J-F Dupuy

J-M Loubes

E Maza

Non parametric estimation of the structural expectation of a stochastic increasing function

Keywords: Functional data analysis, Non parametric warping model, Structural expectation, Curve registration, . Subject Class. MSC-2000: 62G05, 62G20

This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the small variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So we define the structural expectation of the underlying stochastic function. Then we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.

Statistical model for functional phase variations

Functional data analysis deals with the analysis of experiments where one or several quantities are recorded during a time period for different individuals, resulting in a sample of observed curves. One of the main difficultis is given by the fact that curves usually not only present amplitude variability (a variation in the y-axis) but also time or phase variability (a variation in the x-axis). Hence the classical cross-sectional mean does not make sense and the definition of an appropriate population mean is even not obvious. Giving a sense to the common behaviour of a sample of curves, and finding a meaningful mean curve in this context is thus an important issue, called curve registration, or time warping problem, which first appeared in the engineering literature in [START_REF] Sakoe | Dynamic programming algorithm optimization for spoken word recognition[END_REF].

Several methods have been proposed over the years to estimate the mean pattern of a sample of curves. A popular method consists in i) first aligning the curves to a given template by warping the time axis, ii) then taking the mean of all the dewarped curves. Such methods are increasingly common in statistics, see [START_REF] Ramsay | Applied functional data analysis[END_REF] for a review. A landmark registration methodology is proposed by [START_REF] Kneip | Statistical tools to analyze data representing a sample of curves[END_REF] and further developed by [START_REF] Bigot | A scale-space approach with wavelets to singularity estimation[END_REF]. A non parametric method is investigated in [START_REF] Ramsay | Curve registration[END_REF] and in [START_REF] Kneip | Curve registration by local regression[END_REF], using local regressions. Dynamic time warping methodology is developed by [START_REF] Wang | Synchronizing sample curves nonparametrically[END_REF]. An alternative approach is provided in [START_REF] Gamboa | Semi-parametric estimation of shifts[END_REF], where semi-parametric estimation of shifted curves is studied. But these methods imply choosing a starting curve as a fixed point for the alignment process. This initial choice may either bias the estimation procedure, or imply strong and restrictive identifiability conditions.

In this work, we consider a second point of view. We define an archetype representing the common behaviour of the sample curves directly from the data, without stressing a particular curve. Such a method has the advantage of not assuming any technical restrictions on the data and so, enables to handle a large variety of cases. However, the registration procedure and the common pattern have to be clearly defined.

We observe i = 1, . . . , m curves f i : [a, b] → R at equidistant discrete times t ij ∈ [a, b], j = 1, . . . , n. So the data can be written as Y ij = f i (t ij ) , i = 1, . . . , m, j = 1, . . . , n.

(

So, the registration problem aims at finding a mean pattern f and warping functions h i which align the observed curves, i.e such that ∀i = 1, . . . , m, f = f i • h i . Hence each curve is obtained by warping the original curve f using the warping functions h i .

Defining the registration operator is a difficult task. In this paper we propose a random warping procedure which takes into account the variability of the deformation as a random effect. So, we assume that there exists a random process H such that the data are i.i.d realizations of this process. Let H be this warping stochastic process defined as

H : Ω → C ([a, b]) w → H(w, •), (2)
where (Ω, A, P) is an unknown probability space, and (C ([a, b]) , • ∞ , B) is the set of all real continuous functions defined on the interval [a, b], equipped with the uniform norm and with its Borel algebra. Consider h 1 , . . . , h m i.i.d realizations of the process H(t). h i warps a mean pattern f onto the i-th observation curve f i . Hence, model [START_REF] Bigot | A scale-space approach with wavelets to singularity estimation[END_REF] can be modeled by

Y ij = f i (t ij ) = f • h -1 i (t ij ). (3)
For sake of simplicity, we will write f i (t) = f • h -1 i (t), for all i ∈ {1, . . . , m}. We point out that h -1 i is well defined since the warping processes are assumed to be continuous increasing functions.

Under general assumptions [START_REF] Billingsley | Convergence of probability measures[END_REF], model (3) is not identifiable. More precisely, the unknown function f and the unknown warping process H can not be estimated. Indeed, if h : [a, b] → [a, b] is an increasing continuous function, with h(a) = a and h(b) = b, then, for all i ∈ {1, . . . , m} and all j ∈ {1, . . . , n}, we have that

Y ij = f • h-1 • h • h -1 i (•, t ij ). Hence, the function f • h-1 , associated with the warping process H • h-1 , is also a solution of model (3).
The aim of this paper is to build a new kind of pattern which represents the common functional feature of the data but still that can be estimated together with the warping procedure. For this, let φ(.) be the expectation of the warping process and define the structural expectation f ES as

f ES := f • φ -1 .
The structural expectation is obviously not the function f , but the function f composed with φ -1 , the inverse of the expectation of H. Hence it can be seen as the mean warping of the function f by the stochastic process H. In this article, we aim at studying the properties of the structural expectation, and finding an estimator of the structural expectation and of the warping paths h i .

The article is organized as follows. In Section 1, we introduce a warping functional model with a stochastic phase variation, and we introduce the structural expectation. In Section 2, we define empirical estimators of the structural expectation and of the individual warping functions. Asymptotic properties of these estimators are investigated. Proofs are postponed to Section A. Section 3 investigates some extensions of the proposed methodology, in particular to the case of noisy non increasing functions. The results of a simulation study are reported in Section 4. There, we also apply the proposed estimators to a real data set.

2 Theoretical study of a particular case: warping of strictly increasing functions

First, consider the case where f is a strictly increasing. Hence the inverse function f -1 exists and is also strictly increasing. Moreover, a phase warping of function f (i.e. on x-axis) corresponds to an amplitude warping of function f -1 (i.e. on y-axis). We propose estimators of both the inverse of the structural expectation

f -1 ES = φ • f -1 ,

and also of each individual warping function

φ • h -1
i , for all i ∈ {1, . . . , m}, and finally an of the structural expectation

f ES = f • φ -1 .
Note that all the asymptotic results are taken with respect to m and n, so we recall that u(m, n) ----→ m,n→∞ c if, and only if, we have

∀ǫ > 0, ∃(m 0 , n 0 ) ∈ N 2 , ∀(m, n) ∈ N 2 , m > m 0 and n > n 0 ⇒ |u(m, n) -c| < ǫ.
We will assume the following conditions on the warping process in order to define a good registration procedure. The warping process does not change the timeline (not time inversion) and leaves fixed the two extreme points, so for almost all w ∈ Ω, assume that The following proposition introduces respectively the expectation, the second order moment and the covariance function of H. Proposition 2.1. Under assumption (2), the expectation φ(•), the second order moment γ(•) and the covariance function r(•, •) of the stochastic process H are well defined. φ and γ are also continuous increasing functions. Moreover, we have φ

(a) = a, φ(b) = b, γ(a) = a 2 and γ(b) = b 2 . As a consequence, we have that varH(•, a) = varH(•, b) = 0.
Proof. The process H is bounded and increasing. As a consequence, φ and γ exist. Moreover, H is a continuous increasing process, which leads to continuous and increasing first and second order moments.

