J.-F Coeurjolly

R Drouilhet

P Lafaye De Micheaux

J.-F Robineau

asympTest: an R package for performing parametric statistical tests and confidence intervals based on the central limit theorem

Keywords: parametric tests and confidence intervals, central limit theorem, R package

This paper describes an R package implementing large sample tests and confidence intervals (based on the central limit theorem) for various parameters. The one and two sample mean and variance contexts are considered. The statistics for all the tests are expressed in the same form, which facilitates their presentation. In the variance parameter cases, the asymptotic robustness of the classical tests depends on the departure of the data distribution from normality measured in terms of the kurtosis of the distribution.

Introduction

When you are interested in testing a variance parameter for a large sample in the non Gaussian framework, it is not easy to find a test implemented in the standard statistical software. In fact, we could not find one! The only available tool is the chi-square variance test tailor-made to the Gaussian context. This test is however commonly used by practicians even if the Gaussian assumption fails. We applied it to a data set of size n = 1000 with an empirical distribution very different from a normal distribution. The p-value of the chi-square variance test with alternative hypothesis H 1 : σ 2 < 1 is 4.79% which leads us to accept the alternative hypothesis at level α = 5%. Is it reasonable to use this test when we know that it cannot be used in the non Gaussian framework because of its sensitivity to departures from normality, e.g. [START_REF] Box | Non-normality and tests on variances[END_REF]?

From a mathematical point of view, one may wonder why no alternative test has been yet implemented since the asymptotic properties of the sample variance are well-known. Things are much easier in the sample mean study because the one sample t-test is known to be robust to departures from normality for large samples, e.g. [START_REF] Ozgur | A study of the statistical inference criteria: can we agree on when to use z versus t ?[END_REF]. This results from a direct application of the central limit theorem. The same remarks are valid when comparing the two-sample t-test (difference of means test) which is robust for large samples and the Fisher test (ratio of variances test) which is not.

In the statistical framework, one may be a simple user, a tool developer, a theoretician or any combination of the above. There are natural interactions between these different communities and it is expected that their knowledge should be shared, above all for tasks that have now become very basic. It is well-known for a theoretician, see e.g. [START_REF] Casella | Statistical Inference[END_REF], that a 1 common method for constructing a large sample test statistic may be based on an estimator that has an asymptotic normal distribution. Suppose we wish to test a hypothesis about a parameter θ, and θ n is some estimator of θ based on a sample of size n. If we can prove some form of the central limit theorem to show that as n → +∞, ( θ nθ)/ σ b θ d → N (0, 1) where σ 2 b θ is a convergent (in probability) estimate of V ar( θ n ), then one has the basis for an approximate test. This scheme based on the central limit theorem will be called the CLT procedure. We have already specified that we did not find any alternative to the chi-square variance test for testing a variance when the normality assumption fails. On the contrary, the problem of the robustness (to departures from normality) of tests for comparing two (or more) variances has been widely treated in the literature, see e.g. [START_REF] Box | Non-normality and tests on variances[END_REF], [START_REF] Conover | A comparative study of tests for homogeneity of variances with applications to the outer continental shelf bidding data[END_REF], [START_REF] Tiku | Robust inference. Statistics: Textbooks and Monographs[END_REF], [START_REF] Pan | On a levene type test for equality of two variances[END_REF] and the references therein. Some alternative procedures to the Fisher test are implemented in R: the Bartlett test (bartlett.test), the Fligner test (fligner.test), the Levene test (levene.test available in the lawstat package), etc. However to our best knowledge, for large samples, simple alternatives based on the CLT procedure have never been proposed or implemented.

The main objective of this paper is to propose a unified framework, based on the CLT procedure, for large samples to test various parameters such as the mean, the variance, the difference or ratio of means or variances. This approach also allows direct derivation of asymptotic confidence intervals. Tests and confidence intervals are then implemented in our new R package, called asympTest. This modest contribution also solves the problem of finding a robust (to nonnormality) alternative to the chi-square variance test for large samples. It also provides a very simple alternative to the Fisher test. However, note that the purpose of this paper is not to compare our tests to their competitors in terms of power. Finally, a first course of statistical inference usually presents mean tests in both Gaussian and asymptotical frameworks and variance tests restricted to the Gaussian case. The unified approach presented here is is very similar to the classical t-test from a mathematical point of view and gives us the opportunity to propose a more complete teaching framework with no additional difficulty.

The paper is organized as follows. Section 2 deals with the mathematical concepts and describes our main notation. We also propose a mathematical explanation of the reason why the chi-square variance test and the Fisher test are not appropriate even when the sample size is very large. Finally, a general framework is also proposed that allows us to derive some asymptotic statistical tests for the mean, the variance and the difference (and ratio) of means or variances. In Section 3, the R package asympTest is presented. It notably includes the procedures described in the previous section. Finally, Sections 4, 5 and 6 are devoted to some discussions and the proofs of our results.

Mathematical development 2.1 Notation

For one-sample tests, let us denote by Y = (Y 1 , . . . , Y n ) a sample of n independent and identically distributed random variables with mean µ and variance σ 2 . These parameters are classically estimated by

µ (Y) = Y = 1 n n i=1 Y i and σ 2 (Y) = 1 n -1 n i=1 (Y i -µ (Y)) 2 .
In the two-sample context, let

Y (1) = Y (1) 1 , . . . , Y (1)
n (1) and Y (2) = Y (2) 1 , . . . , Y (2)
n (2) denote two independent samples of n (1) and n (2) random variables with respective means µ (1) and µ (2) and variances σ 2

(1) and σ 2 (2) . We also define the following parameters and their estimated versions by denoting Y = Y (1) , Y (2) :

• Difference of (weighted) means:

d µ = µ (1) -ρ × µ (2) (ρ ∈ R) and d µ (Y) = µ (1) Y (1) - ρ × µ (2) Y (2) .
• Difference of (weighted) variances:

d σ 2 = σ 2 (1) -ρ×σ 2 (2) (ρ ∈ R) and d σ 2 (Y) = σ 2 (1) Y (1) - ρ × σ 2 (2) Y (2) .
• Ratio of means:

r µ = µ (1) µ (2) and r µ (Y) = d µ (1) (Y (1) ) d µ (2) (Y (2)
) .

• Ratio of variances:

r σ 2 = σ 2 (1)
σ 2

(2)

and

r σ 2 (Y) = d σ 2 (1) (Y (1) ) d σ 2 (2) (Y (2) )
.

The known parameter ρ is, in our definition of d µ and d σ 2 , intrinsically nonnegative. But note that there is no mathematical problem to deal with negative values of ρ.

In order to compare the Gaussian framework and the general one, we propose to denote by Y G (resp. Y (1),G , Y (2),G ) a vector (resp. two independent vectors) of n (resp. n (1) and n (2) ) Gaussian random variables with mean µ (resp. µ (1) and µ (2) ) and variance σ 2 (resp. σ 2

(1) and σ 2

(2) ). In the two-sample context, let us also denote

Y G = Y (1),G , Y (2),G .
In the sequel, we will use the notation := to define some quantity. For some random variable Z and some distribution L, Z L (resp. approx L) means that Z follows (resp. approximately follows) the distribution L.

About the Chi-square and Fisher tests

In this section, we concentrate on parameters σ 2 and r σ 2 . The classical statistics of the chi-square test of variance and the Fisher test of ratio of variances are defined by

Λ G c σ 2 ,σ 2 (Y) := (n -1) σ 2 (Y) σ 2 and Λ G d r σ 2 ,r σ 2 (Y) := r σ 2 Y (1) , Y (2) r σ 2 .
The notation Λ G c σ 2 ,σ 2 expresses that the statistic is a measure of the departure of σ 2 from σ 2 in the Gaussian framework. When the data are Gaussian, it is well-known that

Λ G c σ 2 ,σ 2 (Y G ) χ 2 (n -1) and Λ G d r σ 2 ,r σ 2 (Y G ) F n (1) -1, n (2) -1 .
However, as shown in Fig. 1, both results become untrue (even approximately) under nonnormality assumption. The theoretical reason may be explained as follows. Let n (1) = n and assume that there exists α > 0 such that n (2) = αn (1) . Assume also that Y (resp. Y (1) and

Y (2) ) has a finite kurtosis k := E((Y -µ) 4 )/V ar(Y ) (resp. k (1) and k (2) ). Then, as n → +∞, Λ G c σ 2 ,σ 2 (Y) -(n -1) √ n -1 d -→ N (0, k -1) (1) √ n Λ G d r σ 2 ,r σ 2 (Y) -1 d -→ N 0, k (1) -1 + k (2) -1 α . ( 2
)
This result is a consequence of a more general one stated in Section 2.3 and proved in Section 6. Equations ( 1) and ( 2) lead to the following approximations 2) .

Λ G c σ 2 ,σ 2 (Y) approx N (n -1, (k -1)(n -1)) and Λ G d r σ 2 ,r σ 2 (Y) approx N 1, k (1) -1 n (1) + k (2) -1 n (
When data are Gaussian ans when n is large, one obtains the well-known approximations

χ 2 (n -1) ≃ N (n -1, 2(n -1)) and F n (1) , n (2) ≃ N 1, 2 n (1) + 2 n (2) since k = k (1) = k (2) = 3 We can underline that Λ G c σ 2 ,σ 2 (Y) (resp. Λ G d r σ 2 ,r σ 2 (Y)) and Λ G c σ 2 ,σ 2 (Y G ) (resp. Λ G d r σ 2 ,r σ 2 (Y G )
) differ in terms of asymptotical variances. More precisely, the gap between the two frameworks is essentially governed by the kurtosis. Indeed, as n → +∞

V ar(Λ G c σ 2 ,σ 2 (Y)) V ar(Λ G c σ 2 ,σ 2 (Y G )) → k -1 2 and V ar(Λ G d r σ 2 ,r σ 2 (Y)) V ar(Λ G d r σ 2 ,r σ 2 (Y G )) → k (1) -1 + k (2) -1 α 2 1 + 1 α .
Tab. 1 proposes the computations of these asymptotic ratios for different distributions. This allows the reader to assess the risk of using the classical statistics Λ G c σ 2 ,σ 2 and Λ G d r σ 2 ,r σ 2 under the non-normality assumption even when the size of the sample is large.

Y d = Y (1) d = Y (2) L Test L = χ 2 (ν) L = E(λ) L = U([a, b]) Variance test 1 + 6 ν 4 2 5 Ratio of variances test 1 + 6 ν 4 2 5
Table 1: Ratio of asymptotic variances (non gaussian/gaussian) k-1 2 and

k (1) -1+ k (2) -1 α 2(1+ 1 α )
in the case where k = k (1) = k (2) and α = 1.

