Asymptotic efficiency of exponentiality tests based on

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Georgian Math. J. 17 (2010), 749 – 763 DOI 10.1515 / GMJ.2010.034

© de Gruyter 2010

Asymptotic efficiency of exponentiality tests based on order statistics characterization Ya. Yu. Nikitin and Ksenia Yu. Volkova

Abstract. We propose new scale-invariant tests for exponentiality based on the characterization in terms of order statistics. Limiting distributions and large deviations of new statistics are described and their local Bahadur efficiency for common alternatives is calculated. Keywords. Testing of exponentiality, order statistics, U -statistics, Bahadur efficiency. 2010 Mathematics Subject Classification. 60F10, 62F03, 62G20, 62G30. Dedicated to Yu. V. Prokhorov on the occasion of his 80th birthday

1

Introduction

The paper concerns invariant testing of exponentiality using the Ahsanullah’s characterization [2]. Let X1 ; : : : ; Xn be non-negative independent observations having a distribution function (df) F and a density f: We wish to test the composite nullhypothesis H0 W F 2 E./; where E./ denotes the class of exponential distributions with an unknown scale parameter > 0: There exist numerous test statistics for this problem, which are based on various ideas, see the reviews [5], [9], [11] and the books [7], [16] and [18]. Note that some of exponentiality tests are “omnibus”, so that they are consistent against any alternative, while many other tests are designed to discover particular classes of alternatives to exponentiality. Some tests of exponentiality are based on the loss-of-memory property, see [1], [3], [15], and a limited number of tests use other characterizations of exponentiality [8], [20], [23], [24]. Constructing and exploring new tests based on the characterizations are of considerable interest as they may lead to new consistent and efficient tests. The paper was partially supported by the RFBR grant No. 10-01-00154-a and by the grant FZP No. 2010-1.1-111-128-033.



750

Ya. Yu. Nikitin and K. Yu. Volkova

Suppose the df F belongs to the class F of life distributions which satisfy the following conditions: F is strictly monotone and the hazard function f .t /=.1 F .t // is monotonically increasing or decreasing for all t 0: Ahsanullah proved in [2] the following characterization of exponentiality within the class F : Let X1 ; : : : ; Xk be a sample of non-negative rv’s having a distribution from the class F : If, for some i and j , 1 i < j < k, the statistics .k i /.XiC1;k Xi;k / and .k j /.Xj C1;k Xj;k / are identically distributed, then the sample is exponentially distributed. As usual Xj;k denotes here the j -th order statistic of the sample X1 ; : : : ; Xk ; 1 j k: Consider the simplest case of this characterization when k D 3, i D 1 and j D 2. We get the following statement: Suppose that X1 ; X2 and X3 are independent non-negative observations with the df from the class F : If two statistics 2.X2;3 X1;3 / and X3;3 X2;3 have the same distribution, then F 2 E./. The proof of this assertion in [2] uses the properties of the class F , but possibly the result is true for a larger class of life distributions. The aim of the present paper is to test the hypothesis H0 using this characterization against the alternatives from the class F : The equivalent formulation is testing of no ageing against positive (negative) ageing. This problem is of significance and has been intensively investigated. There exist numerous tests of exponentiality against the alternatives with monotone failure rate, see, e.g., [4] and [16] for their description. However, no one test dominates the others, and their efficiencies strongly depend on the presumed alternatives. Therefore it seems reasonable to build and to explore new tests of exponentiality hoping that they turn up efficient and even asymptotically optimal for certain “most favorable” alternatives. The idea of testing statistical hypotheses using the equidistribution of linear forms goes back to Linnik [17]. The use of order statistics in this context is relatively new. P Let Fn .t / D n1 niD1 1¹Xi < t º, t 0; be the usual empirical df. Consider two V -empirical df’s Gn .t / D

n 1 X 1¹2.X2;¹i;j;kº X1;¹i;j;kº / < t º; n3

t 0;

i;j;kD1

Hn .t / D

n 1 X 1¹X3;¹i;j;kº X2;¹i;j;kº < t º; n3

t 0;

i;j;kD1

where Xs;¹i;j;kº denotes the s-th order statistic of the subsample .Xi ; Xj ; Xk /; s D 1; 2; 3: It is known that the properties of V - and U -empirical df’s are similar to the properties of usual empirical df’s, see [10], [13]. Hence for large n the df’s



