm,n or ξm,n among those dominating (x, y) (i.e. with X i ≤ x and Y i ≥ y) in order to learn how to reduce inputs and/or increase outputs. Refined "relevant practices" that might be useful to emulate could be identified as follows: the partial frontiers ξm,n and ξm,n being less sensitive to the choice of the order m as m → ∞, the decrease of the percentage of points outside each frontier becomes approximately stable as m → ∞. In particular, the first value of m from which the two percentage curves are approximately horizontal/stable, corresponds to the frontiers ξm,n and ξm,n that are sensible to the magnitude of the most extreme observations whose outputs are highly desirable, but in the same time, they are resistant to these extremes in the sense that they do not envelope them. Such extreme practices could be emulated by the managers of dominated DMUs to improve their own operations.

Application to postal data: In order to capture in a robust way the shape and curvature of the sample boundary, we compared in Figure 3 (top) both partial boundaries ξm,n and ξm,n for a sequence of large values of m ∈ {100, 200, 1000, 4000}. We observe a distance between the two frontiers, for each order m, due to the presence of outliers above the mfrontiers. Following our practical guidelines, a sensible practice is to remove, in a first step, the identified 22 potential outliers from the sample. Figure 3 (bottom) shows how ξm,n and ξm,n become very close for the resulting sample of size n = 3978. Then, in a second step, we overlay in a same picture the evolution of the percentage of sample points outside each partial frontier with respect to m. As can be seen from Figure 11 (r-h.s), the two decreasing percentage curves cross at m ≈ 100 and become approximately linearly stable from m ≈ 250. The partial frontiers ξm,n and ξm,n for m ∈ {100, 250} are graphed in Figure 12 together with their 95% confidence intervals Q n and C n , respectively. ξ100,n and ξ100,n are the largest order-m and order-α(m) frontiers which provide the most similar estimates. The extreme post offices left outside these frontiers, whose outputs are highly desirable, might be useful to emulate. The partial frontiers ξ250,n and ξ250,n also provide a refined identification of relevant post offices to be emulated and satisfactory estimates of the shape of the sample boundary.

Conclusions

We show that the two classes of partial frontiers, {q α } and {ξ m }, are closely related when α = α(m) = (1/2) 1/m , in the same sense as the mean and median of a same distribution do. This answers in particular the important question of how to choose the order α as a function of m for a possible comparison between order-α and order-m frontiers.

Non of the two classes can be claimed to be preferable in all contexts. A sensible practice is to check whether both partial frontier analyses point toward similar conclusions. Obtaining different results from ξm,n and qα(m),n , for sufficiently large values of m, indicates the presence of suspicious extreme data points that could be outlying or perturbed by noise. Before performing any frontier estimation, a useful empirical strategy is to first detect and remove the anomalous data and then to determine, in a second step, the order m at which the percentage of sample points outside the order-m and order-α(m) frontiers is approximately the same. This value of m corresponds to the largest frontiers ξm,n and qα(m),n having the most similar behaviors. These extreme partial frontiers provide satisfactory estimates of the shape of the sample boundary and identify relevant peers that might be useful to emulate.

The theoretical comparison between the reliability of {q α,n } and { ξm,n } is exploited to derive an appealing identification methodology, very easy and fast to implement and providing very good results. The use of partial frontiers for detecting influential observations is not new. A basic tool can be found in [START_REF] Simar | Detecting Outliers in Frontiers Models: a Simple Approach[END_REF] and [START_REF] Daraio | Advanced Robust and Nonparametric Methods in Efficiency Analysis[END_REF] consisting of a picture showing the evolution of the "proportion" of sample points outside either the order-m or order-α frontier as a function of the order and of another tuning parameter.

Our prescription is rather based on the evolution of the "maximal distance" between the related order-m and order-α(m) frontiers as a function of m. Adapting this tool to the inputorientation is straightforward. Our robustness study provides also a theoretical justification for the descriptive technique of [START_REF] Simar | Detecting Outliers in Frontiers Models: a Simple Approach[END_REF] and [START_REF] Daraio | Advanced Robust and Nonparametric Methods in Efficiency Analysis[END_REF].

We derive, among others, an asymptotic confidence interval C n for q α (x) not requiring estimating the conditional quantile density function. We provide sufficient conditions for ensuring the asymptotic normality of both ξm,n (x) and qα,n (x) for the limiting cases m = m n → ∞ and α = α n → 1 as n → ∞. Instead of the assumption involving the asymptotic variance σ 2 (x, m n ) (see Proposition 4.2(ii)), a main challenge is to get the asymptotic normality of ξmn,n (x) under the more conventional condition (4.4). This problem is worth investigating in future. When estimating the same partial frontier q α (x) = ξ m (x), the empirical study reveals interesting findings regarding the performances of the estimators ξm,n and qα,n and the performances of the confidence intervals Q n and C n . Proof Since φn (x) = T 1,x ((X, Y ) n ) := max i|X i ≤x Y i , there exists j ∈ {1, . . . , n} such that X j ≤ x and Y j = φn (x). Let Y * be any arbitrary point such that Y * > Y j . Then, if we replace the FDH point (X j , Y j ) in the sample ((X 1 , Y 1 ), . . . , (X j , Y j ), . . . , (X n , Y n )) by (X j , Y * ), we get the contaminated FDH estimator T 1,x ((X 1 , Y 1 ), . . . , (X j , Y * ), . . . ,

Appendix: lemmas and proofs A Robustness

(X n , Y n )) = Y * . Hence sup (Z) n 1 |T 1,x {(Z) n 1 } -T 1,x {(Z) n }| ≥ |T 1,x {(X 1 , Y 1 ), . . . , (X j , Y * ), . . . , (X n , Y n )} -T 1,x {(Z) n }|
for all Y * > Y j . Therefore a breakdown occurs as Y * → ∞.

Proof of Theorem 2.1 Let N x = n FX,n (x) be the number of observations (X i , Y i ) with

X i ≤ x and let Y x 1 , . . . , Y x Nx be the Y ′ i s such that X i ≤ x. For i = 1, . . . , N x , denote by Y x (i)
the ith order statistic of the points Y x 1 , . . . , Y x Nx . We have φn (x) = Y x (Nx) and so

ξm,n (x) = S m,x ((X, Y ) n ) := Y x (Nx) - Y x (Nx)

0

[ Fn (y|x)] m dy.