Large sample tests based on the central limit theorem

The parameters σ 2 , d µ and d σ 2 can be viewed as particular means. Therefore, the idea (widely used in asymptotic theory) is to design asymptotic tests and confidence intervals thanks to the central limit theorem. The variables r µ (Y)r µ and r σ 2 (Y)r σ 2 can also be expressed in terms of means to which a central limit theorem can be applied. In order to unify asymptotic results, we define θ as one of the parameters µ, σ 2 , d µ , d σ 2 , r µ and r σ 2 . By applying a central limit theorem, the law of large numbers and Slutsky's theorem (see Section 6), one obtains, as

n → +∞, ∆ b θ,θ (Y) := θ (Y) -θ σ b θ (Y) d -→ N (0, 1), (3)
where σ b θ (Y) is the standard error of θ (Y). The assumptions under which the central limit theorem can be applied and the definition of σ b θ (Y) are stated in Tab. 2. R functions have been implemented to evaluate the standard errors of the estimates of θ, see Tab. 3.

n = n (1) = n (2) = 500, Y d = Y (1) d = Y (2) χ 2 (5) (top), E(1) (middle) and U([0, 5]) (bottom). θ Assumptions σ b θ (Y) µ E Y 2 i < +∞ σ 2 (Y) n σ 2 E Y 4 i < +∞ σ 2 Ÿ Ÿ n d µ E Y (j) i 2 < +∞, j = 1, 2 σ 2 (1) Y (1) n (1) + ρ 2 × σ 2 (2) Y (2) n (2) d σ 2 E Y (j) i 4 < +∞, j = 1, 2 σ 2 Ÿ (1)
Ÿ( 1)

n (1) + ρ 2 × σ 2 Ÿ (2)
Ÿ( 2)

n (2) r µ µ (2) = 0, E Y (j) i 2 < +∞, j = 1, 2 1 ˛d µ (2) (Y (2) ) ˛ d σ 2
(1) (Y (1) )

n (1) + r µ (Y) 2 × d σ 2 (2) (Y (2) ) n (2) r σ 2 σ 2 (2) = 0, E Y (j) i 4 < +∞, j = 1, 2 1 d σ 2 (2) (Y (2) ) σ 2 Ÿ (1) ( Ÿ(1) ) n (1) + r σ 2 (Y) 2 σ 2 Ÿ (2) ( Ÿ(2) ) n (2)
:= (Y -µ) 2 , Ÿ := (Y -µ (Y)) 2 , σ 2 Ÿ := V ar Ÿ and σ 2 Ÿ Ÿ := 1 n-1 n i=1 (Y i -µ (Y)) 2 -σ 2 (Y) 2 . θ Dataset(s) σ b θ (y) in R µ y seMean(y) σ 2 y seVar(y) d µ y1, y2 seDMean(y1,y2,rho=1) d σ 2 y1, y2 seDVar(y1,y2,rho=1) r µ y1, y2 seRMean(y1,y2) r σ 2 y1, y2 seRVar(y1,y2)
Table 3: Standard errors of estimates of θ in R.

Remark 1 The figures Fig. 2, Fig. 3 and Fig. 4 allow the reader to illustrate the mathematical result (3).

Remark 2 The asymptotic result (3) allows us to easily construct statistical hypothesis tests and confidence intervals, see e.g. [START_REF] Casella | Statistical Inference[END_REF] p. 385 for development.

Remark 3 The alternative hypothesis H 1 comparing the ratio of means or variances to some reference value r 0 may be expressed in terms of the comparison of the weighted differences d µ := µ (1)r 0 µ (2) or d σ 2 := σ 2 (1)r 0 σ 2 (2) to 0, the value of ρ being fixed to r 0 .

Using asympTest

The 2004)).

The main arguments of the function asymp.test are:

• x: vector of data values.

• y: optional vector of data values.

• parameter: parameter under testing, must be one of "mean", "var", "dMean", "dVar", "rMean", "rVar".

• alternative: alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".

• reference: reference value of the parameter under the null hypothesis.

• conf.level: confidence level of the interval (default is 0.95). The type-one error is then fixed to 1-conf.level.

• rho: optional parameter (only used for parameters "dMean" and "dVar") for penalization (or enhancement) of the contribution of the second parameter.

The user may only specify the first letters of the parameter or alternative.

In order to illustrate this function, let us consider the iris data available in R. This famous (Fisher's or Anderson's Fisher (1935); [START_REF] Anderson | The irises of the gaspe peninsula[END_REF]) data set gives the measurements (in centimeters) of the four variables sepal length and width, and petal length and width, for 50 flowers from each species of iris: setosa, versicolor, and virginica.

> data(iris) > attach(iris) > names(iris)

[1] "Sepal.Length" "Sepal.Width" "Petal.Length" "Petal.Width" "Species" > levels(iris$Species)

[1] "setosa" "versicolor" "virginica" U([0, 5]), Y (2) E(0.5).

Sepal The following table presents the p-values of all the Shapiro-Wilk normality tests for the different variables and the three species.

Let us concentrate on the variable Petal.Width for which the Gaussian assumption seems to be wrong for each one of the three species. The empirical means and variances are given below: -----------------------------------------------------------Species: versicolor mean var 1.32600000 0.03910612 ----------------------------------------------------------- 4 Type I error risks

Comparison between classical and asymptotic variance tests

In the context of large samples, two simulation studies are proposed in order to show the lack of reliability of the classical tests for variance parameters compared with the asymptotic tests studied in this paper. For each of the following examples, 10000 simulations of samples of size n = 1000 have been performed.

1. Let us consider testing H 0 : σ 2 = 1 versus H 1 : σ 2 < 1 with data sampled from distribution E(1) (i.e., under H 0 ).

In the case where α = 5%, the probability of accepting the alternative hypothesis is 8.91% for the asymptotic test and 20.99% for the chi-square test.

false true false 0.7901 0.0000 true 0.1208 0.0891

: σ 2 (1) = σ 2 (2) versus H 1 : σ 2 (1) = σ 2
(2) with data sampled from distribution U([0, 5]) for both samples (i.e., under H 0 ). false true false 0.9498 0.0481 true 0.0000 0.0021 Table 5: Acceptance of H 1 for the Fisher test (rows) versus the asymptotic test (columns)

In the case where α = 5%, the probability of accepting the alternative hypothesis is 5.02% for the asymptotic test and 0.21% for the Fisher test.

In both cases, the probabilities for Type I errors are worse for the classical tests than for the corresponding well-suited asymptotic tests. This is a direct consequence of the previous results summarized by figure Fig. 1.

Back to the example of the introduction

Now, one may wonder what the consequences of these previous results are for pratical purposes. Let us consider again the example presented in the introduction. In order to illustrate the twosamples case, we propose a second example. In the following R outputs, the data of the first (resp. second) example are denoted by y (resp. y1 and y2). These samples have a size n=1000 and their empirical distributions, proposed below, do not seem to fit normal distributions.

The following output provides the p-value of the chi-square test for the first example. The two decisions do not match, but since the empirical variance is 0.8969455, one may think that σ 2 is slightly inferior to 1. In which case, we have to be cautious because our sample might be of the same kind as the 12.08% of Table 4.

The following output provides the p-value of the Fisher test for the second example. The two decisions do not match, but since the empirical variances are 2.034341 and 2.292458, one may think that σ 2

(1) and σ 2 (2) are slightly different. In which case, we have to be cautious because our sample might be of the same kind as the 4.81% of Table 5.

Discussion

We have presented an R package implementing large sample tests for various parameters. The interesting point is that each test statistic can be written in the same form, as follows:

∆ b θ,θ (Y) := θ (Y) -θ σ b θ (Y)
.

This form clearly expresses the departure of the estimate from the true parameter, normalized by some quantity measuring the precision of the estimate. This approach is then attractive and easy to present. In the Gaussian framework, the one and two-sample t-tests follow this idea whereas the chi-square variance test and the Fisher test do not. One may wonder if it is possible to embed the classical Gaussian framework within this formalism. More precisely, for an estimate θ (Y) of some parameter θ with standard deviation σ b θ = V ar( θ (Y)), let us propose the test statistic

∆ b θ,θ (Y) := θ (Y) -θ σ b θ or θ (Y) -θ σ b θ (4)
in the case where σ b θ or possibly some known asymptotic equivalent σ b θ of σ b θ only depends on θ, or

∆ b θ,θ (Y) := θ (Y) -θ σ b θ (Y) (5)
otherwise, where σ b θ (Y) is a consistent estimate of σ b θ . In the large sample framework, the parameters µ, σ 2 , d µ , r µ , d σ 2 and r σ 2 fall into the case of equation ( 5) and the corresponding statistic approximately follows a N (0, 1) distribution. In this same context, the proportion parameter falls into the case of equation ( 4) with σ b p = p(1p)/n. In the Gaussian framework, the parameters µ and d µ leading to the one-sample and twosample t-tests also fall into the case of equation ( 5). Let us now concentrate on parameters σ 2 and r σ 2 (corresponding to the chi-square variance test and the Fisher test). It is known that the variance of σ 2 Y G and an asymptotic equivalent of the variance of r σ 2 Y G are of the form

σ 2 c σ 2 = 1 n μ4 - n -3 n -1 μ2 2 and σ 2 d r σ 2 = 1 n (1) μ(1) 4 -( μ(1) 2 ) 2 ( μ(2) 2 ) 2 + 1 n (2) r 2 σ 2 μ(2) 4 -( μ(2) 2 ) 2 ( μ(2) 2 ) 2
where μk = E (Y -E(Y )) k is the k-th centered moment of Y (with our notation μ2 = σ 2 ).

For Gaussian variables, μ4 = 3( μ2 ) 2 which leads to 2) . Now, in order to build a test, let us give the distributions of ∆

σ 2 c σ 2 = 2 n -1 σ 4 and σ 2 d r σ 2 = 2r 2 σ 2 1 n (1) + 1 n (
G c σ 2 ,σ 2 (Y G ) and ∆ G d r σ 2 ,r σ 2 (Y G ) expressed in terms of Λ G c σ 2 ,σ 2 (Y G ) and Λ G d r σ 2 ,r σ 2 (Y G ): ∆ G c σ 2 ,σ 2 (Y G ) := σ 2 Y G -σ 2 σ c σ 2 = Λ G d σ 2 ,σ 2 (Y G ) n-1 σ 2 -σ 2 σ 2 2/(n -1) χ 2 (n -1) -(n -1) 2(n -1) and ∆ G d r σ 2 ,r σ 2 (Y G ) := r σ 2 Y G -r σ 2 σ d r σ 2 = r σ 2 Λ G d r σ 2 ,r σ 2 (Y G ) -r σ 2 r σ 2 2/n (1) + 2/n (2) F (n (1) -1, n (2) -1) -1 2/n (1) + 2/n (2) .
One may propose a name for the previous two free distributions: centered reduced chi-square distribution and centered reduced Fisher distribution respectively denoted by χ 2 cr (•) and F cr (•, •). If these distributions were implemented such that one may evaluate quantiles and p-values, one could build two new tests directly based on ∆ G c σ 2 ,σ 2 (Y G ) and ∆ G d r σ 2 ,r σ 2 (Y G ). Of course, these two new tests would be strictly equivalent to the classical chi-square variance test and Fisher test and would then fall into the same formalism.

Let us now comment on the concept of robustness in the Gaussian framework. In the particular case of mean hypothesis testing, this robustness is expressed by the fact that ∆ G

Proofs

In this section, we only prove (3). The results (1) and ( 2) are direct consequences. Recall that for each parameter, some assumptions are needed essentially in order to apply the central limit theorem. They are summarized in Tab. 2.

Parameter µ:

This is a direct application of the central limit theorem.

Parameter σ 2 : By definition

σ 2 (Y) -σ 2 = 1 n -1 n i=1 Y i -Y 2 -σ 2 = 1 n -1 n i=1 (Y i -µ) 2 -σ 2 - n n -1 Y -µ 2 .
From the CLT, the law of large numbers and Slutsky's Theorem (see e.g. [START_REF] Ferguson | A course in large sample theory[END_REF]), it comes that as n → +∞ √ n Yµ Parameter d µ :

Recall that n = n (1) and n (2) = αn (1) . As n → +∞,

√ n d µ (Y) -d µ = √ n µ (1) Y (1) -µ (1) -ρ × √ nα √ α µ (2) Y (2) -µ (2) d -→ N 0, σ 2 (1) + ρ 2 × σ 2
(2) α .

Figure 1 :

 1 Figure 1: Histograms of m = 10000 replications of test statistics of variance test (left) and ratio of variance tests (right) in the Gaussian context. The simulation has been done as follows:

Figure 2 :

 2 Figure 2: Histograms of m = 10000 replications of b θ(Y)-θ c σ b θ (Y) for θ = µ, σ 2 , d µ , d σ 2 , r µ and r σ 2 . The simulation has been done as follows: n = n (1) = n (2) = 500, Y (1)χ 2 (5), Y (2) χ 2 (5).

