exponentiality tests based on ahsanullah's

Journal of Mathematical Sciences, Vol. 000, No. 00, 00, 2013

EXPONENTIALITY TESTS BASED ON AHSANULLAH’S CHARACTERIZATION AND THEIR EFFICIENCY K. Yu. Volkova∗ and Ya. Yu. Nikitin∗

UDC 519.2

We construct integral and supremum type tests of exponentiality based on Ahsanullah’s characterization of the exponential law. We discuss limiting distributions and large deviations of new test statistics under the null-hypothesis and calculate their local Bahadur efficiency under common parametric alternatives. Conditions of local optimality of the new statistics are given. Bibliography: 33 titles.

1. Introduction Exponential distribution plays an important role in probability and mathematical statistics, and models with exponentially distributed observations are often met in applications, such as reliability theory, analysis of data of lifetime, analysis of viability of systems, and so on. For that reason, testing of the exponentiality hypothesis is one of the most important problems in testing of statistical hypotheses. There exist numerous tests for checking exponentiality based on various ideas (see [7, 9, 12, 13, 21, 22] etc.). One can mention several characterizations which were bases for exponentiality tests, such as the characterization based on the loss-of-memory property [3, 6, 19] and others [1, 10, 15, 20, 24, 29–31]. The general statement of the problem of exponentiality checking is formulated as follows. Let X1 , X2 , . . . , Xn be independent, identically distributed random values (i.i.d.r.v.’s) having a continuous distribution function (d.f.) F . We check the complicated exponentiality hypothesis: F (x) is the d.f. of the exponential law with density f (x) = λe−λx , x ≥ 0, where λ is an unknown parameter. Assume that the d.f. F belongs to a class of distributions F that satisfies the following conditions: F is strictly monotonous, and the function of intensity of failures f (t)/(1 − F (t)) either monotonically increases or monotonically decreases for all t ≥ 0. In the paper [4], Ahsanullah proved the following characterization of the exponential law by properties of order statistics within the class F. Let X1 , . . . , Xn be nonnegative i.i.d.r.v.’s with d.f. from the class F. If for some i and j, the statistics (k − i)(Xi+1,k − Xi,k ) and (k − j)(Xj+1,k − Xj,k ) are identically distributed for 1 ≤ i < j < k, then the sample has the exponential law of distribution. Sometimes, the problem of checking exponentiality in the class F is called checking of absence of ageing against the alternative that the positive (negative) ageing is observed. Such problems are important for reliability theory, and the corresponding field is now actively developed, see, e.g., [5, 21]. We consider a particular case of this characterization where k = 2, i = 0, and j = 1. In this case, the characterization takes the following form. Let X and Y be nonnegative i.i.d.r.v.’s with d.f. from the class F. If the statistics |X − Y | and 2 min(X, Y ) are identically distributed, then X has the exponential law of distribution. Consider exponentiality tests using this charactization against alternatives from the class F. ∗

St.-Petersburg State University, St.-Petersburg, Russia, e-mail: [email protected], [email protected].

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 412, 2013, pp. 69–87. Original article submitted February 17, 2013. 1072-3374/13/00000-0001 ©2013 Springer Science+Business Media New York 1

Construct the empirical d.f. Fn (t) = n−1

n  i=1

1{Xi < t}, t ∈ R1 , based on a sample X1 , . . . , Xn .

According to Ahsanullah’s characterization, construct two so-called V -empirical d.f.’s (see [17, 18]) by the formulas n 1  1{|Xi − Xj | < t}, t ≥ 0, Hn (t) = 2 n i,j=1

and Gn (t) =

n 1  1{2 min(Xi , Xj ) < t}, n2

t ≥ 0.

i,j=1

It is known that properties of V - and U -empirical d.f.’s are similar to those of a usual d.f., see [14, 17]. Consequently, for large n, the d.f.’s Hn and Gn must be close under the basic hypothesis H0 , and we can measure their closeness by using some test statistics. To check the exponentiality hypothesis, we consider two new statistics that are invariant with respect to the scale parameter λ: ∞ (Hn (t) − Gn (t)) dFn (t)

In =

(1)

0

and Dn = sup | Hn (t) − Gn (t) | .