If N x = 1, ξm,n (x) = φn (x) and so RB( ξm,n (x), (X, Y ) n ) = 1/n by Lemma A.1. Otherwise,

S m,x ((X, Y ) n ) = Y x (Nx) - Nx-1 i=1 [i/N x ] m (Y x (i+1) -Y x (i) ).
Consider the same contaminated sample (X, Y ) n 1 = ((X 1 , Y 1 ), . . . , (X j , Y * ), . . . , (X n , Y n )) used in the proof of Lemma A.1, obtained by replacing the FDH observation (X j , Y j ) by (X j , Y * ), where Y * is an arbitrary point such that Y * > Y x (Nx) . Then, if

N x = 2, we have S m,x ((X, Y ) n 1 ) = (1 -(1/2) m )Y * + (1/2) m Y x (1)
, and thus a breakdown occurs as

Y * → ∞. Likewise, if N x > 2, we have S m,x ((X, Y ) n 1 ) = (1 -[ Nx-1 Nx ] m )Y * + [ Nx-1 Nx ] m Y x (Nx-1) - Nx-2 i=1 [i/N x ] m (Y x (i+1) -Y x (i) )
X i , Y i ) with X i ≤ x. It is clear that max{0, (k -1)-(n-N x )} ≤ ℓ x ≤ k -1. Let N *
x be the number of points (X * i , Y * i ) such that X * i ≤ x. Then it is easy to see that

N * x ≤ N x + (k -1) -ℓ x . (A.1) Let Y * x 1 , . . . , Y * x N *
x be the points Y * i such that X * i ≤ x, and for i = 1, . . . , N * x , denote by Y * x (i)

the ith order statistic such that

Y * x (1) ≤ . . . ≤ Y * x (N * x ) . Then T α,x ((X, Y ) n k-1 ) = Y * x (αN * x ) if αN * x ∈ N * Y * x ([αN * x ]+1)
otherwise.

Because k -1 = N xj and k -1 ≥ ℓ x , we have N xℓ x ≥ j. Since αN x ≤ j, we obtain N x (1α) ≥ ℓ x and so (2N x -αN xℓ x )α ≤ N xℓ x . Using αN x ≤ j, we get

(N x + (k -1) -ℓ x )α = (2N x -j -ℓ x )α ≤ N x -ℓ x . It follows from (A.1) that αN * x ≤ N x -ℓ x . (A.2) It is then clear that Y * x (αN * x ) ≤ Y * x (Nx-ℓx) if αN * x ∈ N * , otherwise it follows from (A.2) that [αN * x ] + 1 ≤ N x -ℓ x , whence Y * x ([αN * x ]+1) ≤ Y * x (Nx-ℓx) . Thus 0 ≤ T α,x ((X, Y ) n k-1 ) ≤ Y * x (Nx-ℓx) . (A.3)
Since we only replace ℓ x points among the N x observations (X i , Y i ) with inputs X i ≤ x, the remaining N xℓ x non-replaced observations (X i , Y i ) have outputs

Y x i ≤ Y x (Nx) . Since these non-contaminated N x -ℓ x outputs Y x i are contained in the set {Y * x 1 , . . . , Y * x N * x }, we have Y * x (Nx-ℓx) ≤ Y x (Nx) . Therefore T α,x ((X, Y ) n k-1 ) ≤ Y x (Nx) in view of (A.3). Thus |T α,x ((X, Y ) n k-1 ) -T α,x ((X, Y ) n )| ≤ φn (x) for any (X, Y ) n k-1 . Proof of Proposition 2.1 Let (X, Y ) n,y k * -1 = ((X 1 , Y * 1 ), . . . , (X n , Y * n )
) be an arbitrary contaminated sample. Using the notations of the proof of Theorem 2.2, we have here

N * x = N x , Y * x (1) ≤ • • • ≤ Y * x (Nx-ℓx) are the N x -ℓ x non-contaminated Y x i 's and Y * x (Nx-ℓx+1) ≤ • • • ≤ Y * x (Nx)
are the resulting ℓ x outliers in the direction of Y . Since the points

Y * x (1) ≤ • • • ≤ Y * x (j)
belong to the set of non-contaminated Y x i 's, the point Y * x (j) is then larger than or equal to j points among these non-contaminated Y x i 's. Therefore Y x (j) ≤ Y * x (j) . On the other hand, we have T

α,x ((X, Y ) n ) = Y x (j) and T α,x ((X, Y ) n,y k * -1 ) = Y * x (j) since αN * x = αN x . Thus T α,x ((X, Y ) n ) ≤ T α,x ((X, Y ) n,y k * -1 ). The second inequality T α,x ((X, Y ) n,y k * -1 ) ≤ Y x (Nx) = φn (x) is established in the proof of Theorem 2.2.

Proof of Proposition 2.2

The result is immediate since by definition of the median we have ξm (x) = inf{y ≥ 0|F m (y|x) ≥ 1/2} = q (1/2) 1/m (x).

B Asymptotics

Fix m ≥ 1 and x ∈ R p + such that F X (x) > 0. Define the domain D x to be the set of distribution functions G(

•, •) on R p + × R + such that G(x, ∞) > 0 and G -1 (1|x) ≤ ϕ(x) (B.1)
where G -1 (1|x) := inf{y ≥ 0| G(y|x) = 1} stands for the upper boundary of the support of the conditional distribution function

G(•|x) = G(x, •)/G(x, ∞). For any G ∈ D x define m,x φ (G) = ∞ 0 [1 -G m (y|x)]dy
where the integrand is identically zero for y ≥ G -1 (1|x). It follows from (B.1) that

m,x φ (G) = ϕ(x) 0 [1 -G m (y|x)]dy for all G ∈ D x .
In particular, we have (1 -[ Fn (y|x)] m )dy since φn (x) ≤ ϕ(x) with probability 1. The following lemma will be useful for the proof of Proposition 4.1(i).