Figure 3 :

 3 Figure 3: Histograms of m = 10000 replications of b θ(Y)-θ c σ b θ (Y) for θ = µ, σ 2 , d µ , d σ 2 , r µ and r σ 2 . The simulation has been done as follows: n = n (1) = n (2) = 500, Y (1) E(1), Y (2) E(1).

Figure 4 :

 4 Figure 4: Histograms of m = 10000 replications of b θ(Y)-θ c σ b θ (Y) for θ = µ, σ 2 , d µ , d σ 2 , r µ and r σ 2 . The simulation has been done as follows: n = n (1) = n (2) = 500, Y (1)U([0, 5]), Y (2) E(0.5).

>

  by(Petal.Width,Species,function(e) c(mean=mean(e),var=var(e

  apparent non normality of the data, one may prefer to apply the corresponding asymptotic test: > asymp.test(y,par="var",alt="l"

>

  var.test(y1,y2) F test to compare two variances data: y1 and y2 F = 0.8874, num df = 499, denom df = 499, p-value = 0.1825 alternative hypothesis: true ratio of variances is not equal to Due to the apparent non normality of the data, one may prefer to apply the corresponding asymptotic test: > asymp.test(y1,y2,"dVar") Two-sample asymptotic difference of variances test data: y1 and y2 statistic = -2.0925, p-value = 0.03639 alternative hypothesis: true difference of variances is not equal to

  b µ,µ (Y G ) is equal to ∆ b µ,µ (Y G) with the same asymptotic distribution N (0, 1). When considering the twonew cases θ = σ 2 or θ = r σ 2 , ∆ G b θ,θ (Y G ) is no longer equal to ∆ b θ,θ (Y G) but have at least the same asymptotic distribution N (0, 1). However, one can prove that ∆ G b θ,θ (Y G ) and ∆ b θ,θ (Y G ) are asymptotically equivalent in probability, which may be viewed as some kind of robustness.

  n → +∞, √ n σ 2 (Y)σ 2 d -→ N (0, V ar((Yµ) 2 )). Since σ c σ 2 := V ar((Y -µ) 2 )n can be consistently estimated by σ c σ 2

Table 2 :

 2 Standard errors of estimates of θ. For the sake of simplicity we denote by Ÿ

  R package asympTest consists of a main function asymp.test and six auxiliary ones designed to compute standard errors of estimates of different parameters, see Tab. 3. The auxiliary functions should not be very useful for the user, except if he/she wants to compute himself/herself the confidence interval. The function asymp.test has been written in the same spirit as the R functions t.test or var.test. The arguments of asymp.test, its value and the resulting outputs are inspired from the ones of t.test or var.test. In particular, the function asympt.test returns an object of class "htest" (which is the general class of test objects in R, see R Development Core Team (

  -

	95 percent confidence interval:
	0.621274	Inf
	sample estimates:
	difference of means
		0.7
	Is the mean petal width of virginica species 4 times larger than the setosa one ?
	> asymp.test(Petal.Width[Species=="virginica"],
	+ Petal.Width[Species=="setosa"],"rMean","g",4)
	Two-sample asymptotic ratio of means test
	data: Petal.Width[Species == "virginica"] and Petal.Width[Species == "setosa"]
	statistic = 8.0936, p-value = 3.331e-16
	alternative hypothesis: true ratio of means is greater than 4
	95 percent confidence interval:
	7.374946	Inf
	sample estimates:
	ratio of means
	8.235772
	Species: virginica
	mean	var
	2.02600000 0.07543265
	Is the mean petal width of setosa species less than 0.5 ?
	> require(asympTest)
	> asymp.test(Petal.Width[Species=="setosa"],par="mean",alt="l",ref=0.5)
	statistic = 14.6447, p-value < 2.2e-16 One-sample asymptotic mean test alternative hypothesis: true difference of (weighted) means is greater than 0
	95 percent confidence interval: data: Petal.Width[Species == "setosa"] 0.9249653 Inf statistic = -17.0427, p-value < 2.2e-16 sample estimates: alternative hypothesis: true mean is less than 0.5 difference of (weighted) means 95 percent confidence interval: -Inf 0.2705145 1.042
	sample estimates:
	mean
	0.246
	Is the mean petal width of virginica species larger than the versicolor one ?
	> asymp.test(Petal.Width[Species=="virginica"],
	+ Petal.Width[Species=="versicolor"],"dMean","g",0)
	Two-sample asymptotic difference of means test
	data: Petal.Width[Species == "virginica"] and Petal.Width[Species == "versicolor"]
	statistic = 14.6254, p-value < 2.2e-16
	alternative hypothesis: true difference of means is greater than 0

This may also be done via a difference of weighted means test.

> asymp.test(Petal.Width[Species=="virginica"], + Petal.Width[Species=="setosa"],"dMean","g",0,rho=4) Two-sample asymptotic difference of (weighted) means test data: Petal.Width[Species == "virginica"] and Petal.Width[Species == "setosa"]

Table 4 :

 4 Acceptance of H 1 for the chi-square test (rows) versus the asymptotic test (columns) 2. Let us consider test H 0

Acknowledgments

We would like to thank Laurence Pierret for having kindly reread our English. She could in no way be held responsible for any remaining mistakes.

Since σ c dµ := σ 2 (1) +ρ 2 × σ 2

(2) α n can be consistently estimated by

Parameter d σ 2 :

As n → +∞,

n can be consistently estimated by

Ÿ( 1)

Ÿ( 1)

Ÿ( 2)

Parameter r µ :

Using Slutsky's Theorem, one may assert that as n → +∞

(2) α n can be consistently estimated by

we obtain as n → +∞,

Parameter r σ 2 :

The proof follows the ideas developed for the parameters σ 2 , d σ 2 and r µ , and thus is left to the reader.


Térence Bayen
email: terence.bayen@univ-avignon.fr

Anas Bouali
email: anas.bouali@univ-avignon.fr

Loïc Bourdin
email: loic.bourdin@unilim.fr

Optimal control problems with non-control regions: necessary optimality conditions

Keywords: Optimal control, non-control regions, Pontryagin maximum principle

We consider a smooth control system that is subject to loss of control in the sense that the state space is partitioned into several disjoint regions and, in each region, either the system can be controlled, as usual, in a permanent way (that is, one can change the value of the control at any real time), or, on the contrary, the control has to remain constant from the entry time into the region until the exit time. The latter case corresponds to a non-control region. The objective of this paper is to state the necessary optimality conditions for a Mayer optimal control problem in such a setting of loss of control. Our main result is based on a hybrid maximum principle that takes into account a regionally switching parameter.

INTRODUCTION

Optimal control theory has been successfully applied to solve optimal control problems arising in various areas such as engineering, biology, aeronautics, aerospace, etc. The two fundamental principles are namely the Pontryagin maximum principle (see [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]) and the Hamilton-Jacobi-Bellman equation (see [START_REF] Bellman | Dynamic programming[END_REF]) that have been developed at the end of the fifties. Both methods allow to synthesize optimal control strategies that are implemented in real world experiments. Nowadays there is still a need of developing further optimality conditions to cover various situations encountered in practice.

The usual tools of optimal control theory presuppose that it is possible to modify the value of the control at any real time. In this paper, we are interested in studying optimal control problems for which it is not possible to act on the control system everywhere in a permanent way, that is, according to the position of the system in the state space, a loss of control can be suffered. Such a situation typically occurs in aerospace when a spacecraft enters into a shadow zone (see [START_REF] Geffroy | Optimal low-thrust transfers with constraints-generalization of averaging techniques[END_REF]; [START_REF] Trélat | Optimal control and applications to aerospace: Some results and challenges[END_REF] and references therein). Let us also mention time crisis problems in the context of viability theory (see [START_REF] Aubin | Viability theory: new directions[END_REF]). Such a problem consists in minimizing the total time spent by a control system outside a given set K (see [START_REF] Bayen | Second-order analysis for the time crisis problem[END_REF]). Depending on the application model, one important issue is to consider situations where a practitioner cannot modify the value of the control in the set K, or in its complement (such as in population models, see [START_REF] Bayen | Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey-predator model[END_REF]).

To model the loss of control, we consider a partition of the state space into several disjoint regions (in the spirit of [START_REF] Haberkorn | Convergence results for smooth regularizations of hybrid nonlinear optimal control problems[END_REF]) and we suppose that each region can be of type C or N C (where C and N C stand for control region and non-control region). In a region of type C, the system can be controlled, as usual, in a permanent way (that is, one can change the value of the control at any real time), whereas, in a region of type N C, the control is frozen at the entry time into the region until the exit time. In the latter case, the constant value of the control can be chosen (as a parameter), but the control has to remain constant as long as the system belongs to the region.

In this paper, we consider a Mayer optimal control problem in which the control system is governed by a smooth dynamics and is subject to loss of control as explained above. Our objective is to write the corresponding necessary optimality conditions in a Pontryagin form. As far as we know, such a framework has never been considered in the literature. Let us emphasize that the Pontryagin maximum principle (PMP in short) has been adapted to numerous situations, in particular whenever the problem depends on additional parameters that remain constant over the whole time interval. More generally, piecewise constant controls (called sampled-data controls) have been considered in [START_REF] Bourdin | Optimal sampleddata control, and generalizations on time scales[END_REF]; Bourdin andDhar (2019, 2020) in which the control remains constant over time subintervals (respecting fixed or free time partitions). However, note that these settings are not adapted to the present work because, here, the time subintervals over which the control has to remain constant depend on the state position.

Our strategy is as follows. We embed our problem into a hybrid optimal control problem including a parameter that is constant on each region, but can change its value from one region to another. We speak of a regionally switching parameter. The corresponding necessary optimality con-ditions in this more general setting have been derived in a subsequent work (see [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF]). Let us emphasize that, because of the presence of the regionally switching parameter, one cannot apply the hybrid maximum principles available in the literature (such as in [START_REF] Sussmann | A nonsmooth hybrid maximum principle[END_REF]; [START_REF] Riedinger | An optimal control approach for hybrid systems[END_REF]; [START_REF] Garavello | Hybrid necessary principle[END_REF]; [START_REF] Caines | A maximum principle for hybrid optimal control problems with pathwise state constraints[END_REF]; [START_REF] Clarke | Applications of optimal multiprocesses[END_REF]; [START_REF] Haberkorn | Convergence results for smooth regularizations of hybrid nonlinear optimal control problems[END_REF]; [START_REF] Clarke | Functional analysis, calculus of variations and optimal control[END_REF]; [START_REF] Barles | Value function for regional control problems via dynamic programming and Pontryagin maximum principle[END_REF]). Note that our methodology also differs from [START_REF] Dmitruk | The hybrid maximum principle is a consequence of Pontryagin maximum principle[END_REF] which provides a hybrid maximum principle adapted to a dynamics subject to changes at some (fixed or free) instants that are independent of the state position. The techniques we use in [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF] rely on a thorough sensitivity analysis in the context of differential equations that is extended to hybrid systems. Such an analysis allows us to compute variation vectors along the hybrid trajectory when considering perturbations of the parameter and of the control in a given region. A crucial hypothesis is the transversality of the optimal trajectory at each crossing time in order to obtain perturbed trajectories possessing the same structure as the nominal one.

The paper is organized as follows. In Section 2, we introduce the framework and state our main result (Theorem 1) showing that:

• in a region of type C, the usual Hamiltonian maximization condition holds true; • in a region of type N C, an averaged Hamiltonian gradient condition in the spirit of [START_REF] Bourdin | Optimal sampleddata control, and generalizations on time scales[END_REF]; Bourdin andDhar (2019, 2020) is fulfilled.

