Closer estimators of a common mean in the sense of Pitman

Consider the problem of estimating the common mean of two normal populations with different unknown variances. Suppose a random sample of sizem is drawn from the first population and a random sample of sizen is drawn from the second population. The paper gives a family of estimators closer than the sample mean of the first population in the sense of Pitman (1937,Proc. Cambridge Phil. Soc.,33, 212–222). In particular, the Graybill-Deal estimator (1959,Biometrics,15, 543–550) is shown to be closer than each of the sample means ifm5 andn5.     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

Tail Behavior and Breakdown Properties of Equivariant Estimators of Location

For translation and scale equivariant estimators of location, inequalities connecting tail behavior and the finite-sample breakdown point are proved, analogous to those established by He et al. (1990, Econometrika, 58, 1195–1214) for monotone and translation equivariant estimators. Some other inequalities are given as well, enabling to establish refined bounds and in some cases exact values for the tail behavior under heavy- and light-tailed distributions. The inequalities cover translation and scale equivariant estimators in great generality, and they involve new breakdown-related quantities, whose relations to the breakdown point are discussed. The worth of tail-behavior considerations in robustness theory is demonstrated on examples, showing the impact of the basic two techniques in robust estimation: trimming and averaging. The mathematical language employs notions from regular variation theory.     

Estimation of parameters in a two-parameter exponential distribution using ranked set sample

In situations where the experimental or sampling units in a study can be easily ranked than quantified, McIntyre (1952,Aust. J. Agric. Res.,3, 385–390) proposed that the mean ofn units based on aranked set sample (RSS) be used to estimate the population mean, and observed that it provides an unbiased estimator with a smaller variance compared to a simple random sample (SRS) of the same sizen. McIntyre's concept ofRSS is essentially nonparametric in nature in that the underlying population distribution is assumed to be completely unknown. In this paper we further explore the concept ofRSS when the population is partially known and the parameter of interest is not necessarily the mean. To be specific, we address the problem of estimation of the parameters of a two-parameter exponential distribution. It turns out that the use ofRSS and its suitable modifications results in much improved estimators compared to the use of aSRS.     

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.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

Estimation of the Common Mean of a Bivariate Normal Population

In this paper we discuss the problem of estimating the common mean of a bivariate normal population based on paired data as well as data on one of the marginals. Two double sampling schemes with the second stage sampling being either a simple random sampling (SRS) or a ranked set sampling (RSS) are considered. Two common mean estimators are proposed. It is found that under normality, the proposed RSS common mean estimator is always superior to the proposed SRS common mean estimator and other existing estimators such as the RSS regression estimator proposed by Yu and Lam (1997, Biometrics, 53, 1070–1080). The problem of estimating the mean Reid Vapor Pressure (RVP) of regular gasoline based on field and laboratory data is considered.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.     

Approximated Bayes and empirical Bayes confidence intervals—The known variance case

In this paper hierarchical Bayes and empirical Bayes results are used to obtain confidence intervals of the population means in the case of real problems. This is achieved by approximating the posterior distribution with a Pearson distribution. In the first example hierarchical Bayes confidence intervals for the Efron and Morris (1975, J. Amer. Statist. Assoc., 70, 311–319) baseball data are obtained. The same methods are used in the second example to obtain confidence intervals of treatment effects as well as the difference between treatment effects in an analysis of variance experiment. In the third example hierarchical Bayes intervals of treatment effects are obtained and compared with normal approximations in the unequal variance case.Financially supported by the CSIR and the University of the Orange Free State, Central Research Fund.     

Simultaneous estimation of means of classified normal observations

Simultaneous estimation of normal means is considered for observations which are classified into several groups. In a one-way classification case, it is shown that an adaptive shrinkage estimator dominates a Stein-type estimator which shrinks observations towards individual class averages as Stein's (1966,Festschrift for J. Neyman, (ed. F. N. David), 351–366, Wiley, New York) does, and is minimax even if class sizes are small. Simulation results under quadratic loss show that it is slightly better than Stein's (1966) if between variances are larger than within ones. Further this estimator is shown to improve on Stein's (1966) with respect to the Bayes risk. Our estimator is derived by assuming the means to have a one-way classification structure, consisting of three random terms of grand mean, class mean and residual. This technique can be applied to the case where observations are classified into a two-stage hierarchy.     

Run length not required: Optimal-mse dynamic batch means estimators for steady-state simulations

This paper addresses the estimation of the variance of the sample mean from steady-state simulations without requiring the knowledge of simulation run length a priori. Dynamic batch means is a new and useful approach to implementing the traditional batch means in limited memory without the knowledge of the simulation run length. However, existing dynamic batch means estimators do not allow one to control the value of batch size, which is the performance parameter of the batch means estimators. In this work, an algorithm is proposed based on two dynamic batch means estimators to dynamically estimate the optimal batch size as the simulation runs. The simulation results show that the proposed algorithm requires reasonable computation time and possesses good statistical properties such as small mean-squared-error (mse).     

Computing S Estimators for Regression and Multivariate Location/Dispersion

Abstract

An improved resampling algorithm for S estimators reduces the number of times the objective function is evaluated and increases the speed of convergence. With this algorithm, S estimates can be computed in less time than least median squares (LMS) for regression and minimum volume ellipsoid (MVE) for location/scatter estimates with the same accuracy. Here accuracy refers to the randomness due to the algorithm. S estimators are also more statistically efficient than the LMS and MVE estimators, that is, they have less variability due to the randomness of the data.     

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)).     

Some modifications of improved estimators of a normal variance

Consider the problem of estimating a normal variance based on a random sample when the mean is unknown. Scale equivariant estimators which improve upon the best scale and translation equivariant one have been proposed by several authors for various loss functions including quadratic loss. However, at least for quadratic loss function, improvement is not much. Herein, some methods are proposed to construct improving estimators which are not scale equivariant and are expected to be considerably better when the true variance value is close to the specified one. The idea behind the methods is to modify improving equivariant shrinkage estimators, so that the resulting ones shrink little when the usual estimate is less than the specified value and shrink much more otherwise. Sufficient conditions are given for the estimators to dominate the best scale and translation equivariant rule under the quadratic loss and the entropy loss. Further, some results of a Monte Carlo experiment are reported which show the significant improvements by the proposed estimators.     

Multivariate Limited Translation Hierarchical Bayes Estimators

Based on the notion of predictive influence functions, the paper develops multivariate limited translation hierarchical Bayes estimators of the normal mean vector which serve as a compromise between the hierarchical Bayes and maximum likelihood estimators. The paper demonstrates the superiority of the limited translation estimators over the usual hierarchical Bayes estimators in terms of the frequentist risks when the true parameter to be estimated departs widely from the grand average of all the parameters.     

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.     

Kernel Density and Hazard Rate Estimation for Censored Data under α-Mixing Condition

We derive rates of uniform strong convergence for kernel density estimators and hazard rate estimators in the presence of right censoring. It is assumed that the failure times (survival times) form a stationary -mixing sequence. Moreover, we show that, by an appropriate choice of the bandwidth, both estimators attain the optimal strong convergence rate known from independent complete samples. The results represent an improvement over that of Cai's paper (cf. Cai (1998b, J. Multivariate Anal., 67, 23–34)).     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

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.