Asymptotic efficiency of exponentiality tests

751

Gn and Hn should be close under H0 ; and we can measure their closeness using some test statistics. We suggest two scale-invariant statistics Z

1

.Hn .t / Gn .t // dFn .t /;

Sn D

(1)

0

Rn D sup jHn .t / Gn .t /j;

(2)

t0

assuming that their large values are critical. We discuss their limiting distributions under the null hypothesis and calculate their efficiencies against common alternatives from the class F like Weibull, Makeham, Gamma and linear failure rate (LFR) distributions. We use the notion of local exact Bahadur efficiency (BE) [6], [19], as the statistic Rn has the nonnormal limiting distribution, hence the Pitman approach to efficiency is not applicable. However, it is known that the local BE and the limiting Pitman efficiency usually coincide, see [19], [26]. The large deviation asymptotics is the key tool for the evaluation of the exact BE, and we address this question using the results of [22]. Finally, we study the conditions of local optimality of our tests and describe the “most favorable” alternatives for them. We do not discuss the practical implementation of new tests and concentrate on their mathematical aspects.

2

Limiting distribution under exponentiality of the statistic Sn

Without loss of generality, we may assume that D 1. The statistic (1) is a V -statistic and admits the representation Sn D

1 n4

n X

1 n4

1 D 4 n

1¹X3;¹i;j;kº X2;¹i;j;kº < Xl º

i;j;k;lD1 n X

1¹2.X2;¹i;j;kº X1;¹i;j;kº / < Xl º

i;j;k;lD1 n X

ˆ.Xi ; Xj ; Xk ; Xl /;

i;j;k;lD1



752


where the centered kernel ˆ.Xi ; Xj ; Xk ; Xl / of degree 4 satisfies the relation 4ˆ.Xi ; Xj ; Xk ; Xl / D 1¹X3;¹i;j;kº X2;¹i;j;kº < Xl º C 1¹X3;¹j;k;lº X2;¹j;k;lº < Xi º C 1¹X3;¹k;l;iº X2;¹k;l;iº < Xj º C 1¹X3;¹l;i;j º X2;¹l;i;j º < Xk º 1¹2.X2;¹i;j;kº X1;¹i;j;kº / < Xl º 1¹2.X2;¹j;k;lº X1;¹j;k;lº / < Xi º 1¹2.X2;¹k;l;iº X1;¹k;l;iº / < Xj º 1¹2.X2;¹l;i;j º X1;¹l;i;j º / < Xk º: First we calculate the projection of this kernel. Let X; Y; Z; W be independent standard exponential rv’s. Then .s/ D E.ˆ.X; Y; Z; W / j X D s/ 3 D P .X3;¹X;Y;Zº X2;¹X;Y;Zº < W j X D s/ 4 3 P .2.X2;¹X;Y;Zº X1;¹X;Y;Zº / < W j X D s/ 4 1 C P .X3;¹Y;Z;W º X2;¹Y;Z;W º < s/ 4 1 P .2.X2;¹Y;Z;W º X1;¹Y;Z;W º / < s/: 4 It follows from Ahsanullah’s characterization that the last two lines disappear. Let us find P .2.X2;¹X;Y;Zº X1;¹X;Y;Zº / < W j X D s/: Note that P .2.X2;¹X;Y;Zº X1;¹X;Y;Zº / < W j X D s/ D P .2.X2;¹s;Y;Zº X1;¹s;Y;Zº / < W / Z 1 t D P X2;¹s;Y;Zº X1;¹s;Y;Zº < e t dt: 2 0 It is evident that t P X2;¹s;Y;Zº X1;¹s;Y;Zº < 2 t D 2P X2;¹s;Y;Zº X1;¹s;Y;Zº < ; 0 < s < Y < Z 2 t C 2P X2;¹s;Y;Zº X1;¹s;Y;Zº < ; 0 < Y < s < Z 2 t C 2P X2;¹s;Y;Zº X1;¹s;Y;Zº < ; 0 < Y < Z < s : 2



753


After a series of calculations, we obtain that P

t X2;¹s;Y;Zº X1;¹s;Y;Zº < D e 2s .1 e t / 2 ³ ³ ² ² t t t=2 2s t=2 .1 e .1 e 2s /: C e .e 1// C 1 s < C1 s > 2 2