In order to prove asymptotic results, the following technical assumptions on the warping process H and on the function f are needed: Assumptions 1. There exists a constant C 1 > 0 such that, for all (s, t) ∈ [f (a), f (b)] 2 , we have

E |H(s) -EH(s) -(H(t) -EH(t))| 2 C 1 |s -t| 2 . ( 4
)
2. There exists a constant C 2 > 0 such that, for all (s, t) ∈ [f (a), f (b)] 2 , we have

f -1 (s) -f -1 (t) 2 C 2 |s -t| 2 .
(5)

3. There exists a constant C 3 > 0 such that, for all ω ∈ Ω, for all (s, t) ∈ [a, b] 2 , we have

H -1 (ω, s) -H -1 (ω, t) 2 C 3 |s -t| 2 . ( 6
)
2.1 Estimator of the structural expectation

f ES Since f i = f • h -1 i we have f -1 i = h i • f -1 . Hence E(f i ) = (E(H)) • f -1 .
Hence it seems natural to consider the mean of functions f -1 i i∈{1,...,m} in order to estimate the inverse of the structural expectation. For all y ∈ [f (a), f (b)], and for all i ∈ {1, . . . , m}, define j i (y) = arg min j∈{1,...,n}

|Y ij -y| and T i (y) := t ij i (y) . (7)
Then, the empirical estimator of the inverse of the structural expectation is defined by

f -1 ES (y) = 1 m m i=1 T i (y). ( 8
)
The following theorem provides consistency and asymptotic normality of estimator (8). Theorem 2.1 (Consistency of the inverse of the structural expectation). Under Assumption (2),

f -1 ES -f -1 ES ∞ as ----→ m,n→∞ 0.
Moreover, let n = m 1 2 +α with α > 0, and assume Conditions (4) and [START_REF] Kneip | Statistical tools to analyze data representing a sample of curves[END_REF]. Then,

√ m f -1 ES -f -1 ES D ---→ m→∞ G,
where G is a centered Gaussian process with covariance given by: for all (s, t)

∈ [f (a), f (b)] 2 , cov(G(s), G(t)) = r f -1 (s), f -1 (t) .
From ( 8) and ( 7), f -1 ES is an increasing step function with jumps occuring at say,

K(m, n) points v 1 , . . . , v K(m,n) in [f (a), f (b)], such that f (a) = v 0 < v 1 < . . . < v K(m,n) < v K(m,n)+1 = f (b). Hence for all y ∈ [f (a), f (b)]\(v k ) k∈K (K = {0, . . . , K(m, n) + 1}), f -1 ES (y) can be expressed as f -1 ES (y) = K(m,n) k=0 u k 1 (v k ,v k+1 ) (y) with a = u 0 < u 1 < . . . < u K(m,n)-1 < u K(m,n) = b.
A natural estimator of the structural expectation f ES is then obtained by linear interpolation between the points (u k , v k ). For all t ∈ [a, b], let define

f ES (t) = K(m,n)-1 k=0 v k + v k+1 -v k u k+1 -u k (t -u k ) 1 [u k ,u k+1 ) (t) + v K(m,n) 1 {b} (t).
Note that by construction, this estimator is strictly increasing and continuous on [a, b].

The following theorem states its consistency.

Theorem 2.2 (Consistency of the estimator of the Structural Expectation). Under Assumption (2), we have

f ES -f ES ∞ as ----→ m,n→∞ 0.
Obtaining confidence bands for f -1 ES is useful for describing and visualizing the uncertainty in the estimate of f -1 ES . This requires finding an estimator of var(G(y

)) = γ • f -1 (y) -f -1 ES (y) 2 , y ∈ [f (a), f (b)]. Lemma 2.3. Let y ∈ [f (a), f (b)]. Let γ • f -1 (y) = 1 m m i=1
T 2 i (y), with T i (.) defined as in [START_REF] Ledoux | Probability in Banach spaces[END_REF]. Then,

γ • f -1 (y) -f -1 ES (y) 2 as ----→ m,n→∞ var(G(y)).
Proof of this lemma is given in Section 4.2. Combining this lemma with the asymptotic normality result stated by Theorem 2.1 yields a pointwise asymptotic confidence band for f -1 ES . Corollary 2.4. An asymptotic (1α)-level pointwise confidence band for f -1

ES is given by

f -1 ES (y) -u 1-α 2 v ar(G(y)) m , f -1 ES (y) + u 1-α 2 v ar(G(y)) m , where v ar(G(y)) = γ • f -1 (y) -f -1 ES (y) 2
and u 1-α 2 is the quantile of order 1 -α 2 of the standard normal distribution.

Note that the construction of a simultaneous asymptotic confidence band for f -1

ES

would require determination of the distribution of sup f (a) y f (b) |G(y)|. This, however, falls beyond the scope of this paper.

Estimator of an individual warping function

In a warping framework, it is necessary to estimate the mean pattern but also the individual warping functions, i.e. (h -1 i ) i∈{1,...,m} . As previously, we can not directly estimate the functions h -1 i only the functions φ • h -1 i . Let i 0 ∈ {1, . . . , m}. Now we want to compute for all i = i 0 ,

T ⋆ i (t) = f -1 i • f i 0 (t).
For this, define j 0 (t) = arg min j∈{1,...,n}

|t i 0 j -t| . ( 9
)
This point is the obervation time for the i 0 curve which is the closest to t. Note that the index j 0 (t) depends on i 0 but, for sake of simplicity we drop this index in the notations.

Then ∀t ∈ [a, b], and ∀i ∈ {1, . . . , m} i 0 , compute

T i (t) = arg min t j ∈{t i1 ,...,t in } Y ij -Y i 0 j 0 (t) , (10)
an estimate of T ⋆ i . Then for a fixed i 0 , noting that

T ⋆ i = h i • h -1 i 0 ,

we can see that an empirical estimator of each individual warping function

φ • h -1 i 0 is given by φ • h -1 i 0 (t) := 1 m -1 m i=1 i =i 0 T i (t). ( 11
)
The following theorem asserts consistency and asymptotic normality of this estimator.

