Bahadur Sense Research Articles

Granularity and Efficiency

Abstract Hammersley (1950) considered, among other matters, the asymptotic relative efficiency (ARE) of the rounded sample median M ε with respect to the rounded sample mean as estimate of a Normal population mean restricted to a uniform grid of mesh size 2ε. This article extends Hammersley's work to a certain class of two-sided extended increasing failure rate (TEIFR) distributions for which the (grid-valued) population mean and median coincide, their common value designated as μ. The ARE of M ε with respect to , as estimators of μ, is examined for our class via the theory of large deviations. The role of the TEIFR assumption is simply to ensure that the tails of the distribution of X i − μ fall off quickly enough to make comparison of asymptotic probabilities of large (beyond ε) deviations of the location-normalized sample median M − μ and mean relevant to the comparison of their asymptotic variances. Even within our somewhat narrow class, we find the ARE of M ε with respect to surprisingly sensitive to distribution shape, as well as to grid mesh size and the actual definition of ARE. Among our findings is that, in the symmetric TEIFR class, the ARE of M ε with respect to is continuous in ε at ε = 0 under a definition of ARE closely related to the commonly used limiting ratio of equivalent sample sizes, but it is not continuous at ε = 0 under Hammersley's definition of ARE. A related finding is that, within the TEIFR class, the asymptotic effective variance [in the sense of Bahadur (1960)] of the sample median M equals its asymptotic variance as usually defined. Another finding is that, in the case of the Laplace distribution, M ε is asymptotically more efficient than , as estimator of the grid-valued population center μ, when the grid is fine (ε small), but it is asymptotically less efficient when the grid is coarse (ε large). All of these findings stem from the comparison of large-deviation rates that are equally relevant to the comparisons of asymptotic error rates of certain tests using and M as test statistics, a matter mentioned briefly in the last section.

ASYMPTOTIC OPTIMALITY OF LIKELIHOOD RATIO TESTS FOR EXPONENTIAL DISTRIBUTIONS UNDER TYPE II CENSORING

SummaryConsider the model of k two parameter exponential populations under a type II censoring scheme. In this paper we establish optimality in the sense of Bahadur of the likelihood ratio test for an arbitrary testing problem by requiring only Condition A in Hsieh (1979). This result unifies and generalizes the results in Samanta (1986).

On the maximum‐likelihood estimator for the location parameter of a cauchy distribution

AbstractIn the literature of point estimation, the Cauchy distribution with location parameter is often cited as an example for the failure of maximum‐likelihood method and hence the failure of the likelihood principle in general. Contrary to the above notion, we prove that even in this case the likelihood equation has multiple roots and that the maximum‐likelihood estimator (the global maximum) remains as an asymptotically optimal estimator in the Bahadur sense.

The likelihood ratio test for the change point problem for exponentially distributed random variables

Let x 1,…, x n+1 be independent exponentially distributed random variables with intensity λ 1 for i ⩽ τ and λ 2 for i > τ, where τ as well as λ 1 and λ 2 are unknown. By application of theorems concerning the normed uniform quantile process it is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing λ 1 = λ 2 (or, equivalently, τ = 0 or n + 1) is an extreme value distribution. Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the ‘strong invariance principle’. Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications. The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test.

Asyhptotic efficiency of sequential and non-sequential procedures based on fisher's method of combining tests

In this paper we propose test statistics based on Fisher's method of combining tests for hypotheses involving two or more parameters simultaneously, It Is shown that these tests are asymptotically efficient In the sense of Bahadur, It is then shown how these tests can be modified to give sequential test procedures which are efficient in the sense of Berk and Brown (1978). The results in section 3 generalize the work of Perng (1977) and Durairajan (1980).

Bahadur asymptotic efficiency of integral tests for symmetry

One finds rough asymptotics for the probabilities of large deviations of statistics of the omega-square type for the testing of the symmetry hypothesis. This gives the possibility to compute their Bahadur local exact slopes and to obtain some results on the characterization of the distributions for which the considered statistics are locally asymptotically optimal in the Bahadur sense for the shift alternative.

Asymptotic relative efficiency of designs for factorial paired comparison experiments

A general approach for comparing designs of paired comparison experiments on the basis of the asymptotic relative efficiencies, in the Bahadur sense, of their respective likelihood ratio tests is discussed and extended to factorials. Explicit results for comparing five designs of 2 q factorial paired comparison experiments are obtained. These results indicate that some of the designs which require comparison of fewer distinct pairs of treatments than does the completely balanced design are, generally, more efficient for detecting main effects and/or certain interactions. The developments of this paper generalize the work of Littell and Boyett (1977) for comparing two designs of R x C factorial paired comparison experiments.

