THE PROBABILITY OF LARGE DEVIATIONS FOR THE SUM FUNCTIONS OF SPACINGS

Let 0= U0,n ≤ U1,n ≤ ··· ≤ Un−1,n ≤ Un,n = 1 be an ordered sample from uniform [0,1] distribution, and Din = Ui,n −Ui−1,n, i = 1,2, . . . ,n; n = 1,2, . . . , be their spacings, and let f1n, . . . , fnn be a set of measurable functions. In this paper, the probabilities of the moderate and Cramer-type large deviation theorems for statistics Rn(D)= f1n(nD1n) + ···+ fnn(nDnn) are proved. Application of these theorems for determination of the intermediate efficiencies of the tests based on Rn(D)-type statistic is presented here too.

The optimal condition of the asymptotical normality and the lower estimation of the remainder in CLT, Mirakhmedov [11], and Edgworth-type asymptotical expansion, Does et al. [3], and the probability of "supper large" deviation result (i.e., of order c √ n), Zhou and Jammalamadaka [16], have been obtained for r.v.(1.1).But the topics of interest to us here are not readily available in the literature.In the present paper, we will prove a probability of large deviation theorems in moderate zone (i.e., of order c √ lnn) and Cramer's zone (i.e., of order o( √ n)).Many authors have considered spacings-based tests testing hypothesis whether a random sample comes from a specified distribution.Most papers are dedicated to Pitman's approach (see, e.g., Holst and Rao [6], Jammalamadaka et al. [8]).Also in Zhou and Jammalamadaka [16], Bahadur's asymptotic efficiency and in Bartoszewicz [1] Bahadur's and Hodjes-Lehman approximate efficiencies have been studied.In the present paper, the intermediate efficiencies due to Kallenberg [9], see also Ivchenko and Mirakhmedov [7], of the spacings-based tests are presented too.
Thus the results of the present paper are filling existing gaps in the investigation of the large deviation probabilities of the statistics of type (1.1) and efficiencies of the tests based on this statistics.
The method developed in the present paper is based on following property of the uniform spacings.Let Y 1 (λ),Y 2 (λ),... be a sequence of independent r.v.'s with common exponential distribution exp(λ), and let For arbitrary λ > 0, there is a regular variant of the conditional distribution of the random vector Y (λ) given S n (λ) = n such that (nD) = (Y (λ)/S n (λ) = n), where (X) denoted the distribution of the random vector X.This equality is known and usually used at λ = 1 (see, Pyke [13], Holst and Rao [6], Does et al. [3]).Its validity for arbitrary λ > 0 follows from the fact that S n (λ) is a sufficient statistic for parameter λ.Thus for arbitrary measurable function L(x 1 ,...,x n ) of nonnegative arguments and each λ > 0, we have (L(nD 1n ,...,nD nn )) where p n (z,λ) is the density function of r.v.S n (λ).We note that the proof of the Cramertype theorem (Theorem 2.2) rests on a special choice of the parameter λ, see Lemma 3.2.
Few words about notations.Many quantities like g m , ρ, g m depend on n, however for notational convenience, we will suppress this suffix, except cases where it is essential.C i is a positive constant that may not be the same in the different expressions, ε is an arbitrary small positive constant.All asymptotic expressions are considered as n → ∞.
The organization of this paper is as follows.The probabilities of large deviation results are formulated in Section 2. A proof of the theorems of Section 2 is given in Section 3. The application of the theorems of Section 2 to the study of the intermediate efficiency of tests based on spacing statistics are presented in Section 4. Appendix contains proofs of auxiliary lemmas.