Theorem 2.5. Under assumption (2),

φ • h -1 i 0 -φ • h -1 i 0 ∞ as ----→ m,n→∞ 0.
Let n = m 1 2 +α (with α > 0) and assume that ( 4) and ( 6) hold. Then

√ m( φ • h -1 i 0 -φ•h -1 i 0 ) converges weakly to a centered Gaussian process Z, √ m( φ • h -1 i 0 -φ • h -1 i 0 ) D ---→ m→∞ Z,
with covariance function defined for all (s, t)

∈ [a, b] 2 by cov(Z(s), Z(t)) = r h -1 i 0 (s), h -1 i 0 (t) .
We may also compute confidence bands for φ

• h -1 i 0 , based on a consistent estimator of var(Z(t)) = γ • h -1 i 0 (t) -{φ • h -1 i 0 (t)} 2 . Lemma 2.6. Let t ∈ [a, b]. Let γ • h -1 i 0 (t) = 1 m m i=1 T 2 i (t)
, with T i (.) defined by ( 10) and ( 9). Then

γ • h -1 i 0 (t) -φ • h -1 i 0 (t) 2 as ----→ m,n→∞ var(Z(t)).
Proof of this lemma relies on the same arguments as proof of Lemma 2.3 and is outlined in Section 4.2. A pointwise asymptotic confidence band for φ

• h -1 i 0 is now given by Corollary 2.7. An asymptotic (1 -α)-level pointwise confidence band for φ • h -1 i 0 is given by φ • h -1 i 0 (t) -u 1-α 2 v ar(Z(t)) m , φ • h -1 i 0 (t) + u 1-α 2 v ar(Z(t)) m , where v ar(Z(t)) = γ • h -1 i 0 (t) -φ • h -1 i 0 (t) 2 .
3 Extensions to the general case

In the preceding part, we studied the asymptotic behaviour of a new warping methodology. However, drastic restrictions over the class of functions are needed, mostly dealing with monocity of the observed functions and on a non noisy model. In this part, we get rid of such asumptions and provide a pratical way of handling more realistic observations. First, note that the assumptions H(a)

a.s = a et H(b) a.s = b can be weakened into the following assumptions ii ′ ) H -1 (•, a) are H -1 (•, b) random variables compactly supported such that sup w∈Ω H -1 (w, a) a.s < inf w∈Ω H -1 (w, b).
Then, we focus on the other assumptions.

Breaking monotonicity

The main idea is to build a transformation G which turns a nonmonotone function into a monotone function while preserving the warping functions. For sake of simplicity, the observation times will be taken equal for all the curves, hence t ij will be denoted t j . Hence, the observations

Y ij = f • f -1 i (t j ), i = 1, . . . , m, j = 0, • • • , n, are transformed into Z ij = G(f ) • h -1 i (t j ) := g • h -1 i (t j ), i = 1, . . . , m, j = 0, • • • , n, (12)
where g is a monotone function. So, estimating the warping process of the monotonized model can be used to estimate the real warping functions, and then align the original observations Y ij to their structural mean.

For this, consider a nonmonotone function

f : [a, b] → {1, . . . , m} and let a = s 0 < s 1 < • • • < s r < s r+1 = b be the different variational change points, in the sense that ∀k ∈ {0, . . . , r -1}, ∀(t 1 , t 2 ) ∈]s k , s k+1 [, ∀(t 3 , t 4 ) ∈]s k+1 , s k+2 [, t 1 < t 2 et t 3 < t 4 ⇒ (f (t 1 ) -f (t 2 )) (f (t 3 ) -f (t 4 )) < 0. ( 13
)
So, consider functional warping over the subset

F = {f : [a, b] → {1, . . . , m} ⊂ R such that (13) holds }.
Let π :]a, b[ {s 1 , . . . , s r } → {-1, 1} be a tool function which indicates whether, around a given point t, the function is f increasing or decreasing, defined by

π : s l(t) < t < s l(t)+1 → π(t) = -1 si s l(t) -s l(t)+1 > 0, 1 si s l(t) -s l(t)+1 < 0,
with l(t) ∈ {0, . . . , r}.

Monotonizing Operator

Define for all f ∈ F the operator G(., f

) : t ∈]a, b[ {s 0 , . . . , s r+1 } → G(t, f ) by G(t, f ) = f (t)π(t) - r k=0 π(t)f (s k )1 ]s k ,s k+1 [ (t) + f (s 0 ) + r k=1 |f (s k-1 ) -f (s k )|1 ]s k ,b[ (t),
and, for all k ∈ {0, . . . , r + 1},

G(s k , f ) = f (a) + k l=1 |f (s l-1 ) -f (s l )| ,
with the notation 0 l=1 |f (s l-1 )f (s l )| = 0. By construction, it is obvious that t → G(t, f ) is strictly increasing. Moreover, the following proposition proves that the warping functions remain unchanged Proposition 3.1. Consider f ∈ F and warping functions h i , i = 1, . . . , m. Set ∀i = 1, . . . , m, G(., f i ) = g i (.). We have g i = g • h -1 i . Proof. For i ∈ {1, . . . , m}, let us prove that f i ∈ F . Using [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF], consider the change points (s k ) k=1,...,r and set

s i k = h i (s k ). For k ∈ {0, . . . , r -1}, consider (t 1 , t 2 ) ∈ s i k , s i k+1 , (t 3 , t 4 ) ∈ s i k+1 , s i k+2 t 1 < t 2 et t 3 < t 4 , then, (f i (t 1 ) -f i (t 2 )) (f i (t 3 ) -f i (t 4 )) = f • h -1 i (t 1 ) -f • h -1 i (t 2 ) f • h -1 i (t 3 ) -f • h -1 i (t 4 ) . Since h -1 i is strictly increasing, we get s k < h -1 i (t 1 ) < h -1 i (t 2 ) < s k+1 and s k+1 < h -1 i (t 3 ) < h -1 i (t 3 ) < s k+2 . Hence (f i (t 1 ) -f i (t 2 )) (f i (t 3 ) -f i (t 4 )) < 0, which yelds that f i ∈ F . So define g i = G(f i ). For all t ∈]a, b[ {s i 1 , . . . , s i r }, we get g i (t) = f i (t)Π(t, f i )- r k=0 Π(t, f i )f i (s i k )1 ]s i k ,s i k+1 [ (t)+f i (s 0 )+ r k=1 f i (s i k-1 ) -f i (s i k ) 1 ]s i k ,b[ (t), But f i (t) = f • h -1 i (t), Π(t, f i ) = Π(t, f • h -1 i ) = Π(h -1 i (t), f ), 1 ]s i k ,s i k+1 [ (t) = 1 ]h i (s k ),h i (s k+1 )[ (t) = 1 ]s k ,s k+1 [ h -1 i (t) ,
which implies that

g i = g • h -1 i .
The discretization implies however that the Z ij can not be computed directly since the functions f i , i = 1, . . . , m, are known on the grid t j , j = 0, . . . , n, while the values, s k and s i k are unknown. So consider estimates of Z ij defined as follows

Zi0 = Y i0 ∀j ∈ {1, . . . , n}, Zij = Zij-1 + |Y ij -Y ij-1 | . ( 14
)
The following proposition proves the consistency of such estimation procedure.