(2)

t≥0

We describe limit distributions and large deviations for both sequences of statistics under the hypothesis H0 , calculate their local Bahadur efficiency for a series of alternatives, and study for them conditions of local asymptotical optimality. For that, we apply the necessary information of the theory of U - and V -statistics and the theory of Bahadur asymptotic efficiency which can be found in the monographs [8, 11, 18] and [23]. We select as the method of calculation of the asymptotic relative efficiency (ARE) of our tests the Bahadur approach since the statistic Dn of Kolmogorov type is not asymptoticaly normal, and the Pitman approach is not applicable to it. In the case of the integral statistic In , the Bahadur local ARE and the Pitman efficiency coincide, see [23, 33]. 2. Statistics In Without loss of generality, we assume that λ = 1. The statistic In is asymptotically equivalent to the U -statistic of degree 3 with centered kernel, 3Ψ(X, Y, Z) = 1{|X − Y | < Z} + 1{|Y − Z| < X} + 1{|X − Z| < Y } −1{2 min(X, Y ) < Z} − 1{2 min(Y, Z) < X} − 1{2 min(X, Z) < Y }. Let X, Y , and Z be independent r.v.’s having the standard exponential distribution. It is known that nondegenerate U - and V -statistics are asymptotically normal, see [16, 18]. To show that the kernel Ψ(X, Y, Z) is nondegenerate, let us calculate its projection ψ. For a fixed X = s, the projection of the kernel has the form 1 2 ψ(s) = E(Ψ(X, Y, Z) | X = s) = P(|Y − s| < Z) + P(|Y − Z| < s) 3 3 1 2 − P(2 min(s, Y ) < Z) − P(2 min(Y, Z) < s). 3 3 2

It is easy to calculate the first and third probabilities: ∞ P(|Y − s| < Z) =

−z

e

z+s s+z    s  −y −z −y −s 1 +s dz e dy + e dz e dy = e 2 0

s

0

s−z

and P(2 min(s, Y ) < Z) =

2 −3s 1 e + . 3 3

The Ahsanullah characterization implies that P(2 min(Y, Z) < s) = P(|Y − Z| < s) = 1 − e−s . Finally, we get the following expression for the projection of the kernel of Ψ: 4 2 1 ψ(s) = e−s (1 + 2s) − e−3s − . 3 9 9 2 Hence, the dispersion Δ of the projection under the hypothesis H0 equals ∞ 647 ≈ 0.015. Δ2 = ψ 2 (s) ds = 42525

(3)

0

Δ2

the kernel of Ψ is nondegenerate. Applying the Hoeffding theorem (see again Thus, [16, 18]), we conclude that   √ 647 d nIn −→ N 0, 4725 as n → ∞. Let us find the logarithmic asymptotic of the probability of large deviations for the sequences of statistics (1) under the null-hypothesis. The kernel of Ψ is not only centered and nondegenerate, it is also bounded. Applying the theorem on large deviations of nondegenerate U - and V -statistics from [26] (see also [11, 24]), we get the following statement. Theorem 1. If a > 0, then lim n−1 log P(In > a) = −f (a),

n→∞

where the function f is continuous for a small enough, and 4725 2 a2 a as a → 0. = 2 18Δ 1294 As alternatives for the considered characterization we use the following alternatives to the exponentiality hypothesis that are concentrated on the semiaxis x ≥ 0: (i) the Weibull alternative with density f (a) ∼

g1 (x, θ) = (1 + θ)xθ exp(−x1+θ ),

θ ≥ 0,

x ≥ 0;

(ii) the Mackeham alternative with density g2 (x, θ) = (1 + θ(1 − e−x )) exp(−x − θ(e−x − 1 + x)),

θ ≥ 0,

x ≥ 0;

(iii) the gamma alternative with density g3 (x, θ) =

xθ e−x , Γ(θ + 1)

θ ≥ 0,

x ≥ 0;

and 3

(iv) the special alternative with density   4 −3x 2  θ  −x −x e (1 + 2x) − e + θ(x − 1) , − 1+ g4 (x, θ) = e 3 3 3

θ ≥ 0,

x ≥ 0.

