Ya. Yu. Nikitin

Bahadur efficiency and local optimality of a test for the exponential distribution based on the gini statistic

The sample scale-free Gini index is known to be a powerful test of exponentiality against a broad class of alternatives. To understand better the efficiency properties of this test we calculate its Bahadur efficiency for most commonly used parametric alternatives to the exponential distribution. Using variational arguments and the Bahadur-Raghavachari inequality for exact slopes we find the conditions of local Bahadur optimality of the Gini test. It turns out that this property surprisingly holds for a family of alternative distributions including the well-known Gompertz-Makeham distribution.

Bahadur efficiency of a test of exponentiality based on a loss-of-memory type functional equation

A test of exponentiality of Kolmogorov-Smirnov type based on a simplified variant of Ioss-of-memory property and proposed by Angus [3] is considered. We find corresponding large-deviation asymptotics under the null-hypothesis reducing the problem to large deviations of U-statistics with a special kernel. As a corollary. Bahadur local efficiency of Angus test is calculated for most commonly used parametric alternatives to exponentiality.

Asymptotic efficiency of exponentiality tests based on order statistics characterization

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.

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

We find the logarithmic L2-small ball asymptotics of some Gaussian processes related to the fractional Brownian motion (fBm), fractional Ornstein--Uhlenbeck process (fOU), and their integrated analogues. We consider also the multiparameter generalizations.

Large deviations and asymptotic efficiency of a statistic of integral type. II

The rough asymptotic behavior of probabilities of large deviations of integral statistics ofω2 type and also their analogues for a sample of random Poisson size are studied. An approach is employed that goes back to I. N. Sanov according to which the asymptotic behavior indicated is determined by the solution of a certain extremal problem. Methods of bifurcation theory for non linear equations are used to investigate the latter.

Local Bahadur efficiency of some goodness-of-fit tests under skew alternatives

Abstract The efficiency of most known distribution-free goodness-of-fit tests such as Kolmogorov–Smirnov, Cramer–von Mises and their variants was studied mainly under the classical alternatives of location and scale. Hence, it is interesting to compare the efficiencies of these tests under asymmetric alternatives like the skew alternative proposed by Azzalini (Scand. J. Stat. 12 (1985) 171). We calculate and compare local Bahadur efficiencies of many known statistics for skew alternatives and discuss the conditions of their local optimality.

Two-Sample Tests Based on the Integrated Empirical Process

Abstract Let X 1, …, X m and Y 1, …, Y n be independent random variables, where X 1, …, X m are i.i.d. with continuous distribution function (df) F, and Y 1, …, Y n are i.i.d. with continuous df G. For testing the hypothesis H 0: F = G, we introduce and study analogues of the celebrated Kolmogorov–Smirnov and one- and two-sided Cramér-von Mises statistics that are functionals of a suitably integrated two-sample empirical process. Furthermore, we characterize those distributions for which the new tests are locally Bahadur optimal within the setting of shift alternatives.

Exact Small Deviation Asymptotics for Some Brownian Functionals

We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

Bahadur efficiency and local asymptotic optimality of certain nonparametric tests for independence

One computes the Bahadur efficiency of nonparametric tests for the verification of the independence hypothesis, based on a Kolmogorov type statistic.

Limit distributions and comparative asymptotic efficiency of the Smirnov — Kolmogorov statistics with a random index

Limit distributions for certain statistics of Smirnov — Kolmogorov type are obtained which consider the weak convergence of the corresponding empirical process. Approximate and precise asymptotic efficiencies of these statistics are computed. It is shown that they are worse in a certain sense than the classical Kolmogorov — Smirnov statistics.