Proposition 3.2. For f ∈ F , i ∈ {1, . . . , m} and t ∈ [a, b], define a sequence j(n) such that j(n) n ----→ n→+∞ t. Then, Zij(n) -Z ij(n) a.s ----→ n→+∞ 0. Proof. Set t ∈]a, b[ {s i 1 , . . . , s i r }, and l ∈ {0, 1, . . . , r} such that t ∈ s i l , s i l+1 . Consider j(n), j(n) n ----→ n→+∞ t, then ∃n 0 ∈ N / ∀n ∈ N, n n 0 ⇒ j(n) n ∈ s i l , s i l+1 . ∀n n 0 , we have Zij(n) = Zij(n)-1 + Y ij(n) -Y ij(n)-1 = Y i0 + j(n) k=1 |Y ik -Y ik-1 | .
Moreover,

Z ij(n) = g • h -1 i t j(n) = f i t j(n) Π t j(n) , f i + Π t j(n) , f i f i s i l + f i (s 0 ) + l k=1 f i s i k-1 -f i s i k , which yelds that Z ij(n) = Π t j(n) , f i Y ij(n) -f i s i l + Y i0 + l k=1 f i s i k-1 -f i s i k = Y ij(n) -f i s i l + Y i0 + l k=1 f i s i k-1 -f i s i k = A + Y i0 + B. Write ∀k = 1, . . . , l + 1, s i k-1 t j k -p k < • • • < t j k s i k , we get that f i s i k-1 -f i s i k = f i s i k-1 -f i (t j k -p k ) + p k q=1 (f i (t j k -q ) -f i (t j k -q+1 )) + f i (t j k ) -f i s i k = f i s i k-1 -f i (t j k -p k ) + p k q=1 |f i (t j k -q ) -f i (t j k -q+1 )| + f i (t j k ) -f i s i k = f i s i k-1 -Y ij k -p k + p k q=1 |Y ij k -q+1 -Y ij k -q | + Y ij k -f i s i k , Hence B = l k=1 f i s i k-1 -f i s i k = l k=1 f i s i k-1 -Y ij k -p k + l k=1 Y ij k -f i s i k + l k=1 p k q=1 |Y ij k -q+1 -Y ij k -q | = l k=1 f i s i k-1 -Y ij k -p k + l k=1 Y ij k -f i s i k + j l k=1 |Y ik -Y ik-1 | - l-1 k=1 |Y ij k+1 -p k+1 -Y ij k+1 -p k+1 -1 | -|Y i1 -Y i0 |.
With the same ideas, we can write

A = Y ij(n) -f i s i l = p l+1 +1 q=j l+1 -j(n)+1 |Y ij l+1 -q+1 -Y ij l+1 -q | + Y ij l+1 -p l+1 -f i s i l -|Y ij l+1 -p l+1 -Y ij l+1 -p l+1 -1 |.
As a result,

Z ij(n) -Zij(n) = l+1 k=1 f i s i k-1 -Y ij k -p k + l k=1 Y ij k -f i s i k - l k=1 |Y ij k+1 -p k+1 -Y ij k+1 -p k+1 -1 | -|Y i1 -Y i0 |.
By continuity of f , f i is also continuous, hence

Z ij(n) -Zij(n) a.s ----→ n→+∞ 0.
For t ∈ s i 0 , . . . , s i r+1 , we get similar results, leading to the conclusion. As a conclusion, we can extend our results to the case of nonmonotone functions since we transform the problem into a monotone warping problem with the same warping functions. These functions h i i = 1, . . . , m can be estimated by our methodology using the new observations Zij ∀i = 1, . . . , m, ∀j = 0, . . . , n. The estimator can be written in the following form

φ • h -1 i 0 mn (t) = 1 m -1 m i=1 i =i 0 T j i , (15)
with T j i = arg min t j ∈{t 0 ,...,tn} Zij -Zi 0 j 0 , and t j 0 = arg min t j ∈{t 0 ,...,tn} |t j -t|.

Dealing with noisy data

If theoretical asymptotical results are only given in a non noisy framework, we can still handle the case where the data are observed in the standard regression model

Y ij = f • h -1 i (t j ) + ε ij , i = 1, . . . , m, j = 0, . . . , n, (16)
with

ε ij i.i.d
∼ N (0, σ 2 ). To be able to apply our algorithm, we first denoise the data. For this, we estimate separately each function f i , i = 1, . . . , m, by a Kernel estimator. On a practical point of view, we describe the estimation procedure used in the simulations.

1. ∀i ∈ {1, . . . , m}, f i is estimated fi (t 0 ) = 1 m m i=1 Y i0 a.s ----→ m→+∞ f (t 0 ), fi (t n ) = 1 m m i=1 Y in a.s ----→ m→+∞ f (t n ).
Given a Gaussian kernel Φ, ∀j ∈ {1, . . . , n -1}, f i (t j ) is estimated by

fi (t j ) = n k=0 Y ik Φ t k -t j ν i n k=0 Φ t k -t j ν i . ( 17
)
The banwidths ν i are to be properly chosen.

2. The estimation procedure can be conducted using the denoised observations fi (t j ), leading to new estimates f (t) of the structural expectation f • φ -1 .

We point out that the efficiency of the procedure heavily relies on a proper choice of the bandwidths ν i , i = 1, . . . , m. Cross-validation technics do not provide good results since the aim is not to get a good estimation of the function but only a good separation of the different functions. Hence oversmoothing the data is not a drawback in this settings. So, the smoothing parameters ν = ν i , ∀i = 1, . . . , m are obtained by minimizing the following matching criterion

ν = arg min ν∈L m i=1 n j=0 f i (t j ) -f (t j ) ,
over a grid L. Practical applications of this algorithm are given in Figure 4 Numerical Study

In this part, we estimate the structural expectation, using both the proposed method and the analytic registration approach developed in [START_REF] Ramsay | Applied functional data analysis[END_REF]. Note that an alternative to analytic registration is provided by the so-called landmarks registration approach in [START_REF] Kneip | Statistical tools to analyze data representing a sample of curves[END_REF], but this approach requires the determination of landmarks (such as local extrema) which can be difficult in our simulations. Hence, this method is not implemented here. First, results on simulated data are given in order to compare these two methods. Then, an application of our methodology is given for a real data set.