Integrating this expression with the weight exp.t / over RC , we get finally P .2.X2;¹X;Y;Zº X1;¹X;Y;Zº / < W j X D s/ D

1 3 2s 4 3s C e e : 3 2 3

Similar arguments give us P .X3;¹X;Y;Zº X2;¹X;Y;Zº < W j X D s/ D P .X3;¹s;Y;Zº X2;¹s;Y;Zº < W / Z 1 D P .X3;¹s;Y;Zº X2;¹s;Y;Zº < t /e t dt: 0

The probability under the integral sign is treated as above so that after integration we have 3 P .X3;¹X;Y;Zº X2;¹X;Y;Zº < W j X D s/ D e 2s C e s .2s 1/; 2

s 0:

Combining the results obtained, we get the following formula for the projection: 3 1 .s/ D e s .2s 1/ C e 3s ; 4 4 Hence the variance 2 WD E

2 .X /

s 0:

(3)

under H0 is equal to

2 D

19 : 4200

It follows that the kernel of our V -statistic is non-degenerate. We can apply Hoeffding’s theorem [14] on the asymptotic normality of V - and U -statistics which implies that as n ! 1 p

d

nSn ! N

0;

38 : 525



754


Large deviations and local efficiency of the statistic Sn

3

The kernel ˆ is centered, bounded and non-degenerate. According to the large deviation theorem for such statistics from [22], we obtain for a > 0 under H0 : lim n1 ln P .Sn > a/ D fS .a/;

n!1

where the function fS is continuous for sufficiently small a > 0; and fS .a/

a2 525 2 a 2 76 32

as a ! 0:

Suppose that under the alternative H1 the observations have the df G. ; / with the density g. ; /; 0; such that G. ; 0/ 2 E./: The measure of BE for any sequence ¹Tn º of test statistics is the exact slope cT . / describing the rate of exponential decrease for the attained level under the alternative. According to Bahadur theory [6], [19] the exact slopes may be found by using the following result. Proposition. Suppose that the following two conditions hold: a)

P

Tn ! b. /;

> 0; P

where 1 < b. / < 1, and ! denotes convergence in probability under G. I /. b)

lim n1 ln PH0 .Tn t / D h.t /

n!1

for any t in an open interval I; on which h is continuous and ¹b. /; > 0º I . Then cT . / D 2 h.b. //: Note that the exact slopes always satisfy the inequality [6], [19] cT . / 2K. /;

> 0;

(4)

where K. / is the Kullback–Leibler “distance” between the alternative and the null-hypothesis H0 : In our case H0 is composite, hence for any alternative density gj .x; / one has Z 1 lnŒgj .x; /= exp.x/gj .x; / dx: Kj . / D inf >0 0

This quantity can be easily calculated as ! 0 for particular alternatives. According to (4), the local BE of the sequence of statistics Tn is defined as cT . / : 2K. / !0

e B .T / D lim



755


Now we will give some examples of efficiency calculations. First consider the Weibull alternative with the density g1 .x; / D .1 C /x exp.x 1C /;

0; x 0;

and the corresponding df G1 .x; /. According to the Law of Large Numbers for U - and V -statistics [14], the limit in probability under H1 is equal to bS . / D P .X3;3 X2;3 < X4 / P .2.X2;3 X1;3 / < X4 / Z 1Z 1Z 1Z 1 ˆ.x; y; z; w/g1 .x; /g1 .y; /g1 .z; /g1 .w; /dxdydzdw: D 0

0

0

0

R1 It is easy to show (see also [21]) that bS . / 4 0 .s/h1 .s/ds; where h1 .x/ D @ g .x; /jD0 and .s/ is the projection from (3). Therefore for the Weibull @ 1 alternative we have 1

Z bS . / 4 0

3 s 1 s e .2s 1/ C e 3s e .1 C ln s s ln s/ds 4 4 3 D .1 ln 2/ 0:2301 ; ! 0; 4

and the local exact slope of the sequence Sn as ! 0 admits the representation cS . / D bS2 . /=.162 / 0:7317 2 : The Kullback–Leibler “distance” K1 . / between the Weibull distribution and the class E./ satisfies K1 . / 2 2 =12; ! 0: Hence the local BE is equal to cS . / 0:445: !0 2K1 . /

e B .S/ D lim Consider the Makeham density

g2 .x; / D .1 C .1 e x // exp.x .e x 1 C x//;