To keep this work concise, we do not include the proof of the necessary optimality conditions, but we refer to [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF] for details. In Section 3, a simple example is developed for illustration. Finally, the paper is concluded with perspectives for further research works in Section 4.

MAIN RESULT

We first introduce some notations. Throughout this paper, let m, n ∈ N * be two fixed positive integers and let T > 0 be a fixed positive real number. In the sequel, ⟨•, •⟩ denotes the inner product over R n and B[x, r] denotes the closed ball in R n of center x ∈ R n and of radius r > 0. Then we denote by S and ∂S the closure and the boundary of any subset S of R n . Finally, we denote by AC(

[0, T ], R n ) (resp. L ∞ ([0, T ], R m
)) the space of absolutely continuous functions (resp. essentially bounded functions) defined on [0, T ] with values in R n (resp. R m ).

Mayer optimal control problem with non-control regions

Throughout this paper we consider a partition of R n given by

R n = j∈J X j ,
where J is a (possibly infinite) family of indexes and the nonempty connected open subsets X j (called regions) are disjoint. We assume that each region is either of type C, either of type N C (see Introduction for details), and thus, for all j ∈ J , we introduce

q j := 1 if X j is a control region, 0 if X j is a non-control region.
Now we introduce the control system

         (x, u) ∈ AC([0, T ], R n ) × L ∞ ([0, T ], R m ), ẋ(t) = f (x(t), u(t)) a.e. t ∈ [0, T ], x(0) = x init , u is constant in non-control regions, (CS)
where the initial condition x init is fixed with x init ∈ X j1 for some j 1 ∈ J , and the dynamics f :

R n × R m → R n is of class C 1 .
As usual in the literature, x is called state (or trajectory) and u is called control. We now give a precise definition of a solution to (CS). Definition 1. We say that a pair (x, u)

∈ AC([0, T ], R n ) × L ∞ ([0, T ], R m
) is a solution to (CS) if the four following conditions are satisfied:

(1) There exists a partition

T = {t c k } k=0,...,N of [0, T ], with N ∈ N * , such that, for all k ∈ {1, . . . , N }, there exists j(k) ∈ J such that ∀t ∈ (t c k-1 , t c k ), x(t) ∈ X j(k) , with j(k) ̸ = j(k -1)
for all k ∈ {2, . . . , N }. In the sequel, the times t c k , for k ∈ {1, . . . , N -1}, are called crossing times and correspond to the instants at which the trajectory x goes from one region to another (in particular x(t c k ) belongs to the interface ∂X j(k) ∩ ∂X j(k+1) ).

(2) It holds that x(0) ∈ X j(1) and x(T ) ∈ X j(N ) .

(3) The state equation ẋ(t) = f (x(t), u(t)) is satisfied for almost every t ∈ [0, T ] and x(0) = x init (and thus j(1) = j 1 ). ( 4) For all k ∈ {1, . . . , N } such that q j(k) = 0, the control u is constant over (t c k-1 , t c k ) (the constant value being denoted by u k in the sequel).

Our objective in the present work is to derive first-order necessary optimality conditions (in a PMP form) for the Mayer optimal control problem with non-control regions given by minimize ϕ(x(T )), subject to (x, u) solution to (CS),

u(t) ∈ U a.e. t ∈ [0, T ], (OCP)
where the Mayer cost function ϕ : R n → R is of class C1 and the control constraint set U is a nonempty closed convex subset of R m .

Regular solution and necessary optimality conditions

Our main result is based on some regularity assumptions concerning the transverse behavior of the optimal trajectory at the interfaces between regions. The precise hypotheses are provided in the next definition and are standard (see, e.g., [START_REF] Haberkorn | Convergence results for smooth regularizations of hybrid nonlinear optimal control problems[END_REF]; [START_REF] Bayen | Second-order analysis for the time crisis problem[END_REF]

: B[x(t c k ), ν] → R such that    y ∈ X j(k) ⇔ F k (y) < 0, y ∈ ∂X j(k) ∩ ∂X j(k+1) ⇔ F k (y) = 0, y ∈ X j(k+1) ⇔ F k (y) > 0, for all y ∈ B[x(t c k ), ν].
(3) At each crossing time t c k , the transverse conditions depicted in Figure 1 and given by

⟨∇F k (x(t c k )), (f ) -(t c k )⟩ > 0, ⟨∇F k (x(t c k )), (f ) + (t c k )⟩ > 0, are fulfilled, where (f ) ± (t c k ) := f (x(t c k ), u ± (t c k )). X j(k-1) X j(k) X j(k+1)
Fig. 1. Illustration of a regular trajectory (in blue) crossing transversally the interfaces between regions.

Let H : R n × R m × R n → R stand for the Hamiltonian function associated with the Mayer optimal control problem (OCP) defined by

H(x, u, p) := ⟨p, f (x, u)⟩, for all (x, u, p) ∈ R n × R m × R n .
Our main result is as follows.

Theorem 1. Let (x, u) be a solution to (OCP), which is moreover a regular solution to (CS), and consider the notations introduced in Definitions 1 and 2. Then there exists a piecewise absolutely continuous costate p : [0, T ] → R n , respecting the partition T = {t c k } k=0,...,N of [0, T ], such that:

(1) The adjoint equation ṗ

(t) = -∇ x f (x(t), u(t)) ⊤ p(t) is satisfied for almost every t ∈ [0, T ]. (2) The final condition p(T ) = -∇ϕ(x(T )) is satisfied. (3) At each crossing time t c k , the discontinuity condition p + (t c k ) -p -(t c k ) = β k ∇F k (x(t c k )), with β k := - ⟨p + (t c k ), (f ) + (t c k ) -(f ) -(t c k )⟩ ⟨∇F k (x(t c k )), (f ) -(t c k )⟩
, is fulfilled. (4) For all k ∈ {1, ..., N } such that q j(k) = 1, the Hamiltonian maximization condition

u(t) ∈ arg max v∈U H(x(t), v, p(t)),
is fulfilled for almost every t ∈ (t c k-1 , t c k ). ( 5) For all k ∈ {1, ..., N } such that q j(k) = 0, the averaged Hamiltonian gradient condition

t c k t c k-1 ∇ u H(x(s), u k , p(s)) ds ∈ N U (u k ).
holds true, where N U (u k ) stands for the normal cone to U at u k .

The proof of Theorem 1 is a direct application of the hybrid maximum principle developed in [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF] that takes into account a regionally switching parameter. Indeed, one has just to see that the control system (CS) can be rewritten as the hybrid control system given by

                 (x, u) ∈ AC([0, T ], R n ) × L ∞ ([0, T ], R m ), λ : [0, T ] → R m is a regionally switching parameter associated with x, ẋ(t) = h(x(t), λ(t), u(t)) a.e. t ∈ [0, T ], x(0) = x init ,
where the hybrid dynamics h :

R n × R m × R m → R n is defined by h(x, λ, u) := h j (x, λ, u) if x ∈ X j , where h j (x, λ, u) := f (x, u) if q j = 1, f (x, λ) if q j = 0, for all (x, λ, u) ∈ R n × R m × R m
and all j ∈ J . Indeed, let us recall that a regionally switching parameter is a function that remains constant while the state position x stays inside a region, and can switch (that is, can change its value) only when the state position x goes from one region to another. To keep this work concise, we do not include the detailed proof of Theorem 1, but we refer to [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF] for details.

Comments

• To summarize, Theorem 1 shows that, in each region of type C, the usual Hamiltonian maximization condition holds true, whereas, in each region of type N C, a socalled averaged Hamiltonian gradient condition (in the spirit of the one obtained for optimal sampled-data control problems, see [START_REF] Bourdin | Optimal sampleddata control, and generalizations on time scales[END_REF]; Bourdin andDhar (2019, 2020)) holds true. It is worth mentioning that the latter is implicit in general since u k intervenes, not only in both sides of the equation, but moreover in the values of x and p along the interval (t c k-1 , t c k ). Furthermore we do not know in advance the values of t c k-1 and t c k . However, as we will see in Section 3, the averaged Hamiltonian gradient condition can be useful to determine the optimal values of the control in regions of type N C.

• In Theorem 1, and as usual in the literature, the Hamiltonian system ( ẋ, ṗ) = (∇ p H, -∇ x H) is satisfied which implies (together with the other necessary optimality conditions) that the Hamiltonian function t → H(x(t), u(t), p(t)), is constant almost everywhere over [0, T ]. Indeed, in a region of type C (with the Hamiltonian maximization condition), one has just to use the standard argumentation (see [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]). In a region of type N C, the result is straightforward since the control is constant.

Finally the discontinuity conditions ensure the constancy at each crossing time t c k (see [START_REF] Haberkorn | Convergence results for smooth regularizations of hybrid nonlinear optimal control problems[END_REF]). • To benefit the most of Theorem 1 (and avoid unnecessary hypotheses), the partition of R n must be written so that the number of regions involved is as small as possible. The idea is to avoid, for example, trajectories that would go from a region of type C to another one (which would be redundant from a model point of view).

• In Theorem 1, the discontinuity condition at each crossing time t c k is written backward in time. Nonetheless, it can also be written forward in time by noting that

β k = - ⟨p -(t c k ), (f ) + (t c k ) -(f ) -(t c k )⟩ ⟨∇F k (x(t c k )), (f ) + (t c k )⟩ .
• In Definition 2, note that the continuity and limit conditions on the control are superfluous in regions of type N C (since u is constant in such a region).

• Several extensions of Theorem 1 could be of interest and can be easily derived. For the following possible extensions, we refer to [START_REF] Bayen | Hybrid maximum principle with regionally switching parameter[END_REF] for details:

-Theorem 1 can be extended to a non-autonomous setting, as well on the dynamics as on the partition of the state space. -The convexity (resp. closedness) hypothesis on U can be removed by using a generalized version of the normal cone (resp. by assuming that all the limits u -(t c k ) and u + (t c k ) belong to U). -One can consider a control constraint set U j in each region X j . This would allow to impose the control value in non-control regions. For example, to deal with the case where no control input is allowed in non-control regions, take U j = {0 R m } for all j ∈ J such that q j = 0. -One can consider a Bolza cost, involving a Lagrange cost associated with a hybrid Lagrangian function adapted to the partition. This setting would allow to deal with time crisis problems for which the constraint set K is a region of type C (or of type N C).

EXAMPLE

In this section, we highlight the use of Theorem 1 on a simple one-dimensional Mayer optimal control problem with one non-control region. Here n = m = 1 and T = 8.

3.1 Presentation of the example As expected, the cost associated with x is the best, but is not admissible, while the cost associated with x is admissible and better than the other admissible costs associated with x ⊥ and x † . This example shows the relevancy of establishing a PMP in the present context of loss of control since, in general, the optimal constant values in non-control regions do not follow the values of the optimal permanent control obtained with the classical PMP. Furthermore, note that, in contrary to what is usually observed in the classical literature (with permanent controls) when the Hamiltonian is linear with respect to the control, the loss of control can induce optimal constant values in non-control regions that do not saturate the control constraint set U. With this example, we also emphasize that the avegared Hamiltonian gradient condition derived in Theorem 1 allows to determine such optimal values.

x x(•)

x ⊥ (•)

x † (•)

x(•) Fig. 3. Trajectories x, x, x ⊥ and x † from Section 3.3 (zoom on the time interval [0, 3]).