: D x ⊂ L ∞ ( Rp+1 ) -→ [0, ϕ(x)] is Hadamard-differentiable at F with derivative ( m,x φ ) ′ F : h ∈ L ∞ ( Rp+1 ) -→ ( m,x φ ) ′ F (h) = m F X (x) ϕ(x) 0 F m-1 (y|x)[h(x, ∞)F (y|x) -h(x, y)]dy.
Proof Let h ∈ L ∞ ( Rp+1 ) and h t → h uniformly in L ∞ ( Rp+1 ), where F + th t ∈ D x for all small t > 0. Write ξ mt (x) := m,x φ (F + th t ). Following the definition of the Hadamard differentiability (see van der Vaart (1998), p.296), we shall show that (ξ mt (x)ξ m (x))/t converges to ( m,x φ ) ′ F (h) as t ↓ 0. We have

ξ mt (x) -ξ m (x) = ϕ(x) 0 [F (y|x)] m - F (x, y) + th t (x, y) F X (x) + th t (x, ∞) m dy.
By Taylor's formula, for any y ∈ [0, ϕ(x)], there exists a point ζ t,x (y) interior to the interval joining F (y|x) and (F (x, y)

+ th t (x, y))/(F X (x) + th t (x, ∞)) such that [F (y|x)] m - F (x, y) + th t (x, y) F X (x) + th t (x, ∞) m = mtζ m-1 t,x (y) h t (x, ∞)F (y|x) -h t (x, y) F X (x) + th t (x, ∞) . Whence ξ mt (x) -ξ m (x) t = m F X (x) + th t (x, ∞) ϕ(x) 0 ζ m-1 t,x (y)[h t (x, ∞)F (y|x) -h t (x, y)]dy. (B.2)
It follows from the definition of ζ t,x (y) and the uniform convergence

h t → h in L ∞ ( Rp+1 ) that ζ m-1 t,x (y)[h t (x, ∞)F (y|x)-h t (x, y)] converges to F m-1 (y|x)[h(x, ∞)F (y|x)-h(x, y
)] uniformly in y as t ↓ 0. Therefore, we obtain lim t↓0 (ξ mt (x)ξ m (x))/t = ( m,x φ ) ′ F (h).

Proof of Proposition 4.1(i) It is well known that the empirical process

√ n( F -F ) converges in distribution in L ∞ (R p+1 ) to F, a p + 1 dimensional F -
φ ( F )- m,x φ (F )) = ( m,x φ ) ′ F ( √ n( F -F )) + o p (
φ: G → m φ (G) as a map D X ⊂ L ∞ ( Rp+1 ) → L ∞ (X ). We have m φ ( F ) := { m,x φ ( F ); x ∈ X } = { ξm,n (x); x ∈ X } a.s. = { ϕ(x) 0 (1 -[ Fn (y|x)] m )dy; x ∈ X } since P [ φn (x) ≤ ϕ(x), ∀x ∈ X ] = 1.
The following lemma will be useful for the proof of Proposition 4.1(ii).

Lemma B.2. m φ is Hadamard-differentiable at F ∈ D X with derivative ( m φ) ′ F (h) : x ∈ X → ( m,x φ ) ′ F (h), for any h ∈ L ∞ ( Rp+1 ).
Proof It suffices to make the proof of Lemma B.1 uniform in x ∈ X . We use the same notation: let

h t → h in L ∞ ( Rp+1 ),
||( m φ) ′ F (h)|| L ∞ (X ) = sup x∈X |( m,x φ ) ′ F (h)| ≤ 2mν inf x∈X F X (x) ||h|| L ∞ ( Rp+1 ) for any h ∈ L ∞ ( Rp+1 ). Therefore √ n( m φ ( F )- m φ (F )) = ( m φ) ′ F ( √ n( F -F )) + o p (
m -F m (y|x) = mF m-1 (y|x)[ Fn (y|x) -F (y|x)] + (m/2)(m -1)[η x,n (y)] m-2 [ Fn (y|x) -F (y|x)] 2
. By using the fact that ξm,n (x)ξ m (x)

a.s. = ϕ(x) 0 (F m (y|x) -[ Fn (y|x)] m )dy, we get ( ξm,n (x) -ξ m (x)) -m ϕ(x) 0 F m-1 (y|x)[F (y|x) -Fn (y|x)]dy (B.3) a.s. = -(m/2)(m -1) ϕ(x) 0 [η x,n (y)] m-2 [ Fn (y|x) -F (y|x)] 2 dy.
On the other hand, we have by the law of the iterated logarithm (LIL) for empirical processes -→ 0 as n → ∞. Finally, since 0 ≤ η x,n (y) ≤ 1 for all y, we arrive at

sup x | FX,n (x)-F X (x)| = O log log n n 1/2 , sup (
√ n{( ξm,n (x) -ξ m (x)) -m ϕ(x) 0 F m-1 (y|x)[F (y|x) -Fn (y|x)]dy} a.s.
-→ 0.

This gives R m,n (x) a.s.

-→ 0 since FX,n (x)/F X (x) a.s.

-

± √ nΦ m,n (x) (2 log log n) 1/2 = lim sup n→∞ ± (m/F X (x)) (2n log log n) 1/2 n i=1 ϕ(x) 0 F m-1 (y|x) [1I(X i ≤ x)F (y|x) -1I(X i ≤ x, Y i ≤ y)] dy = σ(x, m)
with probability 1. Moreover R m,n (x)/(2 log log n) 1/2 a.s.

-→ 0 as n → ∞. Thus, by combining these results, we get the desired LIL.

The following lemma will be needed to prove Theorem 4.2.

Lemma B.3. Assume that the condition of Theorem 4.2 hold. For any α ∈]α 1 , α 2 [ and any

c ∈ R, let α n = α + c/ n FX,n (x) + o(1/ √ n). Then qαn,n (x)
a.s.

-→ q α (x) and √ n(q αn,n (x)qα,n (x))

p -→ c/ F X (x)f (q α (x)|x) as n → ∞.
Proof Following Serfling (1980, p.6), an equivalent condition for the convergence Z n a.s.

-→ Z to hold is lim n→∞ P(sup m≥n |Z m -Z| > ε) = 0 for every ε > 0, where Z 1 , Z 2 , • • • and Z are random variables on (Ω, A, P). Let ε > 0. By the smoothness of F (•|x) at q α (x) we have F (q α (x) -ε|x) < α < F (q α (x) + ε|x). Since α n a.s.

-→ α, we then have by applying the equivalent condition for the almost sure convergence

P[α m < α + F (q α (x) + ε|x) -α 2 , ∀m ≥ n] → 1, P[α - α -F (q α (x) -ε|x) 2 < α m , ∀m ≥ n] → 1 as n → ∞.
On the other hand, since Fn (q α (x) ± ε|x) a.s.