0; x 0;

as a second alternative to exponentiality. After a series of calculations, we have: 1 bS . / 15 ; ! 0: The local exact slope admits the representation cS . / 2 0:0614 ; ! 0; while K2 . / 2 =24; ! 0: Consequently, the local efficiency of the test equals 0.737. The third alternative is the linear failure rate (LFR) density 1

2

g3 .x; / D .1 C x/e x 2 x ;

0; x 0:



756


3 Omitting the calculations similar to previous cases we get bS . / 16 , cS . / 2 2 0:4857 , ! 0. It is easy to show that K3 . / =2; ! 0. Therefore the local BE is equal to 0.486. The last alternative is the Gamma-density

x e x ; . C 1/

g4 .x; / D

0; x 0: 2

In this case we get bS . / 0:1137 , cS . / 0:1786 2 , K4 . / . 12 12 / 2 , ! 0: Hence the local BE equals 0.277. Table 1 collects the values of local BE. Alternative

Weibull

Makeham

LFR

Gamma

Efficiency

0.445

0.737

0.486

0.277

Table 1. Local Bahadur efficiencies of the statistic Sn :

4

Limiting distribution under exponentiality of the statistic Rn

Now consider the Kolmogorov type statistic (2). In this case, for fixed t 0, the difference Hn .t / Gn .t / is the family of V -statistics with the kernels depending on t 0 W „.X; Y; ZI t / D 1¹X3;¹X;Y;Zº X2;¹X;Y;Zº < t º 1¹2.X2;¹X;Y;Zº X1;¹X;Y;Zº / < t º: Let us calculate the projections of these kernels for fixed t : .sI t / WD E.„.X; Y; ZI t / j X D s/ D P .X3;¹s;Y;Zº X2;¹s;Y;Zº < t / P .2.X2;¹s;Y;Zº X1;¹s;Y;Zº / < t /: The required probabilities have been calculated above. Combining the results obtained, we find that ³ ² t .sI t / D 2e s .1 e s /.1 e t / C 1 s < .2e 2s 2e s / 2 ³ ² t t t < s < t .e 2 e 2s .e 2 2/ 2e s / C1 2 t

t

C 1¹s > t º.e 2 1 e 2s .e 2t C e 2 2/ C 2e s .e t 1//:



757


0.025

0.02

0.015

0.01

0.005

0

2

4

6

8

10

t

Figure 1. Plot of the function ı2 .t /.

Now we calculate ı2 .t / WD E 2 .X I t / under H0 : We get t 3t e t .3 2e 2 C 2e t 14e 2 C 13e 2t 2e 3t /; t 0: 15 It is seen that our family of kernels „.X; Y; ZI t / is non-degenerate in the sense of [20] and ı2 D sup t0 ı2 .t / 0:02646, see Figure 1. This value will be important in the sequel when calculating the large deviation asymptotics. The limiting distribution of the sequence of statistics Rn is unknown. Using the arguments of [25], one can show that the V -empirical process p n .t / D n.Hn .t / Gn .t //; t 0;

ı2 .t / D

converges weakly in D.0; 1/ as n ! 1 to a centered Gaussian process .t / p with complicated covariance. Then the sequence of statistics nRn converges in distribution to the value sup t0 j .t /j; which unfortunately has the intractable distribution. Hence we recommend the evaluation of critical values for Rn by simulation.

5

Large deviations and the local efficiency of statistics Rn

Now we obtain the logarithmic large deviation asymptotics of the sequence of statistics (2) under H0 : The family of kernels „.X; Y; ZI t / is centered, bounded and non-degenerate, hence by Theorem 2.4 of [20] we get for a > 0 lim n1 ln P .Rn > a/ D fR .a/;

n!1



758


where the function fR is continuous for sufficiently small a > 0; and fR .a/ D .18ı2 /1 a2 .1 C o.1// 2:100a2 .1 C o.1//

as a ! 0:

To evaluate the efficiency, first consider the Weibull alternative with the density g1 .x; / given above. By the Glivenko–Cantelli theorem for V -statistics [13] the limit in probability under the alternative for statistics Rn is equal to bR . / WD sup jbR .t; /j D sup jP .X3;3 X2;3 < t / P .2.X2;3 X1;3 / < t /j: t0

t0

It is not difficult to show that

Z

1

bR .t; / 3

.sI t /h1 .s/ds; 0

@ where again h1 .s/ D @ g1 .s; / jD0 and .sI t / is the projection defined above. Hence for the Weibull alternative we have for t 0 W