Denote the d.f.’s of these alternatives by Gj (x, θ), j = 1, 2, 3, 4, respectively. The first three alternatives are standard alternatives to the exponentiality hypothesis. The last alternative does not have an established name. It is easily seen that all the alternatives belong to the class of distributions F. Assume that under the alternative H1 , observations have d.f. G(· , θ) with density g(·, θ), θ ≥ 0, such that G(· , 0) is an exponential d.f. In the Bahadur theory, one takes as the quality measure for a sequence of statistics {Tn } the exact slope cT (θ) which describes the rate of the exponential decrease of their P-values under the alternative. To calculate the exact slope, one uses the following Bahadur theorem. Theorem 2 ([8, Sec. 7]). Assume that a sequence of statistics {Tn } satisfies the following two conditions: P

θ b(θ), (a) Tn −→

(b) lim

n→∞

n−1

θ > 0,

where

−∞ < b(θ) < ∞;

log PH0 (Tn ≥ t ) = −f (t)

for any t from an open interval I, where f is coninuous and {b(θ), θ > 0} ⊂ I. Then the following formula is valid: cT (θ) = 2 f (b(θ)), θ > 0. It is known that the following Bahadur–Ragavachari inequality is always valid: cT (θ) ≤ 2K(θ),

(4)

where K(θ) is the Kullback–Leibler “distance” between the null and alternative hypotheses depending on a real parameter θ. Hence, it is natural to define the local Bahadur efficiency of the sequence Tn by the formula eff(T ) := lim cT (θ)/2K(θ). θ→0

If the relation cT (θ) ∼ 2K(θ), θ → 0, holds in (4), then the sequence Tn is called locally asymptotically optimal (LAO) in the sense of Bahadur [8, 23]. Let us return to the sequence of statistics In . Let us calculate the local Bahadur slope and the local efficiency of the sequence of statistics In for the alternative with d.f. G(x, θ) and density g(x, θ), assuming their regularity and the possibility to differentiate the corresponding integrals with respect to parameter in the integrand. These assumptions are satisfied for all the four alternatives considered. Denote h(x) = gθ (x, 0). Note that

1

(5)

h(x) dx = 0.

0

Previously, we have shown in (1) that the second condition of Theorem 2 is satisfied. To check the first condition of Theorem 2, we have to apply the law of large numbers for U -statistics (see, for example, [18]). Using our notation, it is easy to show (see also [25]) that ∞ (6) bI (θ) ∼ 3θ ψ(s)h(s)ds, 0

4

where ψ(s) is the projection calculated in (3). The Weibull alternative. After simple calculations, we derive from (6) that ∞  bI (θ) ∼ 3θ 0

 4 −3s 2 −s 1 1 −s e (1 + 2s) − e e (1 + log s − s log s) ds = (1 + log 2) θ ≈ 0.4233 θ, − 3 9 9 4 θ → 0,

and the exact local slope of the sequence of statistics In can be represented as cI (θ) = b2I (θ)/(9Δ2 ) ∼ 1.308 θ 2 . The Kullback–Leibler information K(θ) between H0 and H1 for the Weibull alternative is calculated, for example, in [1]; it is shown that K(θ) ∼ π 2 θ 2 /12, θ → 0. Hence, the local Bahadur efficiency of our test equals eff(In ) = lim

θ→0

cI (θ) ≈ 0.795. 2K(θ)

The Mackeham alternative. Our formula (6) implies that bI (θ) ∼

4 θ, 45

θ → 0.