CONCLUSION AND PERSPECTIVES

In this work, we have introduced a new framework in optimal control theory letting the possibility for a control system to be subject to loss of control depending on its position in a partition of the state space. In our approach, the control value has to be fixed to an admissible value as long as the system belongs to a non-control region but we do not know in advance how long the system stays in such a region. The corresponding optimal constant value satisfies (and possibly is determined by) the averaged Hamiltonian gradient condition. We believe that this setting differs from other frameworks covered by hybrid optimal control problems or state constrained optimal control problems (see, e.g., [START_REF] Frankowska | Strong local minimizers in optimal control problems with state constraints: second-order necessary conditions[END_REF]) and that it could be employed in various practical situations such as in aerospace (in particular for the determination of an optimal control strategy when a spacecraft enters into a shadow zone). Future works could focus on the determination of optimal control policies in this framework for SIR models or in population models in the context of time crisis problems when one is unable to control in the non-constraint set (see, e.g., [START_REF] Bayen | Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey-predator model[END_REF]). We are also interested in extending the necessary optimality conditions obtained in this paper to the case of feedback controls in non-control regions (instead of frozen controls) and to the context of final state constraints, and in developing Riccati theory for linear control systems subject to loss of control. Also note that, in this paper, we did not discuss the existence of a solution to (OCP) which may be a difficult question due to the presence of non-control regions. So, in Theorem 1, we have assumed that there exists a solution to (OCP), moreover with a finite number of crossing times, excluding that way other possible solutions with more complicated structures such as chattering, boundary arcs, tangential crossing, etc. On the other hand, note that considering non-control regions may impact the controllability of (CS) but we did not discuss controllability issues here. All these subjects constitute interesting perspectives for further research works.

  At each crossing time t c k , the control u is continuous over [t c k -δ, t c k ) and over (t c k , t c k + δ], and admits left and right limits denoted by u -(t c k ) and u + (t c k ). (2) At each crossing time t c k , there exists a C 1 function F k

). Definition 2. Consider a solution (x, u) to (CS) and the notations introduced in Definition 1. Set α := (1)

3 min k=1,...,N |t k -t k-1 | > 0. We say that (x, u) is regular if there exist δ ∈ (0, α) and ν > 0 such that:

ACKNOWLEDGMENT

We would like to thank N. Augier and E. Trélat for fruitful exchanges on hybrid optimal control problems. This research benefited from the support of the FMJH Program PGMO and from the support to this program from EDF-THALES-ORANGE.

Consider the partition R = X 1 ∪ X 2 ∪ X 3 with

In what follows, we suppose that X 1 and X 3 are of type C (that is q 1 = q 3 = 1) and X 2 is of type N C (that is q 2 = 0). Now consider the Mayer optimal control problem given by minimize -x(8), subject to (x, u) ∈ AC([0, 8], R) × L ∞ ([0, 8], R), ẋ(t) = u(t)x(t) + 1 a.e. t ∈ [0, 8],

The situation is depicted in Figure 2 and the corresponding Hamiltonian is given by H(x, u, p) := p(ux + 1), for all (x, u, p) ∈ R 3 .

x Fig. 2. Illustration of the framework of Section 3.

Synthesis of an optimal control

In this section, we assume that there exists a solution (x, u), that is regular, and we suppose that it admits exactly two (unknown) crossing times 0 < t c 1 < t c 2 < 8 and satisfies the following structure:

] is unknown and assumed to satisfy u 2 ̸ = 0. Now let us denote by p : [0, 8] → R the costate provided by Theorem 1. We proceed to the analysis backward in time.

• Step 1: analysis in the region X 3 . The adjoint equation and final condition give

and the Hamiltonian maximization condition writes

Since p(t) > 0 and x(t) > 0 over (t c 2 , 8], we get that u(t) = 1/2 for almost every t ∈ (t c 2 , 8) and thus, since x(t 2 c ) = 1/2, we get that

Step 2: analysis in the region X 2 . From the discontinuity condition at t c 2 and the adjoint equation, the costate p satisfies

We get that

for all t ∈ (t c 1 , t c 2 ). Since x(t c 1 ) = -1, one deduces that

and the relation between t c 1 and t c 2 given by

.

(2)

•

Step 3: analysis in the region X 1 . From the discontinuity condition at t c 1 and the adjoint equation, the costate p satisfies

and the Hamiltonian maximization condition writes

x(t)p(t)v a.e. t ∈ (0, t c 1 ).

Since p(t) > 0 and x(t) < 0 over [0, t c 1 ], we deduce that u(t) = -3/2 for almost every t ∈ [0, t c 1 ] and thus, since x(0) = -2, we get that

for all t ∈ [0, t c 1 ). • Step 4: global analysis. From x(t c 1 ) = -1, one can easily obtain that t c 1 = 2 3 ln( 8 5 ). Furthermore, we can now determine the value u 2 ∈ [-3 2 , 1 2 ] thanks to the averaged Hamiltonian gradient condition which writes

(3)

Using ( 1) and ( 2), we find that:

2 ) and thus (3) implies that γ(u 2 ) = 0 which amounts to solving the equation

This gives us a unique value for u 2 ∈ (-3 2 , 1 2 ) given approximately by u 2 ≃ -0.75. Finally the optimal control is given by

a.e. t ∈ (t c 2 , 8), with t c 1 = 2 3 ln( 8 5 ) ≃ 0.31 and t c 2 ≃ 1.68.

Comparisons with other control strategies

We end-up this case study with comparisons of the optimal control u obtained in the previous section with different control strategies. To keep this work concise, the computations of this section are omitted.

• First, note that, if the region X 2 was of type C, then the classical PMP would imply that the optimal (permanent) control û (associated with the trajectory x) satisfies û(t) = -3/2 if x(t) < 0, 1/2 if x(t) > 0, for almost every t ∈ [0, 8]. However, since X 2 (of type N C) is a strip containing 0, the control û is not admissible (since it requires to change its value in the non-control region X 2 , see Figure 3).

• Second, from the (nonadmissible) control û, one might consider the admissible control u ⊥ (resp. u † ) given by u ⊥ = -3/2 in both regions X 1 and X 2 (resp. u † = -3/2 in region X 1 ) and by u ⊥ = 1/2 in region X 3 (resp. u † = 1/2 in regions X 2 and X 3 ). The associated trajectory is denoted by x ⊥ (resp. x † ).

On Figure 3, we depict the trajectories x, x, x ⊥ and x † .


Perusahaan Dan

Biaya Agensi

Nur Dicky Chandra

Hadiyazid Rachman

Dan Nuryadi Wijiharjono

Tata Kelola

Keywords: Agency cost, Corporate governance mechanism, BUMD, Panel Data Regression ABSTRAK BUMD dibentuk oleh pemerintah daerah, dimaksudkan memiliki peran ganda, Biaya keagenan, Mekanisme tata kelola perusahaan, BUMD, Regresi Data Panel

The purpose of this study is to investigate the effect of corporate governance mechanisms using agency costs in state-owned companies (BUMD). BUMD is formed by the local government, has a dual role, namely to serve the community as well as make a profit. Not many BUMDs are able to develop in accordance with this assistance. Agency problems arise when the agent deviates from the interests of the principal. Local government control over BUMD is needed so that can be explained in the interests of the principal. The results of the Panel Data Regression Analysis revealed that the variables that were able to reduce agency costs at the expense ratio proxy were the number of directors, total sales, total assets. These three variables agency cost, while the variable that can increase agency costs on the proxies of asset utilization ratios and Return On Assets (ROA) is total sales. If the total sales variables is increasing, it will increase the ratio of asset initialization and ROA.

PENDAHULUAN

Penelitian ini bertujuan untuk menginvestigasi pengaruh mekanisme tata kelola perusahaan dalam meminimalkan biaya keagenan (agency costs) pada BUMD DKI Jakarta. Secara teoretis, melalui pendekatan konsep tata kelola perusahaan yang baik atau Good Corporate Governanve (GCG), penelitian ini diharapkan dapat memberikan kontribusi dalam pengembangan teori keagenan (agency theory) dalam konteks GCG. Sedangkan secara praktis, penelitian ini bermanfaat bagi pemerintah daerah dalam melakukan mitigasi biaya keagenan, termasuk model pengendaliannya.

Dalam laporan OECD (2019) disebutkan bahwa kerangka tata kelola perusahaan (corporate governance framework) pada perusahaan terbuka (go public) lebih banyak ditentukan oleh pemerintah sebagai institusi regulator, pembuat kebijakan. Apalagi bagi perusahaan yang berbentuk BUMD, kebijakan pemerintah baik di tingkat daerah maupun pusat, akan menentukan tata kelola tersebut. Konsep tata kelola perusahaan yang baik diharapkan dapat menekan biayabiaya keagenan yang disebabkan oleh masalah keagenan. Bahkan dalam konteks bisnis global, menurut [START_REF] Choi | Foreign ownership, agency costs, and long-term firm growth: Evidence from Korea[END_REF], kepemilikan asing (investor asing) dapat mengurangi biaya agensi dan dapat meningkatkan nilai perusahaan dalam jangka panjang. Kepemilikan asing pada perusahaan memiliki efek moderasi dalam pembayaran deviden dan pertumbuhan perusahaan dalam jangka panjang.

Dalam perkembangan teori keagenan, biaya keagenan merujuk pada biaya yang berkaitan dengan masalah keagenan. Hubungan antara prinsipal (pemilik) dan manajer (agen), dalam teori keagenan, dapat menimbulkan permasalahan yang disebabkan oleh informasi asimetris dan konflik kepentingan, di mana manajer lebih mengusai informasi tentang perusahaan, termasuk dalam masalah keuangan, dibandingkan dengan pemiliknya (prinsipal). Dalam mengatasi masalah keagenan, akan menimbulkan biaya-biaya agensi (agency costs), yang harus ditanggung oleh kedua belah pihak. Terdapat tiga jenis biaya agensi yang harus ditanggung, yaitu biaya pemantauan (monitoring cost) oleh prinsipal, biaya perikatan (bonding cost) oleh agen, dan kerugian residual (residual loss) sebagai pengurang kekayaan prinsipal (Jansen & Meckling, 1976).

Menurut [START_REF] Panda | Agency theory: Review of theory and evidence on problems and perspectives[END_REF], hanya sedikit akademisi dari negara-negara berkembang yang melakukan kajian tentang teori agensi. Padahal, dalam kajian literatur tentang teori agensi ini, banyak akademisi dari negara-negara maju seperti Amerika Serikat dan Inggris yang berbeda dalam bidang disiplin ilmu, telah menggunakan teori ini untuk menganalisis masalah keagenan pada organisasi maupun perusahaan. [START_REF] Owusu | Agency costs, ownership structure and corporate governance mechanisms in Ghana[END_REF] dalam penelitiannya pada sejumlah perusahaan di Ghana menyimpulkan bahwa, peningkatan kepemilikan manajerial dan institusional secara signifikan mampu mengurangi biaya keagenan. Temuan ini menawarkan dukungan bahwa kepentingan pemegang saham dan manajer dapat diselaraskan melalui peningkatan kepemilikan dan tata kelola perusahaan yang baik.

Penelitian Harry G. Broadman pada model BUMN di China menemukan bahwa, kepemilikan mayoritas oleh pemerintah (negara) terbukti dapat menekan biaya keagenan, dan konflik kepentingan antara pemerintah dan manajemen pun dapat ditekan, karena pemerintah lebih mementingkan kepentingan jangka panjang [START_REF] Broadman | The Chinese State as corporate shareholder[END_REF]. Konsentrasi kepemilikan menjadi penting dalam konteks corporate governance. Kepemilikan institusional dapat meningkatkan kualitas tata kelola perusahaan [START_REF] Chung | Corporate governance and institutional ownership[END_REF].