1 bR .t; / e t .ln t C C 13 ln 2 C 6t ln 2 8 ln 3/ 6 2 2t 2 t=2 3 t e Ei.1; 2t / e Ei.1; 3t / e Ei.1; 3t =2/ ; ! 0; C 2 3 3 R 1 zt k where Ei.k; z/ D 1 e t dt; k 0; Re z > 0; is the exponential integral while 0:5772 is the Euler constant. Thus bR . / D sup t0 jbR .t; /j 0:3309 ; and it follows that the local exact slope of the sequence of statistics Rn admits the representation: 2 . /=.9ı2 / 0:4599 2 ; cR . / bR

! 0: 2 2

The Kullback–Leibler “distance” in this case satisfies K1 . / 12 , ! 0, and the local BE is 0:280: Next we take the Makeham distribution, where the calculations are similar, and the local BE is equal to 0:513: In the case of the LFR density and the Gamma density we find that the local BE’s are 0.362 and 0.174. We collect the values of local BE in the Table 2. Alternative

Weibull

Makeham

LFR

Gamma

Efficiency

0.280

0.513

0.362

0.174

Table 2. Local Bahadur efficiencies of the statistic Rn :

We observe that the efficiencies for the Kolmogorov-type test are lower than for the integral test. However, this is the usual situation when testing goodness-offit [19], [20], [24].




6

759

On the consistency of the proposed tests

Consistency is the important and desirable property of a test. The Kolmogorovtype test of exponentiality based on the statistic Rn is consistent against any alternative from the class F : By the Glivenko–Cantelli theorem for U -empirical df’s [13] the limit in probability of Rn under the alternative G from this class is equal to sup jPG .2.X2;3 X1;3 / < t / PG .X3;3 X2;3 < t /j t0

and is positive due to the Ahsanullah’s characterization of exponentiality within the class F . This obviously implies the consistency, see also [12], §1.3. In terms of Bahadur theory, the consistency under the positive alternative value of is equivalent to the condition b. / > 0; where b. / is defined in the Proposition from Section 3. The integral test based on the statistic Sn is not necessarily consistent for any alternative from the class F , but in fact is consistent for many of them. In Section 3 we have found that the function bS . / is positive for small positive under the Weibull, Makeham, LFR and Gamma alternatives that ensures consistency in all these cases.

7

Conditions of local asymptotic optimality

The efficiency values of our tests for standard alternatives are far from maximal ones. Nevertheless, there exist such special alternatives (we call them most favorable) for which our sequences of statistics Sn and Rn are locally asymptotically optimal (LAO) in Bahadur sense, see the general theory in [19, Ch.6]. Denote by G the class of densities g. ; / with the df G. ; / which satisfy the regularity conditions listed below. Consider the functions H.x/ D

ˇ @ ˇ G.x; /ˇ ; D0 @

h.x/ D

ˇ @ ˇ g.x; /ˇ : D0 @

Suppose also that the following regularity conditions hold: 0

Z

h.x/ D H .x/; @ @

Z

1 0

1

x 0;

ˇ ˇ xg.x; /dx ˇ

D0

0

h2 .x/e x dx < 1;

Z

1

D

xh.x/dx: 0



760


It is easy to show, see also [23], that under these conditions ²Z 1 Z 1 2 ³ 2K. / 2; h2 .x/e x dx xh.x/dx 0

! 0:

0

R1 We recall that for the integral statistic (1) we have bS . / 4 0 Let us introduce the auxiliary function Z 1 h0 .x/ D h.x/ .x 1/ exp.x/ uh.u/du:

.s/h.s/ds:

0

It is straightforward that Z Z 1 2 s h .s/e ds 0

Z

2

1

sh.s/ds

Z

1

D

0 1

Z

0 1

.s/h.s/ds D 0

h20 .s/e s ds; .s/h0 .s/ds:

0

Consequently the local BE takes the form e B .S / D lim bS2 . /=.322 K. // !0

Z

1

.s/h0 .s/ds

D

2 Z ı

0

1

2

0

.s/e x dx

Z

1 0

h20 .x/e x dx :