Characterization of distributions by the local asymptotic optimality property of tests statistics

Let X1, X2,... be a sequence of independent, identically distributed random variables with density f(x−θ), θ ε R1. We consider the problem of testing the hypothesis H0∶=0 against H1∶ ≠ 0 on the basis of a sequence of test statistics {Tn(X1,...,Xn). In accordance with the Bahadur theory, a measure of the asymptotic efficiency of {Tn} is its exact slope Ct(θ). One says that {Tn} is locally asymptotically optimal in the Bahadur sense if Ct(θ) ip 2K(θ), θ → 0, where . The purpose of the paper is to characterize densities f for which the property of local asymptotic optimality is shared by such commonly used statistics as the sample mean, the Kolmogorov-Smirnov statistic, the sign statistic, Ω2, etc. Under certain restrictions on f one proves, for example, that the sequence of statistics u is locally asymptotically optimal only for the “hyperbolic cosine” distribution, while the Kolmogorov statistic only for the Laplace distribution. At the end of the paper one obtains similar results for the two-sample case, in particular, for a large class of linear rank statistics.

A Bahadur efficiency comparison between one and two sample rank statistics and their sequential rank statistic analogues

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist. 9 424–436) and Lombard (1981, South African Statist. J. 15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.

Local comparison of linear rank tests, in the Bahadur sense

As continuation ofKremer [1979a] a theory of asymptotic comparison based on local Bahadur efficiency is derived for general linear rank tests of the one-sample symmetry problem and thek-sample problem (k≥2). The results are similar to former considerations based on Pitman efficiency but hold under weaker conditions on the scores-generating functions or local alternatives.

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.

Bahadur deficiency of likelihood ratio tests in exponential families

For many testing problems several different tests may have optimal exact Bahadur slope. The introduction of Bahadur deficiency provides further information about the performance of such tests. Roughly speaking a sequence of tests is deficient in the sense of Bahadur of order o ( h n ) at a fixed alternative θ if the additional number of observations necessary to obtain the same power as the optimal test at θ is of order o ( h n ) as the level of significance tends to zero. In this paper it is shown that in typical testing problems in multivariate exponential families the LR test is deficient in the sense of Bahadur of order o (log n).

Transitivity in Problems of Optimal Stopping

In a sequential decision problem it is usually assumed that the available information is represented by an increasing family $\mathscr{F}$ of $\sigma$-algebras. Often a reduction, e.g., according to principles of sufficiency or invariance, is performed which yields a smaller family $\mathscr{G}$. The consequences of such a reduction for problems of optimal stopping are treated in this paper. It is shown that $\mathscr{G}$ is transitive for $\mathscr{F}$ (in the Bahadur sense) if and only if for any stochastic process adapted to $\mathscr{G}$ the value (i.e., maximal reward by optimal stopping) under $\mathscr{G}$ and the value under $\mathscr{F}$ are equal.

Rate of Convergence of Estimators Based on Sample Mean

The rate of convergence of the point estimators for the parameter based on the sample mean of i.i.d. random vectors is obtained. The result is used to prove the Bahadur's efficiency of the regular best asymptotically normal estimator when the underlying distribution is in the exponential family. An example is given to show that if the distribution is not in the exponential family, then the regular best asymptotically normal estimator is not necessarily efficient in Bahadur's sense.

Bahadur efficiency and probabilities of large deviations

The relative performance of two statistical tests of a hypothesis for large sample sizes is often investigated by means of asymptotic relative efficiencies in the sense of Pitman or Bahadur. Pitman efficiencies are directed at local alternatives, Bahadur efficiencies at fixed (non‐local) alternatives. The present paper reviews some methods and results on Bahadur efficiency with special attention to probabilities of large deviations, which play a key role in the computation of Bahadur efficiencies.

The Rate of Convergence of Consistent Point Estimators

The rate at which the probability $P_\theta\{|t_n - \theta| \geqq \varepsilon\}$ of consistent estimator $t_n$ tends to zero is of great importance in large sample theory of point estimation. The main tools available at present for finding the rate are Bernstein-Chernoff-Bahadur's theorem and Sanov's theorem. In this paper, we give two new techniques for finding the rate of convergence of certain consistent estimators. By using these techniques, we have obtained an upper bound for the rate of convergence of consistent estimators based on sample quantities and proved that the sample median is an asymptotically efficient estimator in Bahadur's sense if and only if the underlying distribution is double-exponential. Furthermore, we have proved that the Bahadur asymptotic relative efficiency of sample mean and sample median coincides with the classical Pitman asymptotic relative efficiency.

Asymptotic Sufficiency of the Vector of Ranks in the Bahadur Sense

We shall consider the hypothesis of randomness under which two samples $X_1, \cdots, X_n$ and $Y_1, \cdots, Y_m$ have an identical but arbitrary continuous distribution. The vector of ranks $(R_1, \cdots, R_{n+m})$ will be shown to be asymptotically sufficient in the Bahadur sense for testing randomness against a general class of two-sample alternatives, simple ones as well as composite ones. In other words, the best exact slope will be attainable by rank statistics, uniformly throughout the alternative.

On a Theorem of Bahadur on the Rate of Convergence of Point Estimators

In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$ Our proof also gives a simple and direct method to verify Bahadur's [2] result.