Simulations

Two simulation studies are carried out in this section. The first one involves a strictly increasing function and the second one involves a non monotone function.

Warped functions. Let f and g (see Figures 1 and2) be defined by ∀t ∈ [0, 1], f (t) = sin(3πt) + 3πt and g(t) = sin(6πt) 6πt .

These two functions will be warped by the following random warping process.

Warping processes. The Stochastic warping functions H i , i = 1, . . . , m, are simulated by the iterative process described below.

Let ǫ > 0. First, for all i = 1, . . . , m, let H (0) i be the identity function. Then, the warping functions H (k+1) i , i = 1, . . . , m, are successively carried out from functions H (k) i , i = 1, . . . , m, by iterating N times the following process :

1. Let U be a uniformly distributed random variable on [10ǫ, 1 -10ǫ].

2. Let V i , i = 1, . . . , m, be independant and identically uniformly distributed variables on [Uǫ, U + ǫ].

3. For all i = 1, . . . , m, the warping function H

(k) i is warped as follows :

H (k+1) i = W i • H (k) i
where W i is defined by

W i (t) = V i U t if 0 t U, 1-V i 1-U t + V i -U 1-U if U < t 1.
We have thus defined stricly increasing stochatic functions

H i : [0, 1] → [0, 1], such that ∀t ∈ [0, 1], E (H i (t)) = φ(t) = t.
Hence, the warping process is centered, in the sense that, the structural expectation f •φ -1 is equal to f . Our simulated warping functions H i , i = 1, . . . , m, have been carried out by using the above iterating process with m = 30, N = 3000 and ǫ = 0.005. These processes are shown on Figure 1 and on Figure 2. In these figure, we can see the very large phase variations. For example, point 0.2 is warped between approximately 0.05 and 0.35 for the first case, and between approximately 0.00 and 0.40 for the second case. Simulated data. Finally, simulated data are carried out on an equally spaced grid as follows :

Y ij = f H -1 i (t j ) and Y ij = g H -1 i (t j ) + ǫ ij with t j = j
n , j = 0, 1, . . . , n, n = 100 points. Simulated warped functions are respectively shown on Figures 1 and2.

Figures 1 and2 show the functions f and g and the mean functions of the warped functions. We can easily see that the classical mean is not adapted to describe the data. In particular, the mean function of the first simulated data does not reflect the flat component of the function f (in the range [0.2, 0.4]) which yet appears in each individual warped function. In the same way, the mean function of the second simulated data set does not reflect the structure of the individual curves. The classical mean attenuated curve variations.

The estimated structural expectations with both the analytic registration approach and the proposed method are shown on Figures 1 and2 (bottom right figures). We can easily see that estimations with our proposed method are closer to the structural expectations f and g. These results can be explained as follows :

• For the first simulated data set, the analytic registration approach does not directly work on the strictly increasing functions but on first derivatives. However, theoretically registering a given function data set is not the same issue as the registration of first derivatives of these functions.

• The analytic registration approach uses the mean curve to register all the functions. Due to the drawbacks of the mean curve when dealing with large deformations (for instance in the second data set, where the result is a very flat mean curve), the structural mean approach provides better results.

• For both first and second simulated data sets, the analytic registration approach works on estimated functions and not directly with given data, which implies an additional source of error.

A concrete application : multiple referees and equity

The field of application of the results presented in this paper is large. Here we consider an example in the academic field : how to guarantee equality in an exam with several different referees?

Consider an examination with a large number of candidates, such that it is impossible to evaluate the candidates one after another. So the students are divised into m groups and a board of examiners is charged to grade the students of one group. The evaluation is performed by assigning a score from 0 to 20. The m different boards of examiners are supposed to behave the same way in order to respect the equality among the candidates. Moreover it is assumed that the sampling of the candidates is perfect in the sense that it is done in such a way that each board of examiners evaluates candidates with the same global level. Hence, if all the examiners had the same requirement levels, the distribution of the ranks would be the same for all the boards of examiners. Here, we aim at balancing the effects of the differences between the examiners, and gaining equity for the candidates, and at studying real data provided by the french competitive teacher exam Agrégation de mathématiques.

This situation can be modeled as follows. Define for each group i = 1, . . . , m, the score obtained by students as X i = X i j ∈ (0, 20), j = 1, . . . , n i . Let f i , i = 1, . . . , m be the repartition function of the group i defined by

f i (t) = 1 n n j=1 1 X i j t .
Under the assumption that all the examiners give the same ranking to two candidates with the same level, which will be denoted H 0 , it implies that the random variables X 1 , . . . , X m are equally distributed. Figure 3 shows the distinct empirical functions for all the m = 13 groups.

First, we test for every couple of sets (i, j) the assumption X i ∼ X j . For this we perform the following homogeneity test, see for instance [START_REF] Bouroche | L'analyse des données. Que Sais-Je ?[END_REF]. Define for all k = 1, . . . , 20,

n i k = m l=1 1 X i l =k and n j k = m l=1 1 X j l =k . Define μk = n i k +n j k 2n . Finally set d k = nμ k and D i n = 20 k=1 (d k -n i k ) 2 d k , D j n = 20 k=1 (d k -n j k ) 2 d k .
Under H 0 , D n = D i n + D j n converges in distribution to a χ 2 distribution and almost surely to +∞ under the assumption that the two laws are different.

We have n = 4000 candidates and m = 13 examiners. We test the assumption H 0 for all the different possibilities. In 60% of cases and for a level equal to 5%, we reject the assumption that the rankings follow the same distribution.

As a consequence the following procedure is proposed. We aim at finding the average way of ranking, with respect to the ranks that were given within the 13 bunchs of candidates. For this, assume that there is such average empirical distribution function and that the empirical distribution function of each group is warped by a random process from this reference distribution function. Hence a good choice is given by the structural expectation, since the functions f j , j = 1, . . . , 13 are increasing so that Theorem 2.1 may apply. Figure 3 shows the different empirical distributions and the structural empirical distribution function. At each dot on the empirical distribution function, corresponding to the rank given within one group, is associated its correspondent structural rank. It is obtained by simply projecting onto the structural empirical distribution function. As a result we obtain rescaled structural ranks corresponding to the rank obtained by each candidate if they could have been judged by an average board of examiner, leading to a more fair rankings. Indeed the difference of judgments between each subgroup are revised according to the average judgment of all the groups of examiners.