The exact local slope admits the representation cI (θ) ∼ 0.058 θ 2 , θ → 0. The Kullback–Leibler information for the Mackeham alternative is known from [1]; it is shown there that K(θ) ∼ θ 2 /24, θ → 0. Hence, the local Bahadur efficiency equals 0.692. The gamma alternative. Similarly to the previous cases, we deduce from (6) that bI (θ) ∼ 2 0.269 θ, cI (θ) ∼ 0.528 θ 2 , K(θ) ∼ ( π12 − 12 ) θ 2 , θ → 0. Hence, the local Bahadur efficiency equals 0.819. The alternative g4 (x, θ). In these case, calculations show that 647 2 θ , θ → 0. 42525 Let us find the expression as θ → 0 for the Kullback–Leibler “distance” between the null hypothesis and the alternative g4 (x, θ) in terms of the function h(x) introduced in (5); in this case, we note that the basis hypothesis is composite. It is easy to show (see also [2, 27]) that  ∞ 2

∞ cI (θ) ∼

h2 (x)ex dx −

2K(θ) ∼ 0

xh(x)dx

θ2,

θ → 0,

0

under our regularity conditions. Hence, in this case the Kullback–Leibler information satisfies the relation 647 2 θ , θ → 0. (7) 2K(θ) ∼ 42525 Hence, the local Bahadur efficiency of our text equals 1, and the integral test is locally optimal in the sense of Bahadur (see [23, Chap. 6]). Later we return to the explanation of this phenomenon. Admittedly, our integral test is consistent and quite effective for all the alternatives considered. 5

Table 1. Local Bahadur efficiency for the statistics In . Alternative Efficiency Weibull 0.795 Mackeham 0.692 Gamma 0.819 g4 1. 000 3. Statistics Dn of Kolmogorov type Now we consider the statistics Dn of Kolmogorov type. In this case, for a fixed t ≥ 0, the differences Hn (t) − Gn (t) are a family of V -statistics whose kernels depend on t ≥ 0: Ξ(X, Y ; t) = 1{|X − Y | < t} − 1{2 min(X, Y ) < t}. Performing simple calculations, we conclude that the projections ξ(s; t) := E(Ξ(X, Y ; t) | X = s) of the kernels are as follows for a fixed t: ⎧ −t−s , ⎪ 0 ≤ s < t/2; ⎨−e −t/2 −t−s (8) ξ(s; t) = e −e , t/2 ≤ s < t; ⎪ ⎩ −s t −t −t/2 − 1, t ≤ s < +∞. e (e − e ) + e Now we calculate the dispersions δ2 (t) of the projections under the hypothesis H0 . As a result, we see that ∞ 1 1 1 ξ 2 (s, t)e−s ds := δ2 (t) = e−t + e−2t + e−3t − e−5t/2 . 3 3 3 0

Hence, our family of kernels Ξ(X, Y ; t) is nondegenerate [24], and δ2 = sup δ2 (t) = 0.115. t

The limit distribution of the statistics Dn is unknown; in principle, it can be found using the methods of the paper [32]. It is proved that the U -empirical process √ ηn (t) = n (Hn (t) − Gn (t)) , t ≥ 0, weakly converges in D(0, ∞) as n → ∞ to a centered Gaussian √ process η(t) whose variation can be found in a closed form. Then the sequence of statistics nDn converges in distribution to the value supt≥0 |η(t)| for which it seems impossible to find the distribution in a closed form. Thus, it it reasonable to search for critical values of the statistics Dn by modeling their sample distribution. The family of kernels {Ξ(X, Y ; t)}, t ≥ 0, is not only centered but also bounded. We apply the theorem on large derivations of the supremum of a family of nondegenerate U - and V statistics [24] to prove the following statement. Theorem 3. For a > 0, lim n−1 log P(Dn > a) = −f (a),

n→∞

where the function f is continuous for a > 0 small enough, and f (a) = (8δ2 )−1 a2 (1 + o(1)) ∼ 1.083 a2 for a → 0. 6

The Weibull alternative. We again consider the Weibull alternative. By the Glivenko– Cantelli theorem for the supremum of a family of U - and V -statistical d.f.’s [17], the limit of Dn in probability equals bD (θ) := sup |bD (t, θ)| = sup | Pθ (|X − Y | < t) − Pθ (2 min(X, Y ) < t) | . t≥0

t≥0

It is easily seen that the following asymptotic is valid for the function bD (t, θ): ∞ bD (t, θ) ∼ 2θ