-→ F (q α (x) ± ε|x), we have

P[α + F (q α (x) + ε|x) -α 2 < Fm (q α (x) + ε|x), ∀m ≥ n] → 1, P[ Fm (q α (x) -ε|x) < α - α -F (q α (x) -ε|x) 2 , ∀m ≥ n] → 1 as n → ∞. It follows P[α m < Fm (q α (x) + ε|x), ∀m ≥ n] → 1 and P[ Fm (q α (x) -ε|x) < α m , ∀m ≥ n] → 1
as n → ∞. Whence P[ Fm (q α (x) -ε|x) < α m < Fm (q α (x) + ε|x), ∀m ≥ n] → 1 as n → ∞. Thus, by applying the fundamental property that the event { Fm (y|x) ≥ α m } is equivalent to {y ≥ qαm,m (x)}, we get P[q α (x)ε < qαm,m (x) ≤ q α (x) + ε, ∀m ≥ n] → 1 as n → ∞.

Therefore P[|q αm,m (x)q α (x)| ≤ ε, ∀m ≥ n] → 1, which is equivalent to qαn,n (x) a.s.

-→ q α (x).

Let us now turn to the second result. Since qα,n (x) and qαn,n (x) a.s.

-→ q α (x) and α n a.s.

-→ α, the interval [a, b] contains both qα,n (x) and qαn,n (x) and the interval [α 1 , α 2 ] contains α n , for n sufficiently large, with probability 1. Hence we have almost surely, for n large enough, √ n(q αn,n (x)qα,n (x)) = √ n{F (q αn,n (x)|x) -F (q α,n (x)|x)}/f (q δn (x)|x)

where min{F (q αn,n (x)|x), F (q α,n (x)|x)} < δ n < max{F (q αn,n (x)|x), F (q α,n (x)|x)}. Define the random function g n : L ∞ ([α 1 , α 2 ]) → R by g n (z) = z(α n )z(α). Putting z n (•) = √ n{F (q •,n (x)|x) -F (q • (x)|x)}, we obtain with probability 1, for all n large enough, √ n(q αn,n (x)qα,n (x)) = [g n (z n ) -√ n(αα n )]/f (q δn (x)|x). -→ α and f (q δn (x)|x) a.s.

-→ f (q α (x)|x). Finally, since √ n(αα n ) a.s.

-→ -c/ F X (x), we get √ n(q αn,n (x)qα,n (x))

p -→ c/ F X (x)f (q α (x)|x).

Proof of Theorem 4.2 Write √ n{q α n1 ,n (x) -(q α,n (x)zσ(α, x)/ √ n)} = √ n(q α n1 ,n (x) - On the other hand, we have P[q α (x) ∈ C n ] = 1 -{P(q α (x) ≤ qα n1 ,n (x)) + P(q α (x) ≥ qα n2 ,n (x))}.

By using (B.7), we obtain P(q α (x) ≤ qα n1 ,n (x)) = P{ √ n(q α,n (x)q α (x)) + o p (1) ≥ zσ(α, x)}. By the asymptotic normality, we have lim n→∞ P(q α (x) ≤ qα n1 ,n (x)) = 1 -Φ(z). Likewise lim n→∞ P(q α (x) ≥ qα n2 ,n (x)) = 1 -Φ(z). Therefore lim n→∞ P[q α (x) ∈ C n ] = 2Φ(z) -1.

Proof of Proposition 4.2 (i) Let σ n = σ(α n , x)/ √ n = α n (1α n )/f (q αn (x)|x) nF X (x).

We shall prove for any real y ∈ R that P[σ -1 n (q αn,n (x)q αn (x)) ≤ y] → Φ(y) as n → ∞. Let n be large enough so that qαn,n (x) belongs to the left neighborhood of ϕ(x) on which F (•|x) is differentiable with a strictly positive derivative f (•|x). We have P[σ -1 n (q αn,n (x) -q αn (x)) ≤ y] =P[q αn,n (x) ≤ q αn (x) + σ n y] = P[ F (q αn (x) + σ n y|x) ≥ α] = P[A n ≥ a n ], where

a n = nF X (x)
α n (1α n ) {α n -F (q αn (x) + σ n y|x)},

A n = nF X (x)
α n (1α n ) { F (q αn (x) + σ n y|x) -F (q αn (x) + σ n y|x)} = F X (x) FX (x) F (q αn (x) + σ n y|x)[1 -F (q αn (x) + σ n y|x)] α n (1

-α n ) 1/2 n i=1 W n,i √ nσ(W n,i )
with W n,i = 1I(X i ≤ x, Y i ≤ q αn (x) + σ n y) -F (q αn (x) + σ n y|x)1I(X i ≤ x) and σ 2 (W n,i ) = F X (x)F (q αn (x) + σ n y|x)[1 -F (q αn (x) + σ n y|x)]. We first need to prove that A n d → N (0, 1) and second we shall show that a n → -y as n → ∞. It is easy to see from (4.4) that q αn (x) = ϕ(x)-1-αn ℓ(x) 1/ρx for n large enough. Likewise since f (y|x) = ρ x ℓ(x){ϕ(x)-y} ρx-1 as y ↑ ϕ(x), we get f (q αn (x)|x) = ρ x ℓ(x){ϕ(x)q αn (x)} ρx-1 = ρ x ℓ(x) 1/ρx (1α n ) (ρx-1)/ρx for n large enough. Then σ n /(ϕ(x)-q αn (x)) = √ α n /ρ x n(1α n )F X (x) → 0 since n(1-α n ) → ∞. It follows that [1 -F (q αn (x) + σ n y|x)]/[1 -F (q αn (x)|x)] = 1σ n y/(ϕ(x)q αn (x)) → 1.

Therefore F (q αn (x) + σ n y|x)[1 -F (q αn (x) + σ n y|x)] ∼ α n (1α n ) as n → ∞. We also have → N (0, 1). Whence A n d → N (0, 1).