In conclusion, structural expectation provides a data-driven pattern, which plays the role of a reference pattern to which all the different curves can be compared. We applied successfully this method to rescale the ranks obtained by candidates evaluated by different boards of examiners. This use is not restrictive since it can be used to provide mean patterns for other types of functional data in various fields such as econometry, biology or data traffic for instance.

A Proofs and technical lemmas

In practice, functions (f i ) i∈{1,...,m} are evaluated on a discrete interval of R as described in Section 1. In order to prove Theorem 2.1 and Theorem 2.5, we first study asymptotic results on the theoretical continuous model, i.e.

f i (t) = f • h -1 i (t), i ∈ {1, . . . , m}, t ∈ [a, b], (18)
where H is defined in the same way as in model [START_REF] Bouroche | L'analyse des données. Que Sais-Je ?[END_REF]. In a second step, we will extend the proofs to the discretized model and prove results of Section 3. So consider that all functions are measured on the entire interval [a, b]. After asymptotic results are proved for this continuous model in Section A.1, we use these results to prove Theorem 2.1 and Theorem 2.5 (and subsequently Theoreme 2.2) in Section A.2. A.1 Asymptotic results for the continuous model For the continuous model (18), we provide asymptotic results (analogous to Theorem 2.1 and Theorem 2.5) and their proofs.

A.1.1 Estimator and asymptotic results of the inverse of the structural expectation

Considering the continuous model (18), we define an empirical estimator of the inverse of the structural expectation in the following way. Set

f -1 ES = 1 m m i=1 f -1 i . (19)
The following theorem gives us consistency and asymptotic normality of this estimator.

Theorem A.1. Under assumption (2), we have that

f -1 ES converges almost surely to f -1 ES f -1 ES -f -1 ES ∞ as ---→ m→∞ 0.
Moreover, let assume that assumptions (4) and ( 5) are allowed. Then, we have that

√ m(f -1 ES -f -1 ES ) converges weakly toward a zero-mean Gaussian process G: √ m f -1 ES -f -1 ES D ---→ m→∞ G,
where the covariance function of G is defined, for all (s, t)

∈ [f (a), f (b)] 2 , by Cov(G(s), G(t)) = r f -1 (s), f -1 (t) .
Proof. Almost sure convergence of estimator f -1 ES is directly deduced from corollary 7.10 (p. 189) in [START_REF] Ledoux | Probability in Banach spaces[END_REF]. This corollary is an extension of the Strong Law of Large Numbers to Banach spaces. For all i ∈ {1, . . . , m}, the functions f -1 i i∈{1,...,m} are obviously strictly increasing, hence f -1 ES is strictly increasing, and we have

f -1 ES = 1 m m i=1 f -1 i = 1 m m i=1 f • h -1 i -1 = 1 m m i=1 h i • f -1 .
For all i ∈ N * , let

X i = h i • f -1 -f -1 ES , and, for all m ∈ N * , let S m = m i=1 X i .
The (X i ) i∈{1,...,m} are B-valued random variables, where B = C ([f (a), f (b)]) is a separable Banach space. Moreover, the dual space of B is the set of bounded measures on [f (a), f (b)] ( [START_REF] Rudin | Real and complex analysis[END_REF]). Hence, our framework corresponds to Chapter 7 of [START_REF] Ledoux | Probability in Banach spaces[END_REF] and thus we can apply Corollary 7.10. Indeed, we have

E ( X 1 ∞ ) < +∞ et E (X 1 ) = 0, then S m m a.s ----→ m→+∞ 0,
which proves almost sure convergence.

We now turn to weak convergence. From the multivariate CLT, for any k ∈ N * and fixed (y 1 , . . . , y

k ) ∈ [f (a), f (b)] k , √ m       f -1 ES (y 1 )
. . .

f -1 ES (y k )    -    f -1 ES (y 1 ) . . . f -1 ES (y k )       D ---→ m→∞ N k (0, Γ) ,
where the covariance matrix Γ = (Γ ij ) i,j is given by Γ ij = cov(F -1 (y i ),

F -1 (y j )) = cov(H(f -1 (y i )), H(f -1 (y j ))), i, j = 1, . . . , k. It remains to show that { √ m(f -1 ES -f -1
ES )} is tight. We verify the moment condition stated by [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] [Example 2.2.12].

E √ m(f -1 ES (s) -f -1 ES (s)) - √ m(f -1 ES (t) -f -1 ES (t)) 2 = E   m 1 m m i=1 f -1 i (s) -EF -1 (s) - 1 m m i=1 f -1 i (t) -EF -1 (t) 2   = E F -1 (s) -EF -1 (s) -F -1 (t) -EF -1 (t) 2 ,
where the last equality follows from the fact that the h i 's are i.i.d. Then, from ( 4) and ( 5), we get that

E √ m(f -1 ES (s) -f -1 ES (s)) - √ m(f -1 ES (t) -f -1 ES (t)) 2 C 1 C 2 |s -t| 2 ,
which completes the proof.

A.1.2 Estimator and asymptotic results of an individual warping function

For the continuous model, we define an empirical estimator of the individual warping function φ • h -1 i 0 (i 0 ∈ {1, . . . , m}) as follows. Conditionally to

F i 0 = f i 0 , for all t ∈ [a, b], let φ • h -1 i 0 (t) = 1 m -1 m i=1 i =i 0 f -1 i • F i 0 (t). ( 20
)
The following theorem gives us consistency and asymptotic normality of this estimator.

Theorem A.2. Under assumption (2), we have that φ

• h -1 i 0 converges almost surely to φ • h -1 i 0 : φ • h -1 i 0 -φ • h -1 i 0 ∞ as ---→ m→∞ 0.
Let n = m 1 2 +α (with α > 0) and assume that (4) and ( 6) hold. Then, we have that

√ m(φ • h -1 i 0 -φ • h -1 i 0 ) converges weakly to a zero-mean Gaussian process Z, √ m(φ • h -1 i 0 -φ • h -1 i 0 ) D ---→ m→∞ Z,
with covariance function defined for all (s, t) ∈ [a, b] 2 by

Cov ( (Z, () s), Z(t)) = r h -1 i 0 (s), h -1 i 0 (t) .
Proof. Let i 0 ∈ {1, . . . , m}.

Given F i 0 = f i 0 , φ • h -1 i 0 = 1 m -1 m i=1 i =i 0 f -1 i • f i 0 . Noting also that φ • h -1 i 0 = f -1 ES • f i 0 , consistency of φ • h -1
i 0 follows by the same arguments as in proof of Theorem A.1.