ξ(s; t)h(s) ds,

(9)

0

 ∂ g(s, θ)θ=0 and ξ(s; t) are the projections calculated in (8). Thus, for the where again h(s) = ∂θ Weibull alternative, t/2 bD (t, θ) ∼ 2θ (−e−t−s )e−s (1 + log s − s log s) ds 0

t + 2θ

(e−t/2 − e−t−s )e−s (1 + log s − s log s) ds

t/2 ∞

(e−s (et − e−t ) + e−t/2 − 1)e−s (1 + log s − s log s) ds

+ 2θ  =

t

 1 t 1 −t e (log t + γ + log 2 + 2t log 2) + e Ei (1, 2t) θ, 2 2

θ → 0,

for t ≥ 0, where γ ≈ 0.5772 is the Euler constant and Ei (k, z) denotes the exponential integral, ∞ Ei (k, z) = e−zt t−k dt, k ≥ 0, Re z > 0. 1

Fig. 1. The graph of the function bD (t, θ) for the Weibull alternative. Using the graph of the function bD (t, θ) (see Fig. 1), we see that bD (θ) = sup |bD (t, θ)| ∼ b(0.692, θ) ≈ 0.585 θ, t≥0

7

and the exact local slope of the sequence of statistics Dn admits the representation cD (θ) ∼ b2D (θ)/(4δ2 ) ∼ 0.741 θ 2 ,

θ → 0.

The Kullback–Leibler information for the Weibull alternative equals K(θ) ∼ the local Bahadur efficiency equals 0.450.

π2 θ2 12 ,

θ → 0; hence,

The Mackeham alternative. In this case, we use formula (9) obtained above; after calculations, we see that   4 −t 2 −2t −3t/2 θ, θ → 0. e + e − 2e bD (t, θ) ∼ 3 3 Thus, applying Fig. 2, we conclude that bD (θ) = b(0.9899, θ) ≈ 0.134 θ. Hence, the exact local slope equals cD (θ) ≈ 0.039 θ 2 . The Kullback–Leibler information for the Mackeham alternative equals K(θ) ∼ and the efficiency of our test equals 0.470.

θ2 24 ,

θ → 0,

Fig. 2. The graph of the function bD (t, θ) for the Mackeham alternative.

The gamma alternative. Similarly to the previous cases, we conclude that bD (t, θ) ∼ (e−t (log t + γ − log 2) + 2e−t/2 Ei (1, t/2) − 2Ei (1, t) + et Ei (1, 2t))θ, θ → 0, where γ is the Euler constant, γ ≈ 0.5772, and ∞ Ei (k, z) =

e−zt t−k dt.

1

Thus, applying Fig. 3, we conclude that bD (θ) = b(0.5457, θ) ≈ 0.379 θ. The exact local slope equals cD (θ) ≈ 0.311 θ 2 . Hence, the local efficiency for this alternative equals 0.482. 8

Fig. 3. The graph of the function bD (t, θ) for the gamma alternative. The alternative g4 (x, θ). Finally, we consider the alternative with density g4 (x, θ). Applying Fig. 4 and performing some calculations, we conclude that the following relations hold as θ → 0:   e−t −t/2 2 −t 2 −3t/2 2 −3t 26 e , and cI (θ) ∼ 0.011 θ 2 . (t + 2) − e (3t + 4) − e + e − bD (t, θ) ∼ θ 3 3 3 15 45 We know that in this case, the Kullback–Leibler information satisfies relation (7). Hence, the local Bahadur efficiency of our test equals 0.740. Note that the efficiency of Kolmogorov type statistics is less than that for the integral statistics In , as this is usual in the problem of hypotheses checking, see also [23].

Fig. 4. The graph of the function bD (t, θ)for the alternative g4 .