Mekanisme internal tata kelola perusahaan meliputi ukuran struktur kepemilikan, dewan komisaris, direktur, remunerasi direktur, komite audit dan transparansi [START_REF] Khan | Corporate governance quality, ownership structure, agency costs and firm performance. Evidence from an emerging economy[END_REF]. Hasil penelitian [START_REF] Javaid | Determining agency theory framework through financial leverage & insider ownership[END_REF] menemukan bahwa praktik tata kelola perusahaan yang baik akan membantu perusahaan dalam meningkatkan kinerja dan membawa kepercayaan investor. Dalam mengukur biaya keagenan, penelitian ini menggunakan rasio biaya, di mana biaya operasi dibandingkan dengan penjualan tahunan. Adapun komponen-komponen tata kelola perusahaan meliputi pengelolaan hutang, kepemilikan manajerial, konsentrasi kepemilikan, dewan direktur, dan kompensasi manajerial [START_REF] Florackis | Agency costs and corporate governance mechanisms: evidence for UK firms[END_REF]. Menurut [START_REF] Chinelo | Ownership structure, corporate governance and agency cost of manufacturing companies in Nigeria[END_REF] dan [START_REF] Roy | Corporate governance and firm performance: A study of Indian listed firms[END_REF], mekanisme tata kelola perusahaan tergambar dalam struktur kepemilikan, dewan direksi, dewan komisaris, komisaris independen, dan komite audit .

Penelitian pada perusahaan agro-industri di Indonesia yang dilakukan oleh [START_REF] Hastori | Agency costs, corporate governance and ownership concentration: The case of agro-industrial companies in Indonesia[END_REF] menunjukkan bahwa biaya keagenan dipengaruhi oleh dewan direksi, dewan komisaris, komisaris independen, dan komite audit. Manajemen dan tata kelola perusahaan memengaruhi dalam mengurangi biaya agensi. Kebijakan pembayaran deviden secara signifikan memiliki hubungan negatif. Artinya, semakin tinggi pembayaran deviden semakin rendah biaya keagenan yang dikeluarkan.

METODE

Penelitian ini menggunakan metode kuantitatif. Metode kuantitatif digunakan untuk menguji mekanisme tata kelola perusahaan BUMD dalam mengurangi biaya keagenan.

Berdasarkan ketersediaan data, maka sumber data pada penelitian ini berasal dari Laporan Keuangan 12 (dua belas) BUMD DKI Jakarta tahun 2015 sampai 2017 yang telah diaudit dan dipublikasikan.

Dalam penelitian ini, Tata Kelola Perusahaan (corporate governance) digunakan sebagai variabel bebas (independent variables). Merujuk Hastori et al. (2015), [START_REF] Hadiprajitno | Struktur kepemilikan, mekanisme tata kelola perusahaan, dan biaya keagenan di Indonesia (studi empirik pada perusahaan di Bursa Efek Indonesia)[END_REF], [START_REF] Mir | Corporate governance structure and firm performance in Pakistanan empirical study[END_REF], dan [START_REF] Singh | Agency Cost, Ownership Structure and Corporate Governance Mechanisms[END_REF], pengukuran tata kelola perusahaan BUMD dalam penelitian ini menggunakan proksi: kepemilikan mayoritas pemerintah atau penyertaan modal daerah (X1), jumlah direksi (X2), jumlah komisaris (X3), jumlah komisaris independen (X4), jumlah komite audit (X5), Total Pengeluaran (X6), Total Penjualan (X7), dan Total Asset (X8) Sedangkan yang digunakan sebagai variabel terikat (dependent variable) adalah biaya keagenan (agency costs). Merujuk pada penelitian [START_REF] Reddy | The relationship between ownership structure, capital structure and corporate governance practices[END_REF][START_REF] Hardesty | Comparative Financial Performance of Agricultural Cooperatives and Investor-Owned Firms[END_REF], [START_REF] Ang | Agency costs and ownership structure[END_REF], [START_REF] Singh | Agency Cost, Ownership Structure and Corporate Governance Mechanisms[END_REF] dan [START_REF] Lerman | Comparative performance of cooperatives and investorowned firms in US food industries[END_REF], analisis biaya keagenan dalam penelitian ini menggunakan tiga alternatif proksi. Pertama (Model A), menggunakan Rasio Beban (expense ratio) sebagai proksi biaya keagenan, yaitu rasio biaya operasi, beban umum dan administrasi (OGA), dibagi dengan total penjualan. Dalam hal ini, Rasio Beban yang tinggi mengindikasikan adanya biaya keagenan yang tinggi, dan berlaku pula sebaliknya. Kedua (Model B), menggunakan Rasio Penggunaan Aset total (asset utilization) sebagai proksi biaya keagenan (total penjualan tahunan dibagi total aset) Artinya, pemanfaatan nilai aset yang tinggi mengindikasikan adanya biaya keagenan yang rendah, demikian pula sebaliknya. Ketiga (Model C), menggunakan ratio Return on Asset (ROA) sebagai proksi biaya keagenan, yaitu laba setelah pajak dibagi dengan total aset. Nilai ROA yang tinggi mengindikasikan bahwa para manager berkinerja baik, dan begitu pula sebaliknya. 1. Pemodelan dengan semua koefisien konstan terhadap waktu dan perusahaan (data waktu dan jenis-jenis perusahaan BUMD di Provinsi DKI Jakarta diabaikan).

2. Pemodelan dengan koefisien slope konstan dan intersep bervariasi antar perusahaan (pengaruh jenis-jenis perusahaan BUMD di Provinsi DKI Jakarta).

3. Pemodelan dengan koefisien slope konstan dan intersep bervariasi antar perusahaan (pengaruh waktu pengumpulan data pada perusahaan BUMD di Provinsi DKI Jakarta).

4. Pemodelan dengan koefisien slope konstan, intersep bervariasi antar perusahaan dan antar waktu (pengaruh jenis-jenis perusahaan BUMD di Provinsi DKI Jakarta dan waktu pengumpulan data)

Pemodelan dengan semua koefisien bervariasi untuk semua jenis perusahaan BUMD di

Provinsi DKI Jakarta.

HASIL DAN PEMBAHASAN

Hasil

Tahap analisis awal dilakukan dengan cara Redundant Fixed Effect-Likelihood Ratio untuk menentukan model apa saja yang dapat digunakan pada data panel BUMD Provinsi DKI Jakarta.

Hasil pengujian terhadap data panel dengan tiga (3) proksi biaya keagenan pada BUMD Provinsi DKI Jakarta disajikan secara berturut-turut pada Tabel 1, Tabel 2, dan Tabel 3 berikut: Model yang terbentuk sudah baik yang ditandai dengan nilai R-Square (R 2 ) sebesar 0.6819.

Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel-variabel yang memengaruhi biaya agensi dengan pendekatan Rasio Beban (RB) adalah sebagai berikut:

1. Penyertaan Modal Daerah (PMD). Semakin tinggi penyertaan modal daerah pada perusahaan BUMD akan memengaruhi secara postif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

2. Jumlah/banyaknya Dewan Direksi (BOD). Semakin banyak jumlah dewan direksi pada perusahaan BUMD akan memengaruhi secara negatif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

3. Jumlah/banyaknya Komite Audit (BOCA). Semakin banyak jumlah komite audit pada perusahaan BUMD akan memengaruhi secara positif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

4. Total Pengeluaran (TE). Semakin tinggi pengeluaran yang terjadi pada perusahaan BUMD akan berpengaruh secara positif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

5. Total Penjualan (TS). Semakin tinggi total penjualan akan memengaruhi secara negatif biaya agensi (rasio beban keuangan) pada BUMD.

6. Total Asset (TA). Semakin besar total asset akan berpengaruh negatif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

b. Hasil pendekatanRasio Asset Utilisation sebagai proksi dari biaya keagenan ditunjukkan pada Tabel 5 di bawah ini, yaitu: Model yang terbentuk sudah baik yang ditandai dengan nilai R-Square (R 2 ) sebesar 0.6922.

Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel-variabel yang memengaruhi biaya agensi dengan pendekatan Rasio Beban (RB) adalah sebagai berikut:

1. Penyertaan Modal Daerah (PMD). Semakin tinggi penyertaan modal daerah setiap tahun pada perusahaan BUMD akan memengaruhi secara postif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

2. Jumlah/banyaknya Dewan Direksi (BOD). Semakin banyak jumlah dewan direksi setiap tahun pada perusahaan BUMD akan memengaruhi secara negatif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

3. Jumlah/banyaknya Komite Audit (BOCA). penambahan jumlah komite audit setiap tahun pada perusahaan BUMD akan memengaruhi secara positif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

Total Pengeluaran (TE). Semakin tinggi TE setiap tahun pada perusahaan BUMD akan

berpengaruh secara positif terhadap biaya agensi (rasio beban keuangan) pada BUMD.

Total Penjualan (TS)

. Semakin tinggi total penjualan setiap tahun akan memengaruhi secara negatif biaya agensi (rasio beban keuangan) pada BUMD. Berdasarkan hasil analisis data pada Tabel 5 dan Tabel 8, hanya variabel total penjualan (TS) yang memengaruhi agency cost. Kemampuan BUMD-BUMD yang ada di Provinsi DKI Jakarta dalam menjual/mengelola asset yang memberikan pemasukan berupa antara lain yaitu:

Total Asset (TA

iklan, sewa ruang pertemuan, sewa hotel, sewa gedung, dan lain lain. Dengan kata lain, sebagai bentuk pemanfaatan aset, meningkatnya penjualan dapat mengakibatkan kenaikan agency cost.

Kenaikan agency cost terjadi disebabkan oleh biaya pemeliharaan, biaya perbaikan dan biaya operasional lainnya yang berkaitan dengan pemanfaatan aset yang dimiliki oleh BUMD Provinsi DKI Jakarta.

Agency Cost dengan proksi Return On Asset (ROA)

Berdasarkan hasil analisis data pada Tabel 6 dan Tabel 9, hanya variabel total penjualan (TS) yang memengaruhi agency cost. Tingginya penjualan produk (layanan) BUMD menunjukkan BUMD-BUMD yang ada di Provinsi DKI Jakarta merupakan perusahaan pilihan para pemangku kepentingan (stakeholders). Keterlibatan dari stakeholders memberikan dampak terhadap tingginya minat membeli produk (layanan) dari BUMD dan atau dapat juga berfungsi sebagai partnerships (rekan kerja) dalam rangka menyediakan produk yang dipasok ke BUMD Provinsi DKI Jakarta. Semakin tinggi tingkat penjualan akan semakin besar pula laba yang diterima oleh BUMD yang akhirnya akan meningkatkan ROA-nya.

SIMPULAN

Berdasarkan analisis data dan pembahasan di atas, penelitian ini menghasilkan beberapa kesimpulan, yaitu:

1. Penggunaan metode pertama (model dengan semua koefisien konstan terhadap waktu dan perusahaan) dan metode kedua (model dengan slope konstan dan intersep beravariasi antar waktu) secara statistik memberikan hasil yang sama.

2. Variabel yang mampu menurunkan biaya keagenan (agency cost) pada proksi rasio beban adalah variabel jumlah direksi, total penjualan, dan total aset. Ketiga variabel ini jika semakin naik atau bertambah akan memberikan dampak yang dapat menurunkan biaya keagenan (agency cost).