Results
For simplicity of notations below, we put Y m = Y m (1), S n = S n (1).Also we let (2.1) be the standard normal distribution function.
for some δ > 0. Then for all x such that 0 ≤ x ≤ √ δ lnn, 3) be fulfilled and for some H > 0, Then for x ≥ 0 and x = o( √ n), where , where δ = min(1,δ).Using this and well-known relation it is easy to check that Theorem 2.1 holds true for 0 ≤ x ≤ √ δ lnn/2.From now on, we suppose that We introduce the truncated functions g m (u .., where ε > 0 will be chosen sufficiently small later on.Putting T m (D) = n m=1 g m (nD m ), ( (i) Estimation of ∇ 1 .For a complex variable z, we denote ϕ n (z) = E exp{z T n (D)}.Let G n be the Cramer's transform with parameter h = x/σ n of the r.v.T n (D).Note that ϕ n (h) ≥ 1/2 for sufficiently large n.Estimation of the ∇ 1 rests on the following lemma, proof of which (being long) is given in the appendix.
We denote the first and second summands inside the square brackets by A 1 and A 2 , respectively.It can be readily shown that A 1 = Φ(−x)exp{x 2 /2}.In A 2 , first of all we integrate by part, and after this we use first assertion of Lemma 3.
Proof of Theorem 2.2.Let us prove the first relation and for x > 1 only.The case x < 1 is not of interest here.Second relation can be obtained from first by substituting −g m (u) instead of g m (u), m = 1,2,....For any λ ∈ J ε = (1 − ε,1 + ε), ε > 0, and |h| ≤ H 1 < min(1 − ε,H)/4 using Holder's inequality and inequality a j ≤ j!exp{H|a|}/H j , a ∈ R, j ≥ 0, we have Putting Y m (λ) = Y m (λ) − 1, we introduce functions of the complex variables u and v: We will consider an analytical continuity of the function K m (u,v,λ) as a function of the variable λ into disk J ε = {λ : |λ − 1| < ε} and we will use the same denotation for it.Put Hence, also the function K m (u,v,λ) = lnK m (u,v,λ) is analytic in the region S(H 2 ,ε) and for each m = 1,...,n.
We will choose λ and h according to Lemmas 3.2 and 3.3.
Lemma 3.2.Under conditions of Theorem 2.2, there exists a unique solution of the equation This solution can be represented as the power series which is convergent for |h| ≤ H 0 , where H 0 > 0 does not depend on n and

.22)
In particular, In what follows, h and λ are roots of (3.20) and (3.23).It is evident that for each i and j, the function λ)/∂u i ∂v j is analytic in the region S(H 2 ,ε), and hence can be represented as power series in a neighborhood of the point (0,0,1).Due to (3.16), (3.21), (3.22), and Cauchy's inequality, we have for each i and j, Mirakhmedov 9 From (3.27) taking into account (2.2), we get