We now turn to weak convergence. From the multivariate CLT, for any k ∈ N * and fixed (t 1 , . . . , t

k ) ∈ [a, b] k , √ m       φ • h -1 i 0 (t 1 ) . . . φ • h -1 i 0 (t k )    -    φ • h -1 i 0 (t 1 ) . . . φ • h -1 i 0 (t k )       D -→ N k (0, Γ 0 ) ,
where the covariance matrix Γ 0 = (Γ 0,ij ) i,j is given by Γ 0,ij = cov(H(h

-1 i 0 (t i )), H(h -1 i 0 (t j ))) = r(h -1 i 0 (t i ), h -1 i 0 (t j )), i, j = 1, . . . , k. It remains to show that { √ m(φ • h -1 i 0 -φ • h -1 i 0 )} is tight.
Again, we verify the moment condition stated by [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] [Example 2.2.12].

E √ m(φ • h -1 i 0 (s) -φ • h -1 i 0 (s)) - √ m(φ • h -1 i 0 (t) -φ • h -1 i 0 (t)) 2 = m (m -1) 2 E    m i=1 i =i 0 f -1 i (f i 0 (s)) -Ef -1 i (f i 0 (s)) -f -1 i (f i 0 (t)) -Ef -1 i (f i 0 (t)) 2    = m m -1 E H(h -1 i 0 (s)) -EH(h -1 i 0 (s)) -H(h -1 i 0 (t)) -EH(h -1 i 0 (t)) 2 .
Now, from assumptions (4) and ( 6), we get that

E √ m(φ • h -1 i 0 (s) -φ • h -1 i 0 (s)) - √ m(φ • h -1 i 0 (t) -φ • h -1 i 0 (t)) 2 2C 1 C 3 |s -t| 2 ,
which completes the proof.

A.2 Proofs of asymptotic results

We now use Theorem A.1 (given for the continuous model) to prove Theorem 2.1.

Proof of Theorem 2.1 Let y ∈ [f (a), f (b)].
The n observation times are equidistant and for each i = 1, . . . , m, f i is almost surely increasing, hence f

-1 i (y) -1 n T i (y) f -1 i (y) + 1 n . This implies that almost surely, f -1 ES (y) - 1 n f -1 ES (y) f -1 ES (y) + 1 n . (21)
Since Let n = m 1 2 +α (α > 0). From (21), we get that almost surely,

f -1 ES + 1 n -f -1 ES ∞ f -1 ES -f -1 ES ∞ + 1 n , we get that f -1 ES + 1 n -f -1 ES ∞ as ----→ m,n→∞ 0
√ m( f -1 ES -f -1 ES ) ∞ 1 m α . Since √ m( f -1 ES -f -1 ES ) ∞ = √ m( f -1 ES -f -1 ES ) - √ m(f -1 ES -f -1 ES ) ∞ , we get that √ m( f -1 ES -f -1 ES ) - √ m(f -1 ES -f -1 ES ) ∞
converges almost surely to 0 as m tends to infinity.

Combining Theorem A.1 and Theorem 4.1 in [START_REF] Billingsley | Convergence of probability measures[END_REF], it follows that

√ m( f -1 ES -f -1 ES ) D ---→ m→∞ G.
We now turn to proof of Lemma 2.3.

Proof of Lemma 2.3 Consider first the continuous model (18), and define

γ • f -1 = 1 m m i=1 f -1 i 2 .
Using similar arguments as in proof of Theorem A.1, we get that γ

• f -1 -γ • f -1 ∞ as ---→ m→∞ 0. Now, since |T i (y) -f -1 i (y)| 1 n , we obtain by straightforward calculations that almost surely, m m i=1 T i .f -1 i (y) + γ • f -1 (y) 1 n 2 , from which we deduce that that γ • f -1 (y) as ----→ m,n→∞ γ •f -1 (y).
Combining this with Theorem 2.1 completes the proof.

Proof of Theorem 2.2. For F an arbitrary function, let F -1 denote its generalized inverse, defined by F -1 (t) = inf{y :

F (y) t}. By Theorem 2.1, for all y ∈ [f (a), f (b)], f -1 ES (y) as ----→ m,n→∞ f -1 ES (y). By Lemma 21.2 in [12], ( f -1 ES ) -1 (t) as ----→ m,n→∞ (f -1 ES ) -1 (t) at every t where (f -1 ES ) -1 is continuous. Since f -1
ES is continuous and stricly increasing, (f -1 ES ) -1 is a proper inverse and is equal to f

• φ -1 , hence for all t ∈ [a, b], ( f -1 ES ) -1 (t) as ----→ m,n→∞ f • φ -1 (t). Now, for all t ∈ [a, b], ( f -1 ES ) -1 (t) can be rewritten as ( f -1 ES ) -1 (t) = v 0 1 {a} (t) + K(m,n)-1 k=0 v k+1 1 (u k ,u k+1 ] (t),
and by construction, letting t

∈ (u k(m,n) , u k(m,n)+1 ] (k(m, n) ∈ K), we have v k(m,n) f ES (t) v k(m,n)+1
. Combining this and the above equality yields that for all t ∈ We now use Theorem A.2 (given for the continuous model) to prove Theorem 2.5. A preliminary definition and a lemma are needed.

(u k(m,n) , u k(m,n)+1 ], | f ES (t) -( f -1 ES ) -1 (t)| v k(m,n)+1 -v k(m,n) . Since f ES is continuous, v k(m,n)+1 -v k(m,n) ----→ m,n→∞ 0, and | f ES (t) -( f -1 ES ) -1 (t)| as ----→ m,n→∞ 0
Let define the modulus of continuity of H as

K H (δ) = sup (s,t)∈[a,b] 2 |s-t| δ H 1 • H -1 2 (s) -H 1 • H -1 2 (t) (δ 0),
where H 1 and H 2 are two independent copies of H. Since for all δ 0, 0

K H (δ) b -a almost surely, we can define L 1 H (H 2 , δ) and L 2 H (H 2 , δ) as L 1 H (H 2 , δ) = E[K H (δ)|H 2 ] and L 2 H (H 2 , δ) = E[K 2 H (δ)|H 2 ] (δ 0), and we have L 1 H (H 2 , 0) = L 2 H (H 2 , 0) = 0. From continuity of H, it holds that L 1 H (H 2 , •) and L 2 H (H 2 ,
•) are right-continuous at 0. Given H 2 = h, this implies that for ǫ > 0, there exists η ǫ > 0 such that L 1 H (h, η ǫ ) < ǫ. We shall use this result in proving the following lemma: Lemma A.

φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) as ----→ m,n→∞ 0. Proof. Letting t ∈ [a, b], one easily shows that φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) 1 m -1 m i=1 i =i 0 h i • h -1 i 0 t + 1 n -h i • h -1 i 0 (t) .
This in turn implies that sup

t∈[a,b] φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) 1 m -1 m i=1 i =i 0 sup (s,t)∈[a,b] 2 |s-t| 1 n h i • h -1 i 0 (s) -h i • h -1 i 0 (t) = 1 m -1 m i=1 i =i 0 Ki H 1 n ,
where the Ki

φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) 1 m -1 m i=1 i =i 0 Ki H (η ǫ ) .
φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) L 1 H (h i 0 , η ǫ ) < ǫ.
This holds for any ǫ > 0, hence

sup t∈[a,b] φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t)
• f i 0 (t j 0 ) -1 n T i f -1 i
• f i 0 (t j 0 ) + 1 n . Recall that t j 0 = arg min j∈{1,...,n} |t j -t|, hence t -1 n t j 0 t + 1 n . Thus, combining the above two inequalities, we get that almost surely f -1 i

• f i 0 (t -1 n ) -1 n T i f -1 i • f i 0 (t + 1 n ) + 1
n , from which we deduce:

φ • h -1 i 0 t - 1 n - 1 n φ • h -1 i 0 (t) φ • h -1 i 0 t + 1 n + 1 n . ( 22
)
We shall focus on the upper bound in this inequality, since the same kind of argument holds for the lower bound. Straightforward calculation yields

sup t∈[a,b] φ • h -1 i 0 t + 1 n + 1 n -φ • h -1 i 0 (t) sup t∈[a,b] φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) + 1 n + φ • h -1 i 0 -φ • h -1 i 0 ∞
, where the RHS of this inequality tends to 0 as m and n tend to infinity, by Lemma A.3 and Theorem A.2 in [START_REF] Billingsley | Convergence of probability measures[END_REF]. It follows that φ

• h -1 i 0 -φ • h -1 i 0 ∞ as ----→ m,n→∞ 0.
We now turn to weak convergence. Assume that n = m 1 2 +α (α > 0). From (22), it holds

√ m φ • h -1 i 0 -φ • h -1 i 0 ∞ sup t∈[a,b] √ m φ • h -1 i 0 t + 1 n -φ • h -1 i 0 (t) + sup t∈[a,b] √ m φ • h -1 i 0 t - 1 n -φ • h -1 i 0 (t) + 2 m α 2Z m + 2 m α ,
where √ m( φ

Z m = √ m 1 m-1 m i=1 i =i 0 Ki H 1 n . Since var(Z m ) = m m -1 L 2 H h i 0 , 1 n -L 1 H h i 0 , 1
• h -1 i 0 -φ • h -1 i 0 ) D ---→ m→∞ Z.
Proof of Lemma 2.6 Consider first the continuous model (18), and define conditionally to

F i 0 = f i 0 γ • h -1 i 0 = 1 m -1 m i=1 i =i 0 f -1 i • f i 0 2 .
Using similar arguments as in proof of Theorem A.2, we get that

γ • h -1 i 0 -γ • h -1 i 0 ∞ as ---→ m→∞ 0. Now, from f -1 i • f i 0 (t -1 n ) -1 n T i f -1 i • f i 0 (t + 1 n ) + 1
n , we get the following inequality

|T i -f -1 i •f i 0 (t)| f -1 i • f i 0 t - 1 n -f -1 i • f i 0 (t) + f -1 i • f i 0 t + 1 n -f -1 i • f i 0 (t) + 2 n .
Acting as in proof of Lemma 2.3, it is fairly straightforward to show that for t ∈ [a, b],

1 m-1 m i=1 i =i 0 T i .f -1 i • f i 0 (t) as ----→ m,n→∞ γ • h -1
i 0 (t). From the above inequality, we also obtain that

|T i -f -1 i • f i 0 (t)| 2 Ki H 1 n + 2 n .
Summing over i ∈ {1, . . . , m}\i 0 , and then using convergence of 1 m-1

i =i 0 T i .f -1 i • f i 0 (t), of γ • h -1
i 0 (t), together with right-continuity of L 1 H (h i 0 , •) and L 2 H (h i 0 , •) at 0, yield that γ • h -1 i 0 (t) as ----→ m,n→∞ γ • h -1 i 0 (t). This and convergence of φ • h -1 i 0 (t) complete the proof of Lemma 2.6.

  i) H(w, •) is an increasing function, ii) H(w, a) = a and H(w, b) = b.

Figure 1 :

 1 Figure 1: Function f is shown on top left figure (solid line). Simulated warping processes are shown on top right figure. Simulated warped functions are shown on bottom left figure. The classical mean of these functions is draw on top left figure (dotted line). Finally, estimations of f with the Analytic registration procedure (dashed line) and with our method (dotted-dashed line) are shown on bottom right figure.

Figure 2 :

 2 Figure 2: Function g is shown on top left figure (solid line). Simulated warping processes are shown on top right figure. Simulated warped functions are shown on bottom left figure. The classical mean of these functions is draw on top left figure (dotted line). Finally, estimations of g with the Analytic registration procedure (dashed line) and with our method (dotted-dashed line) are shown on bottom right figure.

Figure 3 :

 3 Figure 3: Empirical distribution functions (left figure) and structural expectation estimation (right figure).

  , by Theorem A.1. A similar argument holds for LHS(21) and finally, f -1 ESf -1 ES ∞ as ----→ m,n→∞ 0.

3 .

 3 Let t ∈ [a, b]. Under assumption (2), the following holds: sup t∈[a,b]

H 1 n 1 n

 11 (i ∈ {1, . . . , m}\i 0 ) are independent random variables distributed asK H |H 2 = h i 0 . Since K H (•) is increasing, if n ∈ N is sufficiently large so that 1n < η ǫ , we get that sup t∈[a,b]

  By the law of large numbers, we get that almost surely,

2 . 5

 25 Let t ∈ [a, b]. For each i ∈ {1, . . . , m}, f i is almost surely increasing, hence almost surely f -1 i

n 2 ---→ m→∞ 0 (recall that n = m 1 2

 201 +α and that L 1H (h i 0 , •) and L 2 H (h i 0 , •) are right-continuous at 0), we get that √ m φ • h -1 i 0φ • h -1

  . It follows that almost surely, f ES (t) converges to f ES (t). Uniform convergence finally follows from Dini's theorem.