3. Variabel yang dapat menaikkan agency cost pada proksi rasio asset utilasation dan Return On Asset (ROA) adalah total penjualan. Variabel total penjualan jika semakin bertambah maka akan meningkatkan rasio asset utilisation dan Return On Asset (ROA). Kemampuan BUMD dalam menjual/menawarkan produk (layanan) seperti iklan, sewa ruang pertemuan, sewa hotel, sewa

Tabel 1. Hasil Pengujian Redundant Fixed Effect-Likelihood Ratio Agency Cost dengan Pendekatan Rasio Beban

	Tabel 2.	TE	3.80E-12	7.26E-13	5.230517	0.0000
		TS	-2.10E-13	3.16E-14	-6.633946	0.0000
	TA	-1.80E-14	7.03E-15	-2.553670	0.0166
	R-squared		0.681878 Mean dependent var	0.213056
	Adjusted R-squared	0.587619 S.D. dependent var	0.186881
	S.E. of regression	0.120009 Akaike info criterion	-1.190174
	Sum squared resid	0.388861 Schwarz criterion	-0.794295
	Log likelihood	30.42314 Hannan-Quinn criter.	-1.052002
	F-statistic		7.234126 Durbin-Watson stat	1.276748
	Prob(F-statistic)	0.000042
	Sumber: Data diolah (2020)
	Redundant Fixed Effects Tests
	Pool: POOL01
	Test period fixed effects
	Effects Test		Statistic	d.f.	Prob.
	Period F			0.420948	(2,25)	0.6610
	Period Chi-square		1.192365	2	0.5509
	Period fixed effects test equation:
	Dependent Variable: Y1 (Agency Cost dengan pendekatan Rasio Beban)
	Method: Panel Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
		C	-1.474789	0.821669	-1.794869	0.0839
	PMD	2.143975	0.804454	2.665132	0.0128
	BOD	-0.165288	0.051008	-3.240413	0.0032
	BOC	0.020446	0.016967	1.205007	0.2387
	BOIC	-0.021467	0.168582	-0.127341	0.8996
	BOCA	0.191129	0.083826	2.280077	0.0307

Hasil Pengujian Redundant Fixed Effect-Likelihood Ratio Agency Cost dengan Pendekatan Rasio Asset Utilisation

	Tabel 3.
	Period fixed effects test equation:
	Dependent Variable: Y2 (Agency Cost dengan pendekatan Rasio Asset Utilisation)
	Method: Panel Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
	C	2.600797	5.152985	0.504717	0.6179
	PMD	-2.206220	5.045021	-0.437306	0.6654
	BOD	0.032156	0.319892	0.100522	0.9207
	BOC	-0.002131	0.106407	-0.020024	0.9842
	BOIC	0.079998	1.057240	0.075667	0.9402
	BOCA	-0.309182	0.525702	-0.588131	0.5613
	TE	-5.37E-12	4.55E-12	-1.179788	0.2484
	TS	9.13E-13	1.98E-13	4.600260	0.0001
	TA	-2.88E-14	4.41E-14	-0.653993	0.5186
	R-squared	0.497811 Mean dependent var	0.595278
	Adjusted R-squared	0.349014 S.D. dependent var	0.932807
	S.E. of regression	0.752623 Akaike info criterion	2.481813
	Sum squared resid	15.29391 Schwarz criterion	2.877693
	Log likelihood	-35.67263 Hannan-Quinn criter.	2.619986
	F-statistic	3.345578 Durbin-Watson stat	1.103022
	Prob(F-statistic)	0.008640
	Sumber: Data diolah (2020)

Hasil Pengujian Redundant Fixed Effect-Likelihood Ratio Agency Cost dengan Pendekatan Return On Asset (ROA)

	Tabel 4.
	Dari hasil pengujian Redundant Fixed Effect-Likelihood Ratio tersebut di atas
	menunjukkan bahwa jenis-jenis perusahaan BUMD di Provinsi DKI Jakarta tidak memiliki
	pengaruh, sebaliknya untuk waktu (time/period effect) memiliki pengaruh, sehingga
	Redundant Fixed Effects Tests Pool: POOL01 pemodelan/analisis yang terbentuk hanya ada 2 (dua) jenis pemodelan, yaitu:
	Test period fixed effects
	1. Model pertama; Model dengan semua koefisien konstan terhadap waktu dan perusahaan. Pada
	Effects Test pemodelan ini, hasil yang diperoleh adalah sebagai berikut: Statistic	d.f.	Prob.
	Period F a. Hasil pendekatan Rasio Beban sebagai proksi dari biaya keagenan ditunjukkan pada Tabel 0.070426 (2,25) 0.9322
	Period Chi-square 4 di bawah ini, yaitu:		0.202259	2	0.9038
	Period fixed effects test equation:
	Dependent Variable: Y3 (Agency Cost dengan pendekatan Return On Asset)
	Method: Panel Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
	C	0.288023	0.211804	1.359858	0.1851
	PMD	-0.300052	0.207366	-1.446967	0.1594
	BOD	0.011336	0.013149	0.862147	0.3962
	BOC	-0.004867	0.004374	-1.112768	0.2756
	BOIC	-0.033687	0.043456	-0.775195	0.4450
	BOCA	-0.001811	0.021608	-0.083814	0.9338
	TE	-1.75E-13	1.87E-13	-0.936385	0.3574
	TS	4.01E-14	8.16E-15	4.915804	0.0000
	TA	-1.01E-15	1.81E-15	-0.555209	0.5833
	R-squared	0.551998 Mean dependent var	0.032500
	Adjusted R-squared	0.419257 S.D. dependent var	0.040594
	S.E. of regression	0.030935 Akaike info criterion	-3.901532
	Sum squared resid	0.025838 Schwarz criterion	-3.505652
	Log likelihood	79.22757 Hannan-Quinn criter.	-3.763359
	F-statistic	4.158454 Durbin-Watson stat	0.924663
	Prob(F-statistic)	0.002417
	Sumber: Data diolah (2020)

Hasil Pengujian dengan Pemodelan Koefisien Konstan (Agency Cost dengan Pendekatan Rasio Beban)

	Dependent Variable: Y1 (Agency Cost dengan pendekatan Rasio Beban)
	Method: Pooled Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
	C	-1.474789	0.821669	-1.794869	0.0839
	PMD	2.143975	0.804454	2.665132	0.0128
	BOD	-0.165288	0.051008	-3.240413	0.0032
	BOC	0.020446	0.016967	1.205007	0.2387
	BOIC	-0.021467	0.168582	-0.127341	0.8996
	BOCA	0.191129	0.083826	2.280077	0.0307
	TE	3.80E-12	7.26E-13	5.230517	0.0000
	TS	-2.10E-13	3.16E-14	-6.633946	0.0000
	TA	-1.80E-14	7.03E-15	-2.553670	0.0166
	R-squared	0.681878 Mean dependent var		0.213056
	Adjusted R-squared	0.587619 S.D. dependent var		0.186881
	S.E. of regression	0.120009 Akaike info criterion		-1.190174
	Sum squared resid	0.388861 Schwarz criterion		-0.794295
	Log likelihood	30.42314 Hannan-Quinn criter.		-1.052002
	F-statistic	7.234126 Durbin-Watson stat		1.276748
	Prob(F-statistic)	0.000042
	Sumber: Data diolah (2020)

Tabel 5. Hasil Pengujian dengan Pemodelan Koefisien Konstan (Agency Cost dengan Pendekatan Rasio Asset Utilisation)

	TS	9.13E-13	1.98E-13	4.600260	0.0001
	TA	-2.88E-14	4.41E-14	-0.653993	0.5186
	R-squared	0.497811 Mean dependent var		0.595278
	Adjusted R-squared	0.349014 S.D. dependent var		0.932807
	S.E. of regression	0.752623 Akaike info criterion		2.481813
	Sum squared resid	15.29391 Schwarz criterion		2.877693
	Log likelihood	-35.67263 Hannan-Quinn criter.		2.619986
	F-statistic	3.345578 Durbin-Watson stat		1.103022
	Prob(F-statistic)	0.008640
	Sumber: Data diolah (2020)
	Method: Pooled Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
	C	2.600797	5.152985	0.504717	0.6179
	PMD	-2.206220	5.045021	-0.437306	0.6654
	BOD	0.032156	0.319892	0.100522	0.9207
	BOC	-0.002131	0.106407	-0.020024	0.9842
	BOIC	0.079998	1.057240	0.075667	0.9402
	BOCA	-0.309182	0.525702	-0.588131	0.5613
	TE	-5.37E-12	4.55E-12	-1.179788	0.2484

Dependent Variable: Y2 (Agency Cost dengan pendekatan Rasio Asset Utilisation) Model yang terbentuk sudah cukup baik yang ditandai dengan nilai R-Square (R 2 ) sebesar 0.4978. Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel yang memengaruhi biaya agensi dengan pendekatan Rasio Asset Utilisation (RP) adalah penjualan total (TS). c. Hasil pendekatan Return on Asset sebagai proksi dari biaya keagenan ditunjukkan pada Tabel 6 di bawah ini, yaitu: Tabel 6.

Hasil Pengujian dengan Pemodelan Koefisien Konstan (Agency Cost dengan Pendekatan Return On Asset)

	Adjusted R-squared	0.419257 S.D. dependent var	0.040594
	S.E. of regression	0.030935 Akaike info criterion	-3.901532
	Sum squared resid	0.025838 Schwarz criterion	-3.505652
	Log likelihood	79.22757 Hannan-Quinn criter.	-3.763359
	F-statistic	4.158454 Durbin-Watson stat	0.924663
	Prob(F-statistic)	0.002417
	Sumber: Data diolah (2020)
	Model yang terbentuk sudah cukup baik yang ditandai dengan nilai R-Square (R 2 ) sebesar
	0.5520. Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel yang memengaruhi
	biaya agensi dengan pendekatan Return On Asset (ROA) adalah penjualan total (TS).
	Dependent Variable: Y3 (Agency Cost dengan pendekatan Return On Asset)
	Method: Pooled Least Squares
	Variable	Coefficient	Std. Error	t-Statistic	Prob.
	C	0.288023	0.211804	1.359858	0.1851
	PMD	-0.300052	0.207366	-1.446967	0.1594
	BOD	0.011336	0.013149	0.862147	0.3962
	BOC	-0.004867	0.004374	-1.112768	0.2756
	BOIC	-0.033687	0.043456	-0.775195	0.4450
	BOCA	-0.001811	0.021608	-0.083814	0.9338
	TE	-1.75E-13	1.87E-13	-0.936385	0.3574
	TS	4.01E-14	8.16E-15	4.915804	0.0000
	TA	-1.01E-15	1.81E-15	-0.555209	0.5833
	R-squared	0.551998 Mean dependent var	0.032500

2. Model kedua; Model Time Effects (pemodelan dengan slope konstan dan intersep beravariasi antar waktu). Pada pemodelan ini, hasil yang diperoleh adalah sebagai berikut: a. Hasil pengujian model Time Effects pada penelitian ini dengan pendekatan Rasio Beban sebagai proksi dari biaya keagenan ditunjukkan pada Tabel 7 di bawah ini, yaitu: Tabel 7.