F X (
Therefore the monotone function S n (•) = P[A n ≥ •] converges pointwise to 1 -Φ(•) which is continuous. By Dini's Theorem, S n also converges uniformly to 1 -Φ. Finally it suffices to show that a n → -y to conclude that P[A n ≥ a n ] → Φ(y). First we have a n = -yσ n nF X (x)f (δ n |x)/ α n (1α n ) = -yf (δ n |x)/f (q αn (x)|x) for a real δ n lying between q αn (x) and q αn (x)+σ n y. Second, since f (δn|x) f (qα n (x)|x) = 1 + qα n (x)-δn ϕ(x)-qα n (x)

ρx-1 for all n large enough, and qα n (x)-δn ϕ(x)-qα n (x) ≤ |y|σn ϕ(x)-qα n (x) → 0, we get f (δ n |x)/f (q αn (x)|x) → 1 and a n → -y. = O (log log n/n) 1/2 in view of (B.4). For y ∈]0, ϕ(x)[ we have 0 < η x,n (y) < 1 and [η x,n (y)] m(n)-2 a.s.

→ 0 when n → ∞, so using the dominated convergence theorem we get

  y a ) is the unique outlier in the sample. If this point is far enough from the cloud of data points, then the local distance | ξm,n (x a ) -ξm,n (x a )| coincides for all m ≥ 1 with the global distance d(m).

  i. for m < m a , both ξm,n (x a ) and ξm,n (x b ) are unaffected by the two outliers, while ξm,n (x a ) is more attracted by these outliers than ξm,n (x b ) due to the left-border effect. This implies that | ξm,n (x a ) -ξm,n (x a )| ≥ | ξm,n (x b ) -ξm,n (x b )| as m increases, whence d(m) = | ξm,n (x a ) -ξm,n (x a )| as m ↑ m a . Therefore the graph of d(m) should have a sharp positive slope as m ↑ m a ; ii. once m exceeds m a , the local distance | ξm,n (x a ) -ξm,n (x a )| decreases smoothly to zero (breakdown of ξm,n (x a )), while | ξm,n (x b ) -ξm,n (x b )| still increases rapidly as m ↑ m b . Let m a,b be the value of m at which | ξm,n (x b ) -ξm,n (x b )| exceeds | ξm,n (x a ) -ξm,n (x a )|. Then m a ≤ m a,b < m b . If m a = m a,b , then d(m) = | ξm,n (x b ) -ξm,n (x b )| for m ≥ m a . Whence d(m) increases rapidly as m ↑ m b and decreases smoothly as m ≥ m b . In contrast, if m a < m a,b < m b , then d(m) = | ξm,n (x a ) -ξm,n (x a )| decreases smoothly for m ∈ [m a , m a,b ) whereas d(m) = | ξm,n (x b )-ξm,n (x b )| increases rapidly for m ∈ [m a,b , m b ) and decreases smoothly for m ≥ m b .

[ 1 ]

 1 Plot the graph of d(m) and its concave envelopment for m = 1, [ n 10 ], [ 2n 10 ], . . . , [ 9n 10 ], n.

Proposition 4 . 1 .

 41 (i) For all m ≥ 1 and any

( 4 . 4 )

 44 . Likewise, in this restrictive setting,[START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF] recover the same asymptotic Weibull distribution of φn (x) for the estimator ξmn,n (x) provided that m n = O(n log n).

Figure 1 :

 1 Figure 1: The true frontiers ξ m and ξm for several values of m (Cobb-Douglas model).

  ure 2 where the frontiers ξm,n and ξm,n are plotted in absence of outliers (bottom: n = 100, 200, 300) and in presence of the 5 outliers (top: n = 105, 205, 305). Moreover, as pointed in Remark 2.2, once ξm,n (x) breaks down, it becomes less resistant to the influential outliers than ξm,n (x) as m increases. This is exactly what happens for ξ25,105 and ξ25,205 at x = 0.3, where these order-α(25) frontiers are clearly more influenced than ξ25,105 and ξ25,205 (here m = 25). In contrast, before breaking down at the point x = 0.3, we see that ξ10,105 and ξ10,205 (here m = 10) are rather more robust than ξ10,105 and ξ10,205 , respectively.

Figure 2 :

 2 Figure 2: In each picture, ξ10,n and ξ25,n (respectively: ξ10,n and ξ25,n ) in solid and dotted blue (respectively: red) lines. From left to right and from top to bottom: n = 105, 205, 305, 100, 200, 300.

A

  real data set: To further illustrate the sensitivity and resistance properties of the empirical partial frontiers ξm,n and ξm,n , we use the real data example of[START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF] and[START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] on the frontier analysis of 9521 French post offices observed in 1994, with X as the quantity of labor and Y as the volume of delivered mail. In this illustration, we only consider the n = 4000 observed post offices with the smallest levels x i . We compared ξm,n and ξm,n for different orders m ∈ {100, 200, 1000, 4000}. The cloud of points and the resulting estimates are provided in Figure3(top): for m large enough (m ∈ {100, 200}), the quantile-based frontier ξm,n is clearly more resistant to the extreme points than the expected maximal output frontier ξm,n . But for m too large (i.e. m ∈ {1000, 4000}), both partial boundaries ξm,n and ξm,n are drastically influenced by the few ostensible FDH points. Nevertheless, while ξ4000,n coincides overall with the FDH frontier, ξ4000,n has the advantage to be still resistant to this envelopment frontier. These results are expected in our theory.

Figure 3 :

 3 Figure 3: Top (full sample): ξm,n and ξm,n for m = 100, 200, 1000, 4000. Bottom: as above without the anomalous data indicated by circles.

Figure 4 :

 4 Figure 4: 95% confidence intervals Q n (red) and C n (blue) of ξ m = q α (Simulated example).

Figure 5 :

 5 Figure 5: Simulated example. The graph of d(m) in solid blue line and its concave envelopment in dotted red line.

  Application to postal data: We test our procedure on the French post offices data set which contains several outlying points in the output-orientation. Proceeding to step[START_REF] Aragon | Nonparametric Frontier Estimation: A Conditional Quantile-based Approach[END_REF] and

Figure 6 :

 6 Figure 6: Potential outlying post offices detected by the semi-automatic procedure.

Figure 7 :

 7 Figure 7: The resulting pictures for m = 1, 400, 800, . . . , 4000 (French post offices).

Figure 8 :

 8 Figure 8: As above with m = 1, 40, . . . , 400 (l-h.s) and m = 1, 8, . . . , 80 (Middle and r-h.s).

Figure 9 :

 9 Figure 9: The resulting pictures for m = 1, 10, 20, . . . , 100 (multivariate simulated data).

Figure 10 :

 10 Figure 10: The resulting pictures for m = 1, 7, 14, 21, . . . , 70 (PFT data).

Figure 11 :

 11 Figure 11: (l-h.s) The difference d(m * )d(1) for the suspicious observations. (r-h.s) Evolution of the % of sample points outside the partial frontiers ξm,3978 and ξm,3978 .

