A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

r-k Class estimation in regression model with concomitant variables

We treat with the r-k class estimation in a regression model, which includes the ordinary least squares estimator, the ordinary ridge regression estimator and the principal component regression estimator as special cases of the r-k class estimator. Many papers compared total mean square error of these estimators. Sarkar (1989, Ann. Inst. Statist. Math., 41, 717–724) asserts that the results of this comparison are still valid in a misspecified linear model. We point out some confusions of Sarkar and show additional conditions under which his assertion holds.     

Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters

We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution.     

On the estimation of entropy

Motivated by recent work of Joe (1989,Ann. Inst. Statist. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. Our estimators are different from Joe's, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothness and dimensionality also determines the convergence rate of Joe's estimator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing parameter.     

Consistent estimation of location region

Asymmetric multivariate probability distributions can be difficult to characterize in terms of their location. The works of Doksum (1975, Scand. J. Statist., 2, 11–22) and Blough (1985, Ann. Inst. Statist. Math., 37, 545–555) provide the construction of a location region for a given distribution. Any point in this closed, convex region will serve as a location parameter. It is the purpose of this paper to obtain a consistent estimator of the location region. Consistency is defined in terms of an appropriate pseudometric.     

Limiting properties of some measures of information

In this paper we investigate the limiting behaviour of the measures of information due to Csiszár, Rényi and Fisher. Conditions for convergence of measures of information and for convergence of Radon-Nikodym derivatives are obtained. Our results extend the results of Kullback (1959,Information Theory and Statistics, Wiley, New York) and Kirmani (1971,Ann. Inst. Statist. Math.,23, 157–162).     

On Adaptive Combination of Regression Estimators

Consider the problem of choosing between two estimators of the regression function, where one estimator is based on stronger assumptions than the other and thus the rates of convergence are different. We propose a linear combination of the estimators where the weights are estimated by Mallows' C _L . The adaptive estimator retains the optimal rates of convergence and is an extension of Stein-type estimators considered by Li and Hwang (1984, Ann. Statist., 12, 887-897) and related to an estimator in Burman and Chaudhuri (1999, Ann. Inst. Statist. Math. (to appear)).     

On a Multiparameter Version of Tukey's Linear Sensitivity Measure and its Properties

A multiparameter version of Tukey's (1965, Proc. Nat. Acad. Sci. U.S.A., 53, 127–134) linear sensitivity measure, as a measure of informativeness in the joint distribution of a given set of random variables, is proposed. The proposed sensitivity measure, under some conditions, is a matrix which is non-negative definite, weakly additive, monotone and convex. Its relation to Fisher information matrix and the best linear unbiased estimator (BLUE) are investigated. The results are applied to the location-scale model and it is observed that the dispersion matrix of the BLUE of the vector location-scale parameter is the inverse of the sensitivity measure. A similar property was established by Nagaraja (1994, Ann. Inst. Statist. Math., 46, 757–768) for the single parameter case when applied to the location and scale models. Two illustrative examples are included.     

Identifiability of mixtures of power-series distributions and related characterizations

The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the power-series distributions, is given. Specifically, by using probabilistic arguments, an elementary and shorter proof of the Lüxmann-Ellinghaus's (1987,Statist. Probab. Lett.,5, 375–378) result is obtained. Moreover, it is shown that this result is a special case of a stronger result connected with the Stieltjes moment problem. Some recent observations due to Singh and Vasudeva (1984,J. Indian Statist. Assoc.,22, 93–96) and Johnson and Kotz (1989,Ann. Inst. Statist. Math.,41, 13–17) concerning characterizations based on conditional distributions are also revealed as special cases of this latter result. Exploiting the notion of the identifiability of power-series mixtures, characterizations based on regression functions (posterior expectations) are obtained. Finally, multivariate generalizations of the preceding results have also been addressed.     