The local Bahadur asymptotic optimality means that the expression on the righthand side is equal to 1. It follows from the Cauchy–Schwarz inequality, see also [21], that this happens iff h0 .x/ D e x .C1 .x/ C C2 .x 1// for some constants C1 > 0 and C2 ; so that h.x/ D g .x; 0/ takes the same form. Such distributions constitute the LAO class in the class G . Now consider the Kolmogorov-type statistic (2) with the family of kernels „.X; Y; ZI t / and the projection .sI t /: In this case the efficiency is equal to 2 e B .R/ D lim bR . /= sup.18ı2 .t //K. / !0

Z

2 .sI t /h0 .s/ds

D sup t0

t0

1

0

ı

Z

1

sup t0

2 .s; t /e x dx

0

Z 0

1

h20 e x dx :

It follows that the sequence of statistics Rn is locally optimal iff h.x/ D e x .C1 .x; t0 / C C2 .x 1// for t0 D arg max t0 ı2 .t / 1:1508 and some constants C1 > 0; C2 : The distributions having such a function h.x/ form the domain of LAO in the corresponding class. Acknowledgments. The authors are grateful to the referee for his/her useful comments and valuable suggestions.




761

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[14] V. S. Korolyuk and Yu. V. Borovskikh, Theory of U -Statistics, Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors. Mathematics and its Applications, 273. Kluwer Academic Publishers Group, Dordrecht, 1994. [15] H. L. Koul, A test for new better than used, Comm. Statist.—Theory Methods A6:6 (1977), 563–573. [16] C.-D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability, Springer, New York, 2006. [17] Yu. V. Linnik, Linear forms and statistical criteria, I, II, Selected Transl. Math. Statist. and Prob., Vol. 3, pp. 1–40 and 41–90; Amer. Math. Soc., Providence, R.I., 1962; Ukrain. Mat. Zh. 5(1953), 207–243 and 247–290 (in Russian). [18] P. Nabendu, J. Chun and R. Crouse, Handbook of Exponential and Related distributions for Engineers and Scientists, Chapman and Hall, 2002. [19] Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press, Cambridge, 1995; Russian original: Probability Theory and Mathematical Statistics, 45. Fizmatlit “Nauka”, Moscow, 1995. [20] Ya. Yu. Nikitin, Large deviations of U -empirical Kolmogorov-Smirnov tests and their efficiency. Journ. of Nonparam. Statist. 22:4 (2010), 649–668. [21] Ya. Yu. Nikitin and I. Peaucelle, Efficiency and local optimality of nonparametric tests based on U - and V -statistics, Metron 62:2 (2004), 185–200. [22] Ya. Yu. Nikitin and E. V. Ponikarov, Rough asymptotics of probabilities of large deviations of Chernoff type for von Mises functionals and U -statistics, (Russian) in: Proceedings of the St. Petersburg Mathematical Society, Vol. 7 (Russian), 124– 167, Tr. St.-Peterbg. Mat. Obshch., 7, Nauchn. Kniga, Novosibirsk, 1999; English translation in: Proceedings of the St. Petersburg Mathematical Society, Vol. VII, 107–146, Amer. Math. Soc. Transl. Ser. 2, 203, Amer. Math. Soc., Providence, RI, 2001. [23] Ya. Yu. Nikitin and A. V. Tchirina, Bahadur efficiency of a test of exponentiality based on a loss-of-memory type functional equation, J. Nonparametr. Statist. 6:1 (1996), 13–26. [24] R. F. Rank, Statistische Anpassungstests und Wahrscheinlichkeiten grosser Abweichungen, Dissertation, Hannover, 1999, 240 pp. [25] B. W. Silverman, Convergence of a class of empirical distribution functions of dependent random variables, Ann. Probab. 11:3 (1983), 745–751. [26] H. S. Wieand, A condition under which the Pitman and Bahadur approaches to efficiency coincide, Ann. Statist. 4:5 (1976), 1003–1011.

Received April 12, 2010; revised July 29, 2010.



Asymptotic efficiency of exponentiality tests Author information Ya. Yu. Nikitin, Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, Stary Peterhof, 198504, Russia. E-mail: [email protected] Ksenia Yu. Volkova, Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, Stary Peterhof, 198504, Russia. E-mail: [email protected]


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Asymptotic efficiency of exponentiality tests based on

Asymptotic efficiency of exponentiality tests based on

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