Figure 12 :

 12 Figure 12: 95% confidence intervals Q n (left) and C n (right). Here n = 3978 (without anomalous data). From top to bottom: m = 100 and m = 250 (French post offices).

Lemma A. 1 .

 1 Let x ∈ R p + such that FX,n (x) > 0. Then RB( φn (x), (X, Y ) n ) = 1/n.

  and thus a breakdown occurs as Y * → ∞.Proof of Theorem 2.2 The quantile qα,n (x) = T α,x ((X, Y ) n ) of the sample (X, Y ) n is given by (2.1). Denote by N * the set of all positive integers. In what follows the index j is suchthat T α,x ((X, Y ) n ) = Y x (j) , i.e., j = αN x if αN x ∈ N * and j = [αN x ] + 1 otherwise. (i) First let us show that k = N xj + 1 points are sufficient for breakdown of qα,n (x): If we replace, among the observations (X i , Y i ) with X i ≤ x, the k largest outputs Y x (j) , . . . , Y x (Nx)by an arbitrary point Y * > Y x (Nx) without replacing their corresponding inputs X i , then the X ′ i s of the obtained contaminated sample (X, Y ) n k such that X i ≤ x are the same as those of the initial sample (X, Y ) n and their corresponding orderedY ′ i s are Y x (1) ≤ . . . ≤ Y x (j-1) ≤ Y * ≤ . . . ≤ Y * ,where Y * occurs k times. Hence the αth quantile of (X, Y ) n k , defined as the jth order statistic, is T α,x ((X, Y ) n k ) = Y * . Therefore a breakdown occurs as Y * → ∞. (ii) Let us now show that k -1 = N x -j points are not sufficient for breakdown of qα,n (x):Let (X, Y ) n k-1 = ((X * 1 , Y * 1 ), . . . , (X * n , Y * n )) be a contaminated sample by replacing k -1 points of (X, Y ) n with arbitrary values in R p + × R + . Let ℓ x be the number of replaced points among the observations (

( 1 -

 1 [ Fn (y|x)] m )dy = ξm,n (x)

Lemma B. 1 .

 1 The map m,x φ

  1). Let us now consider √ n( ξm,n (x)ξ m (x)) as a process indexed by x ∈ X , an arbitrarily fixed set such that inf x∈X F X (x) > 0. Here m ≥ 1 is still fixed. Define the domain D X to be the set of distribution functions G on R p+1 + such that G ∈ D x for all x ∈ X . Let ν be the finite upper boundary of the support of Y and define, for any G ∈ D X , the map m φ (G) : x → m,x φ (G) as a map X -→ [0, ν]. Finally, define the functional

  m

  1) by Theorem 20.8 in van der Vaart (1998, p.297). Proof of Theorem 4.1 Write R m,n (x) := √ n( ξm,n (x)ξ m (x)) -√ nΦ m,n (x). By Taylor's formula, for any y ∈ [0, ϕ(x)], there exists a point η x,n (y) interior to the interval joining F (y|x) and Fn (y|x) such that [ Fn (y|x)]

n n 1 / 2 (B. 4 ) with probability 1 .

 1241 x,y) | F (x, y)-F (x, y)| = O log log It follows that sup y | Fn (y|x) -F (y|x)| = O (log log n/n) 1/2 with probability 1, whence sup y { √ n[ Fn (y|x)-F (y|x)] 2 } a.s.

(B. 5 )

 5 Let us show that z n converges in distribution in L ∞ ([α 1 , α 2 ]) to a process z with continuous paths at α: let D 1 be the set of all restrictions of distribution functions on R to [a, b], andfor any G ∈ D 1 , let G -1 : ]0, 1[-→ R denotes the generalized inverse map α → G -1 (α) := inf{y|G(y) ≥ α}. Then by Lemma 3.3 in Daouia (2005), the inverse map φ 1 : G → G -1 as a map D 1 ⊂ D([a, b]) -→ L ∞ ([α 1 , α 2 ]) is Hadamard differentiable at F (•|x) tangentially to C([a, b]) with derivative φ ′ 1,F (•|x) : h -→ -h(F -1 (•|x))/f (F -1 (•|x)|x).We also havez n = √ n{F (φ 1 ( Fn (•|x))|x) -F (φ 1 (F (•|x))|x)} = √ n{φ 2 • φ 1 ( Fn (•|x))φ 2 • φ 1 (F (•|x))}, (B.6)whereφ 2 : G -1 -→ F (•|x) • G -1 . Let us show that φ 2 as a map φ 1 (D 1 ) ⊂ L ∞ ([α 1 , α 2 ]) -→ L ∞ ([α 1 , α 2 ]) is Hadamard differentiable at φ 1 (F (•|x)) = F -1 (•|x) = q • (x) tangentially to φ ′ 1,F (•|x) (C([a, b])). Let H = φ ′ 1,F (•|x) (h) with h ∈ C([a, b]) and take an arbitrary converging pathH t → H in L ∞ ([α 1 , α 2 ]) such that F -1 (•|x) + tH t ∈ φ 1 (D 1) for all small t > 0. By the smoothness of F (•|x), it can be easily seen that[F (F -1 (β|x) + tH t (β)|x) -F (F -1 (β|x)|x)]/t -→ H(β)f (F -1 (β|x)|x) as t → 0 uniformly in β ∈ [α 1 , α 2 ]. Then φ 2 is Hadamard differentiable at φ 1 (F (•|x)) with derivative φ ′ 2,φ 1 (F (•|x)) : H -→ H × f (q • (x)|x) = -h(q • (x)).Hence by the chain rule (see van der Vaart 1998, Theorem 20.9, p.298), we haveφ 2 • φ 1 : D 1 -→ L ∞ ([α 1 , α 2 ]) is Hadamard differentiable at F (•|x) tangentially to C([a, b]) with derivative (φ 2 • φ 1 ) ′ F (•|x) = φ ′ 2,φ 1 (F (•|x)) • φ ′ 1,F (•|x). With this result and the representation (B.6) of z n , we can apply immediately the functional delta method (van der Vaart 1998, Theorem 20.8, p.297) in conjunction with Theorem 3.1 in[START_REF] Daouia | Asymptotic Representation Theory for Nonstandard Conditional Quantiles[END_REF] to obtain the convergence in distribution ofz n in L ∞ ([α 1 , α 2 ]) to z = (φ 2 • φ 1 ) ′ F (•|x) W • F (•|x)/ F X (x) = -W/ F X (x)where W (•) denotes the standard Brownian bridge. Moreover the process z has continuous paths. Sinceg n (z n ) d -→ 0 whenever z n d -→ z in L ∞ ([α1 , α 2 ]) for a process z with continuous paths at α (see van der Vaart 1998, Proof of Lemma 21.7, p.308), we conclude that g n (z n ) in (B.5) converges in distribution to 0. On the other hand, by the smoothness of F (•|x), we have δ n a.s.