A general ratio estimator and its application in model based inference

A general ratio estimator of a population total is proposed as an approximation to the estimator introduced by Srivastava (1985,Bull. Internat. Statist. Inst.,51(10.3), 1–16). This estimator incorporates additional information gathered during the survey in a new way. Statistical properties of the general ratio estimator are given and its relationship to the estimator proposed by Srivastava is explored. A special kind of general ratio estimator is suggested and it turns out to be very efficient in a simulation study when compared to several other commonly used estimators.The work of this author was supported by AFOSR grant #830080.     

Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

Triple stage point estimation for the exponential location parameter

This paper deals with the problem of estimating the minimum lifetime (guarantee time) of the two parameter exponential distribution through a three-stage sampling procedure. Several forms of loss functions are considered. The regret associated with each loss function is determined. The results in this paper generalize the basic results of Hall (1981, Ann. Statist., 9, 1229–1238).     

Average worth and simultaneous estimation of the selected subset

Suppose a subset of populations is selected from the given k gamma G( _i,p ) (i = 1,2,...,k)populations, using Gupta's rule (1963, Ann. Inst. Statist. Math., 14, 199–216). The problem of estimating the average worth of the selected subset is first considered. The natural estimator is shown to be positively biased and the UMVUE is obtained using Robbins' UV method of estimation (1988, Statistical Decision Theory and Related Topics IV, Vol. 1 (eds. S. S. Gupta and J. O. Berger), 265–270, Springer, New York). A class of estimators that dominate the natural estimator for an arbitrary k is derived. Similar results are observed for the simultaneous estimation of the selected subset.     

Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2 Distribution

Consider the test problem about matrix normal mean M with the null hypothesis M = O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379–1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of ² distributions referred to as X² distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the X² distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).     

Sampling designs for regression coefficient estimation with correlated errors

The problem of estimating regression coefficients from observations at a finite number of properly designed sampling points is considered when the error process has correlated values and no quadratic mean derivative. Sacks and Ylvisaker (1966,Ann. Math. Statist.,39, 66–89) found an asymptotically optimal design for the best linear unbiased estimator (BLUE). Here, the goal is to find an asymptotically optimal design for a simpler estimator. This is achieved by properly adjusting the median sampling design and the simpler estimator introduced by Schoenfelder (1978, Institute of Statistics Mimeo Series No. 1201, University of North Carolina, Chapel Hill). Examples with stationary (Gauss-Markov) and nonstationary (Wiener) error processes and with linear and nonlinear regression functions are considered both analytically and numerically.Research supported by the Air Force Office of Scientific Research Contract No. 91-0030.     

Asymptotic Properties of Self-Consistent Estimators with Mixed Interval-Censored Data

Mixed interval-censored (MIC) data consist of n intervals with endpoints L _i and R _i , i = 1, ..., n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time X _i is only known to lie between L _i and R _i , i = 1, 2, ..., n. Peto (1973, Applied Statistics, 22, 86–91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290–295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.     

Estimation of two normal means which may be common

Consider the problem of estimating the mean of a normal population when independent samples from this as well as a second normal population are available. Pre-test estimators which combine the two sample means if a test of the hypothesis of equal population means accepts but otherwise use only the first sample mean, are compared to limited translation estimators which are derived in the spirit of Bickel (1984, Ann. Statist., 12, 864–879) (we also cover the cases of unknown variances). Our conclusion is that if the accuracy with which the second population mean can be estimated is of the same or better order of magnitude as teh accuracy with which the first can be estimated, then the limited translation estimators largely dominate the pre-test estimators in terms of mean square error loss.This research was supported by grants from the FRD of the CSIR of South Africa.     

Robust estimation of common regression coefficients under spherical symmetry

Consider the problem of estimating the common regression coefficients of two linear regression models where the two distributions of the errors may be different and unknown. Under the spherical symmetry assumption, the paper proves the superiority of a Graybill-Deal type combined estimator and the further improvement by the Stein effect which were exhibited by Shinozaki (1978, Comm. Statist. Theory Methods, 7, 1421–1432) in the normal case. This shows the robustness of the dominations since the conditions for the dominations are independent of the errors distributions.Research supported by NSERC Grant No. A3088 and GR-5 Grant from Faculty of Graduate Studies, Carleton University, Ottawa, Canada.