4. Conditions of local asymptotic optimality Let us find conditions of local asymptotic optimality for the integral and supremum statistics, i.e., let us describe families of alternatives for which the studied statistics are locally the best in the sense of Bahadur. Such problems have been considered in the study of classical nonparametric tests in [23, Chap. 6], where variational methods were used. 9

Table 2. Local Bahadur efficiency for the statistic Dn . Alternative Efficency Weibull 0.450 Mackeham 0.470 Gamma 0.482 g4 0.740 Consider the functions

  ∂ ∂ G(x, θ)θ=0 , h(x) = g(x, θ)θ=0 . ∂θ ∂θ Assume that the following regularity conditions are satisfied (see [27]): ∞  h2 (x)ex dx < ∞, h(x) = H (x) for x ≥ 0, H(x) =

(10)

0

and ∂ ∂θ

∞

 xg(x, θ) dx

∞ θ=0

0

=

xh(x)dx.

(11)

0

Denote by G the class of densities g(x, θ) with d.f. G(x, θ) that satisfy the regularity conditions (10) and (11). We obtain optimality conditions in terms of the function h(x). As was noted, under the regularity conditions formulated above, alternative densities from the class G satisfy the following relation:

∞  ∞ 2 h2 (x)ex dx − xh(x) dx θ 2 , θ → 0. 2K(θ) ∼ 0

0

We start with the integral statistic In with kernel Ψ(X, Y, Z) and projections ψ(x). Recall that the following asymptotic is valid for the function bI (θ): ∞ bI (θ) ∼ 3θ ψ(x)h(x) dx. 0

h(x)ex

is an element of the Hilbert space of functions for which their We also assume that squares are integrable on [0, ∞] with weight e−x :

∞ h(x)ex ∈ L2 ≡ q(x) : q 2 (x)e−x dx < ∞ . 0

Consider an auxiliary function ∞ h0 (x) = h(x) − (x − 1) exp(−x)

uh(u) du. 0

The following relations are obvious:  ∞ 2 ∞ ∞ 2 x h (x)e dx − xh(x) dx = h20 (x)ex dx 0

10

0

0

and

∞

∞ ψ(x)h(x) dx =

0

ψ(x)h0 (x) dx. 0

Hence, the local asymptotic efficiency takes the form  ∞ 2  ∞ ∞    ψ(x)h0 (x) dx ψ 2 (x)e−x dx · h20 (x)ex dx . eff I = lim b2I (θ)/ 9Δ2 2K(θ) = θ→0

0

0

0

The local asymptotic optimality in the sense of Bahadur for the statistic In means that the expression on the right equals 1. It is directly checked that the function ψ(x) satisfies the regularity conditions (10) and (11); hence, this function belongs to the class L2 . The Cauchy– Bunyakovskii–Schwarz inequality implies that this is so if and only if h0 (x)ex = C1 ψ(x) with some constant C1 = 0. At the same time, the consistency for small θ > 0 is guaranteed if C1 > 0. Hence, the function h(x) = gθ (x, 0) takes the form h(x) = e−x (C1 ψ(x) + C2 (x − 1)) with some constants C1 > 0 and C2 . The set of distributions for which the function h(x) takes this form is the LAO domain in the class G. An example of such an alternative is given by the alternative whose density g(x, θ) for all θ → +0 is given by the fomula     4 −3x 2 θ −x −x e (1 + 2x) − e + θ(x − 1) , x ≥ 0. − 1+ g(x, θ) = e 3 3 3 Recall that the considered Ahsanullah characterization is fulfilled for functions from the class F. It is easily seen that the function g(x, θ) belongs to the required class. In particular, the alternative g4 satisfies the formulated condition; this explains why the integral test is locally optimal for it. Now we consider the statistic Dn of Kolmogorov type with families of kernels Ξ(X, Y ; t) and projections ξ(x; t). In this case, the function b(θ) has the following asymptotic:   ∞     (12) bD (θ) ∼ 2θ sup  ξ(x; t)h0 (x) dx.  t≥0  0

Hence, the local asymptotic efficiency takes the form   eff D = lim b2D (θ)/ sup 4δ2 (t) 2K(θ) θ→0

t≥0

2 

 ∞ ξ(x; t)h0 (x) dx

= sup t≥0

0

 ∞ sup t≥0

ξ 2 (x, t)e−x dx ·

0

∞

h20 ex dx .