  qα,n (x)) + zσ(α, x). It follows from Lemma B.3 that√ n{q α n1 ,n (x) -(q α,n (x)zσ(α, x)/ √ n)} p -→ 0 as n → ∞. (B.7) Likewise √ n{q α n2 ,n (x) -(q α,n (x) + zσ(α, x)/ √ n)} p -→ 0 as n → ∞. Hence √ n{(q α n2 ,n (x)qα n1 ,n (x)) -2zσ(α, x)/ √ n} p -→ 0 as n → ∞.

(

  ii) We know by the proof of Theorem 4.1 (see Equation (B.3)) that √ n( ξm,n (x)ξ m (x)) a.s. = (F X (x)/ FX,n (x)) √ nΦ m,n (x) ,n (y)] m-2 [ Fn (y|x) -F (y|x)] 2 dy and that sup y | Fn (y|x) -F (y|x)| a.s.

ϕ(x) 0 [

 0 η x,n (y)] m-2 dy a.s. → 0. Since √ nm(m -1)/σ(x, m) = O(n/ log log n), we obtain √ n σ(x, m) (m/2)(m -1) ϕ(x) 0 [η x,n (y)] m-2 [ Fn (y|x) -F (y|x)] i √ nσ(Z n,i )whereZ n,i = (m/F X (x))1I(X i ≤ x) ϕ(x) 0

F m- 1

 1 (y|x)[F (y|x) -1I(Y i ≤ y)]dy and its varianceσ 2 (Z n,i ) = σ 2 (x, m). We have nE[|Z n,1 | 3 ]/{nσ 2 (Z n,1 )} 3/2 ≤ mϕ(x)/F X (x) √ nσ(Z n,1 ) → 0 since m/ √ nσ(x, m) → 0. Hence Lyapounov's Theorem gives √ nσ -1 (x, m)Φ m,n (x) d → N (0, 1). Therefore √ nσ -1 (x, m)( ξm,n (x)ξ m (x))d → N (0, 1).

Table 1 :

 1 The values n × RB( ξm,n (x)) with n = 105.

Table 2 :

 2 1000 Monte-Carlo simulations, n = 1000 (l-h.s) with 5 outliers added (r-h.s).

			n = 1000				n = 1005
		MSE	Bias	MSE	Bias
	x	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)	ξ10,n (x)
	0.15	0.0001	0.0001	-0.0011	-0.0009	0.0002	0.0001	0.0105	0.0023
	0.35	0.0001	0.0001	-0.0010	-0.0007	0.0002	0.0001	0.0103	0.0045
	0.55	0.0001	0.0001	-0.0002	0	0.0001	0.0001	0.0066	0.0040
	0.75	0.0001	0.0001	-0.0009	-0.0007	0.0001	0.0001	0.0058	0.0021
	0.95	0.0001	0.0001	-0.0008	-0.0007	0.0001	0.0001	0.0022	0.0017
	x	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)	ξ15,n (x)
	0.15	0.0001	0.0001	-0.0012	-0.0008	0.0003	0.0001	0.0147	0.0032
	0.35	0.0001	0.0001	-0.0009	-0.0008	0.0002	0.0001	0.0128	0.0037
	0.55	0.0001	0.0001	-0.0011	-0.0009	0.0001	0.0001	0.0087	0.0045
	0.75	0.0001	0.0001	-0.0006	-0.0004	0.0001	0.0001	0.0076	0.0029
	0.95	0	0.0001	-0.0004	0.0002	0.0001	0.0001	0.0028	0.0022
	x	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)	ξ20,n (x)
	0.15	0.0001	0.0001	-0.0011	-0.0008	0.0004	0.0001	0.0186	0.0030
	0.35	0.0001	0.0001	-0.0011	-0.0008	0.0003	0.0001	0.0159	0.0047
	0.55	0.0001	0.0001	-0.0004	-0.0001	0.0001	0.0001	0.0099	0.0036
	0.75	0	0.0001	-0.0007	-0.0006	0.0001	0.0001	0.0085	0.0025
	0.95	0.0001	0.0001	-0.0007	-0.0006	0.0001	0.0001	0.0033	0.0031

Table 3

 3

	provides the average lengths and the achieved coverages of the 95% asymptotic
	confidence intervals C n and Q n computed over 1000 random replications, for sample sizes

Table 3 :

 3 Average Lengths (avl) and Coverages (cov) of the 95% confidence intervals C n and Q n , sample sizes n = 100 and n = 1000.