Hasil Pengujian dengan Pemodelan Time Effects (Agency Cost dengan Pendekatan Rasio Beban)

	3-C	-0.028535
		Effects Specification
	Period fixed (dummy variables)
	R-squared	0.692242	Mean dependent var	0.213056
	Adjusted R-squared	0.569138	S.D. dependent var	0.186881
	S.E. of regression	0.122669	Akaike info criterion	-1.112185
	Sum squared resid	0.376193	Schwarz criterion	-0.628332
	Log likelihood	31.01932	Hannan-Quinn criter.	-0.943307
	F-statistic	5.623257	Durbin-Watson stat	1.212347
	Prob(F-statistic)	0.000220
	Sumber: Data diolah (2020)
	Dependent Variable: Y1 (Agency Cost dengan pendekatan Rasio Beban)
	Method: Pooled Least Squares
	Variable	Coefficient Std. Error	t-Statistic	Prob.
	C	-1.259067	0.881089	-1.428989	0.1654
	PMD	1.931389	0.865438	2.231690	0.0348
	BOD	-0.167215	0.052494	-3.185438	0.0039
	BOC	0.019325	0.017398	1.110779	0.2772
	BOIC	-0.035214	0.183505	-0.191898	0.8494
	BOCA	0.183936	0.089635	2.052063	0.0508
	TE	3.75E-12	7.53E-13	4.980903	0.0000
	TS	-2.06E-13	3.27E-14	-6.283047	0.0000
	TA	-1.61E-14	7.51E-15	-2.138338	0.0424
	Fixed Effects (Period)
	1-C	0.014371
	2-C	0.014163

Hasil Pengujian dengan Pemodelan Time Effects (Agency Cost dengan Pendekatan Rasio Asset Utilisation) Dependent Variable: Y2 (Agency Cost dengan pendekatan Rasio Asset Utilisation) Method: Pooled Least Squares

  . Hasil pendekatan Rasio Asset Utilisation sebagai proksi dari biaya keagenan ditunjukkan pada Tabel 8 di bawah ini, yaitu:

	Model yang terbentuk sudah cukup baik yang ditandai dengan nilai R-Square (R 2 ) sebesar
	0.5183. Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel yang memengaruhi
	biaya agensi dengan pendekatan Rasio Asset Utilisation (RP) adalah penjualan total (TS). Semakin tinggi penjualan setiap tahunnya akan meningkatkan biaya agensi (Rasio Asset Utilisation) pada BUMD. c. Hasil pendekatan Return On Asset sebagai proksi dari biaya keagenan ditunjukkan pada Tabel Tabel 8. Variable Coefficient Std. Error t-Statistic Prob. 9 di bawah ini, yaitu:
	C	Tabel 9.	2.609800	5.502323	0.474309	0.6394
	PMD	-2.129389 5.404582	-0.393997	0.6969
	BOD	-0.006647 0.327818	-0.020277	0.9840
	BOC	-0.002257 0.108647	-0.020770	0.9836
	BOIC	0.381379	1.145971	0.332799	0.7421
	BOCA	-0.475404 0.559761	-0.849298	0.4038
	TE		-4.87E-12 4.70E-12	-1.035600	0.3103
	TS		9.16E-13	2.04E-13	4.484508	0.0001
	TA		-2.67E-14 4.69E-14	-0.570210	0.5736
	Fixed Effects (Period)
	1-C	0.205291
	2-C	-0.106475
	3-C	-0.098816
			Effects Specification
	Period fixed (dummy variables)
	R-squared	0.518262	Mean dependent var	0.595278
	Adjusted R-squared	0.325566	S.D. dependent var	0.932807
	S.E. of regression	0.766057	Akaike info criterion	2.551349
	Sum squared resid	14.67110	Schwarz criterion	3.035202
	Log likelihood	-34.92427 Hannan-Quinn criter.	2.720226
	F-statistic	2.689540	Durbin-Watson stat	1.065674
	Prob(F-statistic)	0.021785

). Semakin besar total aset setiap tahun akan berpengaruh negatif terhadap biaya agensi (rasio beban keuangan) pada BUMD. b

Hasil Pengujian dengan Pemodelan Time Effects (Agency Cost dengan Pendekatan Return on Asset)

  Hasil yang diperoleh pada output di atas menunjukkan bahwa variabel yang memengaruhi biaya agensi dengan pendekatan Return on Asset (ROA) adalah penjualan total (TS).Pemilihan model terbaik antara fixed effects atau random effects (baik komponen maupun waktu) dapat dicari dengan menggunakan alat analisis Pengujian Hausmann. Tabel 10 Berikut di bawah ini merupakan hasil Pengujian Hausmann terhadap berbagai pendekatan biaya agensi.Berdasarkan hasil "Cross-section random effects test comparisons" di atas, menunjukkan bahwa tidak ada perbedaan yang signifikan baik menggunakan model fixed effects maupun model random effects, sehingga penggunaan kedua model ini secara statistika bisa dianggap sama. BUMD yang ada di provinsi DKI Jakarta dalam mengelola asset yang produktif. Pengelolaan asset-asset ini berupa antara lain yaitu: iklan, sewa ruang pertemuan, sewa hotel, sewa gedung, dan lain lain sesuai dengan peruntukan dan klaster industri BUMD di Provinsi DKI Jakarta. Hal inilah yang menyebabkan biaya keagenan (agency cost) menjadi turun.2. Agency Cost dengan proksi Rasio Penggunaan Aset (Asset Utilisation)

	F-statistic mengingat fungsi BUMD Provinsi DKI Jakarta yang berperan bukan saja sebagai agen 3.111777 Durbin-Watson stat 0.926601 TA -0.000000 -0.000000 0.000000 0.3268 Chi-Sq.
	Prob(F-statistic) Cross-section random effects test equation: 0.010311 Test Summary yang bersifat profit oriented melainkan juga social oriented (public services). Statistic Chi-Sq. d.f. Prob.
	Sumber: Data diolah (2020) Dependent Variable: Y1 (Agency Cost dengan pendekatan Rasio Beban) Cross-section random 0.000000 6 b. Jumlah Dewan Direksi (BOD); Sebagai perpanjangan tangan Gubernur DKI Jakarta, 1.0000
	Method: Panel Least Squares Dewan Direksi dapat membuat garis kebijakan perusahaan sesuai mandat yang
	Model yang terbentuk sudah cukup baik yang ditandai dengan nilai R-Square (R 2 ) sebesar Variable Coefficient Std. Error t-Statistic Prob. * Cross-section test variance is invalid. Hausman statistic set to zero. diterimanya. Keberadaan direksi pada BUMD DKI Jakarta dinilai mampu
	C PMD BOD Cross-section random effects test comparisons: -0.474660 3.017825 0.127850 3.063104 NA NA Variable Fixed Random merepresentasikan keterwakilan gubernur, sehingga mampu menekan biaya agensi yang -0.157285 0.8770 0.041739 0.9672 NA NA Var(Diff.) Prob. dikeluarkan. c. Jumlah Komite Audit (BOCA); Kinerja komite audit diperlukan untuk mengawasi dalam 0.5545. Pemilihan Model Terbaik BOC 0.385118 0.253830 1.517228 0.1487 PMD 0.666977 -2.304294 478.152022 0.8919 proses kegiatan, administrasi keuangan dan pembuatan laporan pertanggungjawaban
	BOIC BOC keuangan tahunan BUMD, meskipun jika terjadi penambahan jumlah/personel komite NA NA NA NA 0.141495 -0.003359 3.427523 0.9376
	BOCA BOCA audit mengakibatkan kenaikan biaya agensi dan menjadi beban bagi BUMD. 0.129434 0.093580 1.383131 0.1856 0.071563 -0.227623 0.239038 0.5406
	Dependent Variable: Y3 (Agency Cost dengan pendekatan Return on Asset) TE 3.11E-12 1.39E-12 2.241541 0.0395 TE 0.000000 -0.000000 0.000000 0.4876 d. Total Pengeluaran (TE); Peningkatan peran serta yang bersumber dari modal daerah
	Method: Pooled Least Squares Variable Coefficient Tabel 10. Hasil Pengujian Hausmann TS -2.20E-13 TA -3.44E-14 (APBD) Provinsi DKI Jakarta bagi BUMD terus meningkat sebagai upaya untuk Std. Error t-Statistic Prob. 1.58E-13 -1.390379 0.1835 1.75E-14 -1.965269 0.0670 Sumber: Data diolah (2020) membiayai kegiatan-kegiatannya. Penyebab besarnya pengeluaran BUMD disebabkan
	C oleh biaya operasional BUMD Provinsi DKI Jakarta mengalami peningkatan dari tahun ke 0.307356 0.230265 1.334792 Effects Specification 0.1940 PMD -0.320501 0.226175 -1.417050 0.1688 BOD 0.011794 0.013719 0.859735 Correlated Random Effects -Hausman Test Cross-section fixed (dummy variables) tahun sehingga menjadi beban biaya tersendiri bagi BUMD Provinsi DKI Jakarta. Pool: POOL01 0.3981 BOC -0.004966 0.004547 -1.092209 0.2852 BOIC -0.039841 0.047957 -0.830757 0.4140 BOCA 0.000249 0.023425 0.010631 0.9916 TE -1.88E-13 1.97E-13 -0.953833 e. Total Penjualan (TS); Tingginya penjualan produk (layanan) BUMD Provinsi DKI Jakarta Test cross-section random effects Test Summary R-squared 0.893652 Mean dependent var 0.213056 Adjusted R-squared 0.767364 S.D. dependent var tidak terlepas dari peran serta masyarakat yang sering memanfaatkan produk-produk yang 0.186881 Chi-Sq. Statistic Chi-Sq. d.f. Prob. S.E. of regression 0.090137 Akaike info criterion -1.674786 dikeluarkan oleh BUMD dengan harga yang sangat terjangkau. Hal ini tentu berdampak Pembahasan 0.3493 TS 4.04E-14 8.55E-15 4.725623 0.0001 TA -8.69E-16 1.96E-15 -0.442892 0.6617 Fixed Effects (Period) 1-C -0.002049 2-C 0.003015 Cross-section random 0.000000 6 1.0000 * Cross-section test variance is invalid. Hausman statistic set to zero. ** WARNING: estimated cross-section random effects variance is zero. Sum squared resid 0.129996 Schwarz criterion -0.795053 Log likelihood 50.14614 Hannan-Quinn criter. kepada penerimaan yang diperoleh oleh BUMD sehingga pada akhirnya biaya keagenan Berdasarkan kriteria pemilihan model terbaik pada bagian Tabel 10, baik pada model -1.367736 F-statistic 7.076312 Durbin-Watson stat yang menjadi beban keuangan BUMD menjadi semakin mengecil. pertama maupun model kedua memberikan hasil yang sama secara statistik pada masing-masing 2.786547 Prob(F-statistic) f. Total Asset (TA); Tingginya penjualan dengan memanfaatkan aset-aset yang dimiliki oleh proksi agency cost, sehingga hasil penelitian yang mengacu pada Tabel 4, Tabel 5, Tabel 6, Tabel 0.000126 7, Tabel 8, dan Tabel 9 di atas, dapat diuraikan sebagai berikut: BUMD menunjukkan kemampuan BUMD-
	3-C Cross-section random effects test comparisons: -0.000966 1. Agency Cost dengan proksi Rasio Beban
	Effects Specification Berdasarkan hasil analisis data pada Tabel 4 dan 7, diperoleh hasil variabel-variabel yang
	Period fixed (dummy variables) Variable memengaruhi Agency Cost dengan proksi Rasio Beban adalah sebagai berikut: Fixed Random Var(Diff.) Prob.
	PMD a. Penyertaaan Modal Daerah (PMD); Penyertaan modal daerah oleh Pemda DKI Jakarta 0.127850 2.143975 9.058094 0.5029
	R-squared Adjusted R-squared BOC Correlated Random Effects -Hausman Test 0.554508 Mean dependent var 0.376312 S.D. dependent var 0.385118 0.020446 0.064285 memiliki target/tujuan yang relevan dan harus dijalankan oleh BUMD. Besarnya 0.032500 0.1504 0.040594 S.E. of regression 0.032059 Akaike info criterion BOCA 0.129434 0.191129 0.005234 0.3938 Pool: POOL01 Penyertaan Modal Daerah (PMD) yang dilakukan oleh Pemda DKI Jakarta kepada BUMD -3.796039 Sum squared resid 0.025694 Schwarz criterion -3.312186 TE 0.000000 0.000000 0.000000 0.5940 Test cross-section random effects memiliki konsekuensi praktis akan menimbulkan kenaikan biaya keagenan (agency cost)
	Log likelihood TS	79.32870 Hannan-Quinn criter. -0.000000 -0.000000 0.000000	-3.627161 0.9476
		20