Asymptotical relative indeterminate efficiency
Let X 1n ,X 2n ,...,X n−1,n be an ordered sample from distribution F(x), and W kn = X kn − X k−1,n , k = 1,...,n, with notations X on = 0 and X nn = 1.We wish to test null hypothesis H 0 : F(x) = x, 0 ≤ x ≤ 1, versus sequence of alternatives Sherzod Mira'zam Mirakhmedov 11 where δ(n) → 0, as n → ∞, and function L(x) satisfy smoothness conditions, under which W kn can be related to the uniform spacings D kn by the relation (cf.[6, (3.2)]) , where l(u) = L (u), r kn = (k − 0.5)/n, and o p (•) is uniform in k.Hence, Theorems 2.1 and 2.2 can be applied for test statistics of type We assume that function f (u, y) is defined on [0,∞] × [0,1] and has continuous derivatives of first and second orders with respect to u.
We will apply Theorems 2.1 and 2.2 to the analysis of the intermediate asymptotic efficiency (IAE) due to Kallenberg [9] (see, also Ivchenko and Mirakhmedov [7]) of the f -tests.Note that asymptotical properties of the f -tests are different for nonsymmetric and symmetric tests.
In what follows P i , E i , Var i denote the probability, expectation, and variance under H i , i = 0,1.Let A in and B 2 in stand for the asymptotic values of n −1 E i R n and n −1 Var i R n , receptively, and x n = √ n(A 1n − A 0n )/B 0n .We briefly outline here the scheme of asymptotic study of ω-IAE, the β-IAE is analyzed similarly.Performance of statistic R n will be measured by the asymptotic value of n is determined by the asymptotic behavior of large deviation probabilities for R n under H 0 , which is given by Theorems 2.1 and 2.2.In agreement with Kallenberg [9], this efficiency will be called weak ω-IAE This efficiency will be weak ω-IAE when ), it will be ω-IAE, and strong ω-IAE when 4   √ nδ(n) = o( 4√ n).
The properties of the correlation coefficient imply that c 2 ( f ) ≤1 and c 2 ( f ) = 1 if and only if f (y) = ay 2 + by + c (which is equivalent to f (y) = y 2 in terms of the test statistic).Thus, within the class of symmetric tests, the Greenwood statistic G n is most efficient in the sense of weak ω-IAE (as it was in Pitman's sense) since it satisfies condition (i) of Theorem 4.2.However, for more distant alternatives when δ(n) 4 n/ logn → ∞, ω-IAE of G n remains an open question since this statistic does not satisfy condition (ii) of Theorem 4.2.The optimality of symmetrical tests in the sense of strong ω-IAE can be deduced from Theorem 4.2 only for some subclasses of statistics satisfying Cramer's condition (ii).
Remark 4.3.The central limit theorem for R n implies that for δ(n) 4   √ n → χ > 0 (i.e., for Pitman alternatives), one has P 0 {R n ≥ A 1n } = Φ(−χ 2 • c( f ) 1 0 l 2 (u)du), that is, the asymptotic efficiency of test statistic R n is still determined by the functional c 2 ( f ).In other Sherzod Mira'zam Mirakhmedov 13 words, a Pitman efficient remains optimal in the sense of ω-IAE as long as Theorems 2.1 and 2.2 hold true for test statistic.
Remark 4.4.The Greenwood statistic satisfies the well-known Linnik's condition with parameter α = 1/6, hence relations (2.5) should be true for x = o(n 1/6 ) (but as yet do not have a proof).Therefore, we suspect that Greenwood test must still be most efficient in middle ω-IAE sense.
The proofs of Theorems 4.1 and 4.2 follow from Theorems 2.1 and 2.2 and calculations analogous to that presented by Holst and Rao [6] and Ivchenko and Mirakhmedov [7].
The analysis of β-IAE goes along the same lines.According to the general principle stated above, the efficiency of f -test is measured by the asymptotic value of e β n( f ) = − lnP 1 (R n (W) > nA on ).We have where f mn nD mn , f mn (x) = f mn x 1 − l r mn δ(n) + o δ(n) .(4.9) Omitting summand o(δ(n)) for simplicity of notes and putting z = l(r mn )δ(n), we get mn ≤ εxσ n ∪ T n (D)>u, n m=1 g m nD mn >εxσ n ⊆ T n (D) > u n m=1 g m nD mn > εxσ n .
exp H f mn (Y ) .

( 4 . 10 )
Thus we can use Theorems 2.1 and 2.2 to asymptotic analysis of e β n( f ).Corresponding calculation shows that for e β n( f ), the assertions of Theorems 4.1 and 4.2 are still true, that is, e β n( f ) and e ω n ( f ) are asymptotically equivalent.
v=0 is analytic in the region S(H 2 ,ε).Also by(3.16) and Cauchy's inequality, | Λ(h,λ)| ≤ ln2/H 2 .Hence in the region S(H 2 ,ε), the function Λ(h,λ) is bounded uniformly with respect to h, λ, n, and it can where h 1n = 1, and there is H 0 > 0 such that Proof.Putting A(h) = n −1 A(h,λ(h)), we rewrite (3.23) as A(h) = n −1 xσ n .Obviously, Thus, within the class of nonsymmetric tests, the linear test (based on statistics L n ) is most efficient in the of ω-IAE (all three types) as it was in Pitman's sense, since it satisfies condition (ii) of this theorem.As follows from Theorem 4.1, nonsymmetric tests are "thin directed" in sense that each nonsymmetric f -test essentially depends on alternative H 1 .Note that for linear test, ρ 2 ( f ) = Symmetric tests cannot distinguish alternative H 1 (4.1) that is at a "distance" δ(n) = n −1/2 , and can distinguish more distant (4.1) alternatives with δ(n) = n −1/4 .Moreover within the class of symmetric tests, the optimal test in Pitman's sense is Greenwood's test based on statistics:G n = W 2 1n + W 2 2n + ••• + W 2 nn ,see Holst and Rao [6].For symmetric tests, intermediate alternatives H 1