0

It is easily checked that for a fixed t, the function ξ(x, t) satisfies the regularity conditions (10) and (11); hence, ξ(x, t) ∈ L2 . Denote t0 = arg sup δ2 (t). We apply the Cauchy–Bunyakovskii– t≥0

Schwarz inequality to the integral in (12) and calculate the supremum in t (see also [2, 28]) to get the condition in the form 2 ∞  ∞ ∞ 2 −x ξ(x; t0 )h0 (x) dx = ξ (x, t0 )e dx · h20 ex dx; 0

0

0

the equality holds only if h0 (x) = C1 e−x ξ(x, t0 ). 11

Clearly, for a function h0 (x) of this form, the local asymptotic efficiency of the statistic Dn equals 1. Thus, the sequence of statistics {Dn } is locally optimal if and only if the corresponding function h(x) satisfies the condition h(x) = e−x (C1 ξ(x, t0 ) + C2 (x − 1)) for t0 = arg sup δ2 (t) t≥0

and some constants C1 > 0 and C2 . The set of distributions for which the function h(x) has this form is the LAO domain in the corresponding class G. We have not succeded in construction of examples of alternatives which form the LAO domain since the characterization imposes additional restrictions on the distribution functions of observations, which makes their construction very complicated. 5. Conclusion We have suggested two new exponentiality tests based on the known Ahsanullah characterization. We have described their limit distributions and large deviations; their local Bahadur efficiency has been calculated for several alternatives. We have found the form of alternatives which form the LAO domains for our tests. The statistics turned out to be quite effective for the alternatives considered, and they can be recommended for practical applications. On the whole, we believe that the study of U -empirical tests based on characterizations of the exponential law is really perspective. This research was supported by the RFBR (project 13-01-00172a, by the Program “Leading Scientific Schools” (project 1216.2012.1), and by the research program 6.38.672.2013 of SPbGU. Translated by S. Yu. Pilyugin. REFERENCES 1. V. V. Litvinova, Asymptotic properties of symmetry and goodness-of-fit texts based on characterizations, Candidate dissertation, SPbGU (2004). 2. A. V. Chirina, Asymptotic efficiency of exponentiality tests that are free of scale parameter, Candidate dissertation, SPbGU (2006). 3. I. Ahmad and I. Alwasel, “A goodness-of-fit test for exponentiality based on the memoryless property,” J. Roy. Statist. Soc., 61, 681–689 (1999). 4. M. Ahsanullah, “On a characterization of the exponential distribution by spacings,” Ann. Inst. Statist. Math., 30, 163–166 (1978). 5. M. Z. Anis and S. Dutta, “Recent tests of exponentiality against IFR alternatives: a survey,” J. Statist. Comput. Simul., 80, 1373–1387 (2010). 6. J. E. Angus, “Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation,” J. Statist. Plann. Infer., 6, 241–251 (1982). 7. S. Asher, “A survey of tests for exponentiality,” Commun. Statist. Theory Meth., 19, 1811– 1825 (1990). 8. R. R. Bahadur, Some Limit Theorems in Statistics, SIAM, Philadelphia (1971). 9. N. Balakrishnan and A. Basu, The Exponential Distribution: Theory, Methods and Applications, Gordon and Breach, Langhorne, PA (1995). 10. L. Baringhaus and N. Henze, “Tests of fit for exponentiality based on a characterization via the mean residual life function,” Statist. Papers, 41, 225–236 (2000). 11. A. DasGupta, Asymptotic Theory of Statistics and Probability, Springer, New York (2008). 12. K. A. Doksum and B. S. Yandell, “Tests of exponentiality,” Handbook of Statist., 4, 579–612 (2000). 13. A. Grane and J. Fortiana, “A location- and scale-free goodness-of-fit statistic for the exponential distribution based on maximum correlations,” Statist., 43, 1–12 (2009). 12

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