Efficiency of exponentiality tests based on a special property of exponential distribution

New goodness-of-fit tests for exponentiality based on a particular property of exponential law are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, the second one is a Kolmogorov type statistic.We show that the kernels corresponding to our statistics are nondegenerate. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and is comparedwith simulated powers of new tests. Conditions of local optimality of new statistics in Bahadur sense are discussed and examples of “most favorable” alternatives are given. New tests are applied to reject the hypothesis of exponentiality for the length of reigns of Roman emperors which was intensively discussed in recent years.     

On the minimal time function for distributed control systems in Banach spaces

In this paper, it is shown that the minimal time function is locally Lipschitz continuous for the control systemx=Ax+u in a Banach spadeE, under either of two conditions:A is linear and generates aC ₀-semigroup of bounded linear operators; orA is nonlinear, possibly multivalued, and dissipative. The main tool used for the nonlinear case is a result of Barbu concerning the null controllability of the system.     

Estimation of a non-negative location parameter with unknown scale

For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\) . We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

Dual divergence estimators and tests: Robustness results

The class of dual ?-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.     

Robust and Efficient Estimation for the Generalized Pareto Distribution

In this article we implement the minimum density power divergence estimator (MDPDE) for the shape and scale parameters of the generalized Pareto distribution (GPD). The MDPDE is indexed by a constant 0 that controls the trade-off between robustness and efficiency. As increases, robustness increases and efficiency decreases. For = 0 the MDPDE is equivalent to the maximum likelihood estimator (MLE). We show that for > 0 the MDPDE for the GPD has a bounded influence function. For < 0.2 the MDPDE maintains good asymptotic relative efficiencies, usually above 90%. The results from a Monte Carlo study agree with these asymptotic calculations. The MDPDE is asymptotically normally distributed if the shape parameter is less than (1 + )/(2 + ), and estimators for standard errors are readily computed under this restriction. We compare the MDPDE, MLE, Dupuis optimally-biased robust estimator (OBRE), and Peng and Welshs Medians estimator for the parameters. The simulations indicate that the MLE has the highest efficiency under uncontaminated GPDs. However, for the GPD contaminated with gross errors OBRE and MDPDE are more efficient than the MLE. For all the simulated models that we studied the Medians estimator had poor performance.AMS 2000 Subject Classification. Primary—62F35, Secondary—62G35     

Sensitivity analysis of M-estimates

Bahadur representation of the difference of estimators of regression coefficients for the full data set and for the set from which one observation was deleted is given for the M-estimators which are generated by a continuous -function. The representation is invariant with respect to the scale of residuals and it indicates that the bound of the norm of the difference is proportional to the gross error sensitivity. Then for the -function which corresponds to the median it is shown that the difference of the estimates for the full data and for data without one observation, although being bounded in probability, can be much larger than indicated by the gross error sensitivity.     

Local Bahadur efficiency of rank tests for the independence problem

In previous papers [Approximate and local Bahadur efficiency of linear rank tests in the two-sample problem, Ann. Statist.7, 1246–1255, 1979; Local comparison of linear rank tests in the Bahadur sense, Metrika, 1979] the author developed for linear rank tests of the one-sample symmetry and the k-sample problem (k ≥ 2) a theory of local comparison, based on the concept of Bahadur efficiency. In the present article this theory is carried over to rank tests of the independence problem.     

Two families of normality tests based on Polya-type characterization and their efficiencies

We introduce two families of statistics based on the extended Pólya characterization of the normal law, to be used for testing of normality. The first family depends on a parameter a ∈ (0, 1), and for any a its members are asymptotically normal and consistent for many alternatives of interest. We study the local Bahadur efficiency of these statistics as a function of a and find that for common alternatives the Pólya case is the worst and the maximum of efficiency is attained for a close to 0 or 1. The second family depends on a positive integer m, and the efficiency increases as m grows. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 147–159.     

Asymptotic Properties of Exponentiality Tests Based on L-Statistics

We study a class of scale-free exponentiality tests based on linear combinations of order statistics divided by the sample mean. For this class we find large deviations under null hypothesis and exact Bahadur efficiency, obtain conditions of consistency against the IFR and DFR classes of alternatives and describe domains of local Bahadur optimality.     

On Bahadur asymptotic efficiency in a semiparametric regression model

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A robust estimator of multivariate location based on projection

This paper presents an estimator of location vector based on one-dimensional projection of high dimensional data. The properties of the new estimator including consistency ,asymptotic normality and robustness are discussed. It is proved that the estimator is not only stronglyconsistent and asymptotically normal but also with a breakdown point 1/2 and a bounded influence function.     

Some test statistics based on the martingale term of the empirical distribution function

Summary It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated. The Institute of Statistical Mathematics     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

Efficient and robust tests for semiparametric models

In this paper, we investigate a hypothesis testing problem in regular semiparametric models using the Hellinger distance approach. Specifically, given a sample from a semiparametric family of \(\nu \)-densities of the form \(\{f_{\theta ,\eta }:\theta \in \Theta ,\eta \in \Gamma \},\) we consider the problem of testing a null hypothesis \(H_{0}:\theta \in \Theta _{0}\) against an alternative hypothesis \(H_{1}:\theta \in \Theta _{1},\) where \(\eta \) is a nuisance parameter (possibly of infinite dimensional), \(\nu \) is a \(\sigma \)-finite measure, \(\Theta \) is a bounded open subset of \(\mathbb {R}^{p}\), and \(\Gamma \) is a subset of some Banach or Hilbert space. We employ the Hellinger distance to construct a test statistic. The proposed method results in an explicit form of the test statistic. We show that the proposed test is asymptotically optimal (i.e., locally uniformly most powerful) and has some desirable robustness properties, such as resistance to deviations from the postulated model and in the presence of outliers.     

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions. Research supported by NSERC of Canada.     

Cramér-von Mises tests for independence

In this paper recent results of Gregory [(1977)Ann. Statist.5 110–123] are used to obtain the asymptotic null distribution of a weighted Cramér-von Mises type test for independence. We use approximate Bahadur slopes to find good weight functions for certain alternatives. Some percentage points of the asymptotic distribution are given.     

On a Problem of Statistical Inference in Null Recurrent Diffusions

We consider a particular example of statistical inference in null recurrent one-dimensional diffusions. In a first parametric model, we prove local asymptotic mixed normality (LAMN) and efficiency of the sequence of maximum likelihood estimates (MLE): its speed of convergence is n ^/2 with ranging over (0, 1). In a second semiparametric model (where in addition an unknown nuisance function with known compact support is included in the drift), we prove a local asymptotic minimax bound and specify asymptotically efficient estimates for the unknown parameter.     

Robust statistical inference based on the C-divergence family

This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators.

     

Projection based scatter depth functions and associated scatter estimators

In this article, we study a class of projection based scatter depth functions proposed by Zuo [Y. Zuo, Robust location and scatter estimators in multivariate analysis, The Frontiers in Statistics, Imperial College Press, 2005. Invited book chapter to honor Peter Bickel on his 65th Birthday]. In order to use the depth function effectively, some favorable properties are suggested for the common scatter depth functions. We show that the proposed scatter depth totally satisfies these desirable properties and its sample version possess strong and uniform consistency. Under some regularity conditions, the limiting distribution of the empirical process of the scatter depth function is derived. We also found that the aforementioned depth functions assess the bounded influence functions.A maximum depth based affine equivariant scatter estimator is induced. The limiting distributions as well as the strong and consistency of the sample scatter estimators are established. The finite sample performance of the related scatter estimator shows that it has a very high breakdown point and good efficiency.