	n = 100	m = 5 and α = .8557		n = 100	m = 10 and α = .9242
	x	cov Qn	cov Cn	avl Qn	avl Cn	x	cov Qn	cov Cn	avl Qn	avl Cn
	0.4	0.9120	0.9540	0.0903	0.1347	0.65	0.9200	0.9640	0.0889	0.1392
	0.5	0.9310	0.9510	0.0903	0.1302	0.70	0.9080	0.9460	0.0883	0.1324
	0.6	0.9360	0.9500	0.0907	0.1291	0.75	0.9310	0.9510	0.0888	0.1274
	0.7	0.9440	0.9560	0.0910	0.1299	0.80	0.9140	0.9550	0.0878	0.1257
	0.8	0.9450	0.9470	0.0913	0.1283	0.85	0.9110	0.9510	0.0881	0.1300
	0.9	0.9400	0.9540	0.0914	0.1273	0.90	0.9110	0.9500	0.0886	0.1298
	1	0.9380	0.9540	0.0913	0.1295	0.95	0.9250	0.9490	0.0891	0.1239
	n = 1000	m = 5 and α = .8557		n = 1000	m = 10 and α = .9242
	x	cov Qn	cov Cn	avl Qn	avl Cn	x	cov Qn	cov Cn	avl Qn	avl Cn
	0.4	0.9540	0.9560	0.0293	0.0401	0.65	0.9560	0.9480	0.0290	0.0394
	0.5	0.9470	0.9360	0.0293	0.0397	0.70	0.9280	0.9410	0.0291	0.0393
	0.6	0.9570	0.9540	0.0293	0.0402	0.75	0.9350	0.9390	0.0291	0.0395
	0.7	0.9490	0.9530	0.0293	0.0400	0.80	0.9320	0.9500	0.0290	0.0390
	0.8	0.9450	0.9470	0.0294	0.0400	0.85	0.9380	0.9420	0.0290	0.0393
	0.9	0.9360	0.9510	0.0294	0.0401	0.90	0.9280	0.9470	0.0290	0.0392
	1	0.9420	0.9640	0.0293	0.0403	0.95	0.9480	0.9470	0.0290	0.0393

  where F +th t is contained in D X for all small t. Abbreviate th t ) to ξ mt (x). By the uniform convergence of h t and the definition of ζ t,x (y), wehave inf x∈X |F X (x) + th t (x, ∞)| → inf x∈X F X (x) and sup x∈X ,y∈ R |ζ m-1 t,x (y) -F m-1 (y|x)| → 0 as t ↓ 0.By using sup x∈X ,y∈ R |ζ t,x (y)| ≤ 1 and sup x∈X ϕ(x) ≤ ν, it can be easily seen that sup x∈X |(ξ mt (x)ξ m (x))/t -(

	m,x
	φ (F + m,x φ ) ′ F (h)| → 0 as t ↓ 0, which ends the proof.
	Proof of Proposition 4.1(ii) By applying the functional delta method in conjunction with Lemma B.2, it is immediate that √ n(

m φ ( F )-m φ (F )) converges in distribution in L ∞ (X ) to the linear transformation G m = ( m φ) ′

F (F) of the Gaussian process F. Furthermore, the linear operator ( m φ) ′ F (•) is defined and continuous on the whole space L ∞ ( Rp+1 ) since

  → 1. By applying again the classical LIL (see e.g. Serfling 1980, Theorem A, p.35), we obtain for either choice of sign lim sup

	n→∞

  x)/ FX (x) a.s. → 1. Hence to check that A n d → N (0, 1), it is enough to show according Loève's criterion (1963, p.295) that lim n→∞ n |z|≥ε z 2 dF n,1 (z) = 0 for all ε > 0, where F n,1 is the common distribution function of the random variables W n,i / √ nσ(W n,1 ). We have by Chebyshev's inequality. Since σ 2 (W n,1 ) ∼ α n (1α n )F X (x) and n(1α n ) → ∞, we get n |z|≥ε z 2 dF n,1 (z) → 0 and so

	ε	|z|≥ε	z 2 dF n,1 (z) ≤ ≤ P[|W n,1 | ≥ ε R |z| 3 1I(|z| ≥ ε)dF n,1 (z) = E √ nσ(W n,1 )] { √ nσ(W n,1 )} 3 ≤ 1/nε 2 { √	W n,i √ nσ(W n,1 ) nσ(W n,1 )} 3	3	1I	W n,i √ nσ(W n,1 )	≥ ε
			n i=1	W n,i √ nσ(W n,i )	d

Acknowledgements. The authors thank the Editor, Associate Editor and reviewers for their valuable comments which led to a considerable improvement of the manuscript. This research was supported by the Research Fund KULeuven (GOA/07/04-project) and by the IAP research network P6/03, Federal Science Policy, Belgium. Support from the French "Agence Nationale pour la Recherche" (under grant ANR-08-BLAN-0106-01/EPI project) is also acknowledged.


HAL is

Hyperspectral laboratory and airborne measurements as tools for local mapping of swelling soils in Orléans area (France)

Gilles Grandjean, Grégory Dufréchou, Audrey Hohmann

To cite this version:

Gilles Grandjean, Grégory Dufréchou, Audrey Hohmann. Hyperspectral laboratory and airborne measurements as tools for local mapping of swelling soils in Orléans area (France). EGU General Assembly 2013, Apr 2013, Vienne, Austria. hal-00796837

Hyperspectral laboratory and airborne measurements as tools for local mapping of swelling soils in Orléans area (France)

Gilles Grandjean, Gregory Dufrechou, and Audrey Hohmann BRGM, DRP/RIG, Orleans, France (g.grandjean@brgm.fr)

Swelling soils contain clay minerals that change volume with water content and cause extensive and expensive damage on infrastructures. Based on spatial distribution of infrastructure damages and existing geological maps, the Bureau de Recherches Géologiques et Minières (BRGM, the French Geological Survey) published in 2010 a 1:50 000 swelling hazard map of France. This map indexes the territory to low, intermediate, or high swell susceptibility, but does not display smallest and isolated clays lithologies. At local scale, identification of clay minerals and characterization of swell potential of soils using conventional soil analysis (DRX, chemical, and geotechnical analysis) are slow, expensive, and does not permit integrated measurements. Shortwave infrared (SWIR: 1100-2500 nm) spectral domains are characterized by significant spectral absorption bands that provide an underused tool for estimate the swell potential of soils. Reflectance spectroscopy, using an ASD Fieldspec Pro spectrometer, permits a rapid and less expensive measurement of soil reflectance spectra in the field and laboratory. In order to produce high precision map of expansive soils, the BRGM aims to optimize laboratory reflectance spectroscopy for mapping swelling soils. Geotechnical use of laboratory reflectance spectroscopy for local characterization of swell potential of soils could be assessable from an economical point of view. A new high resolution airborne hyperspectral survey (covering ca. 280 km 2 , 380 channels ranging from 400 to 2500 nm) located at the W of Orléans (Loiret, France) will also be combined with field and laboratory measurements to detect and map swelling soils.