Unbiased estimation of the parameter of a selected binomial population

The problem is to estimate the parameter of a selected binomial population. The selction rule is to choose the population with the greatest number of successes and, in the case of a tie, to follow one of two schemes: either choose the population with the smallest index or randomize among the tied populations. Since no unbiased estimator exists in the above case, we employ a second stage of sampling and take additional observations on the selected population. We find the uniformly minimum variance unbiased estimator (UMVUE) under the first tie break scheme and we prove that no UMVUE exists under the second. We find an unbiased estimator with desirable properties in the case where no UMVUE exists.     

On the relationship between levinson recursion and the r and s arrays for arma model identification

Motivated by the papers of Woodward and Gray (1979) and Gray, Kelly and McIntire (1978) on the R and S array approach to ARMA modeling, the authors show that the R and S array algorithm is completely equivalent to Levinson recursion. Since entries in the R and S array can be computed by either algorithm, the equivalence provides greater insight into the R and S methodology as well as its links to Akaike's AIC or FPE. Numerical simulations serve to highlight the differences between the various approaches as well as illustrate the problems associated with exact methods. The K and S array approach is shown to be an effective procedure for determining ARMA model orders.     

Influence analysis in truncated distributions

Conditional bias and asymptotic mean sensitivity curve (AMSC) are useful measures to assess the possible effect of an observation on an estimator when sampling from a parametric model. In this paper we obtain expressions for these measures in truncated distributions and study their theoretical properties. Specific results are given for the UMVUE of a parametric function. We note that the AMSC for the UMVUE in truncated distributions verifies some of the most relevant properties we got in a previous paper for the AMSC of UMVUE in the NEF-QVF case, main differences are also established. As for the conditional bias, since it is a finite sample measure, we include some practical examples to illustrate its behaviour when the sample size increases.     

On the existence of transformations preserving the structure of order statistics in lower dimensions

In a Type-II right censored sample from the standard uniform distribution, several transformations of respective order statistics are examined, which transform the censored sample into a complete sample in a lower dimension. Such transformations have been considered by Lin et al. (2008), Michael and Schucany (1979) and O’Reilly and Stephens (1988) in the context of goodness-of-fit tests. It is shown that by dropping the assumption of an underlying uniform distribution, these transformed random variables can no longer be considered themselves as order statistics, in general. In the case of the transformation of Michael and Schucany, it is shown that the uniform distribution is the only one possessing this property.     

Best unbiased estimation of fixed effects in mixed ANOVA models

In a linear model with an arbitrary variance–covariance matrix, Zyskind (Ann. Math. Statist. 38 (1967) 1092) provided necessary and sufficient conditions for when a given linear function of the fixed-effect parameters has a best linear unbiased estimator (BLUE). If these conditions hold uniformly for all possible variance–covariance parameters (i.e., there is a UBLUE) and if the data are assumed to be normally distributed, these conditions are also necessary and sufficient for the parametric function to have a uniformly minimum variance unbiased estimator (UMVUE). For mixed-effects ANOVA models, we show how these conditions can be translated in terms of the incidence array, which facilitates verification of the UBLUE and UMVUE properties and facilitates construction of designs having such properties.     

Inadmissibility of the uniformly minimum variance unbiased estimator of the inverse gaussian variance

The uniformly minimum variance unbiased estimator (UMVUE) of the variance of the inverse Gaussian distribution is shown to be inadmissible in terms of the mean squared error, and a dominating estimator is given. A dominating estimator to the maximum likelihood estimator (MLE) of the variance and estimators dominating the MLE's and the UMVUE's of other parameters are also given.     

The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

Evaluation and comparison of estimations in the generalized exponential-Poisson distribution

The uniformly minimum variance unbiased, maximum-likelihood, percentile and least-squares estimators of the probability density function and the cumulative distribution function are derived for the generalized exponential-Poisson distribution. This model has shown to be useful in reliability and lifetime data modelling, especially when the hazard rate function has a bathtub shape. Simulation studies are also carried out to show that the maximum-likelihood estimator is better than the uniformly minimum variance unbiased estimator (UMVUE) and that the UMVUE is better than others.     

UMVU estimation of the reliability function of the generalized life distributions

The problems of estimating the reliability function and P=P_rX > Y are considered for the generalized life distributions. Uniformly minimum variance unbiased estimators (UMVUES) of the powers of the parameter involved in the probabilistic model and the probability density function (pdf) at a specified point are derived. The UMVUE of the pdf is utilized to obtain the UMVUE of the reliability function and ‘P’. Our method of obtaining these estimators is quite simple than the traditional approaches. A theoretical method of studying the behaviour of the hazard-rate is provided.     

Simultaneous estimation of the hardy-weinberg proportions

We consider the problem of simultaneously estimating k + 1 related proportions, with a special emphasis on the estimation of Hardy-Weinberg (HW) proportions. We prove that the uniformly minimum-variance unbiased estimator (UMVUE) of two proportions which are individually admissible under squared-error loss are inadmissible in estimating the proportions jointly. Furthermore, rules that dominate the UMVUE are given. A Bayesian analysis is then presented to provide insight into this inadmissibility issue: The UMVUE is undesirable because the two estimators are Bayes rules corresponding to different priors. It is also shown that there does not exist a prior which yields the maximum-likelihood estimators simultaneously. When the risks of several estimators for the HW proportions are compared, it is seen that some Bayesian estimates yield significantly smaller risks over a large portion of the parameter space for small samples. However, the differences in risks become less significant as the sample size gets larger.     

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.     

Estimation of p(y<x) in the gamma case

We consider estimation of P(Y<X) when X?Γr(M,λ) and Y?Γ(N,μ) are independent with M and N known. A concise representation of the UMVUE and several representations for the MLE are derived. Closed-form exact expressions of both MSE's and the bias of the MLE are obtained. Large-sample results are given and numerical comparison of the two point estimators is made. Confidence intervals are given.     

Estimation and testing of P(Y > X) in two-parameter exponential distributions

In this article, we obtain the UMVUE of the reliability function ξ=P(Y>X) and the UMVUE of ξ ^k =[P(Y>X)] ^k in the two-parameter exponential distributions with known scale parameters. We also derive the distribution of the UMVUE of ξ and further considering the tests of hypotheses regarding the reliability function ξ.     

Book Reviews

Book reviewed in this article:
Asymptotic optimal inference for non-ergodic models. By Ishwar V. Basawa and David J. Scott.
Extremes and Related Properties of Random Sequences and Processes. By M. R. Leadbetter.
Directory of Statistical Microcomputer Software, 1985 Edition. By Wayne A. Woodward Alan C. Elliot & Henry L. Gray.     

A fortran implementation of univariate fourier series density estimation

A set of Fortran-77 subroutines is described which compute a nonparametric density estimator expressed as a Fourier series. In addition, a subroutine is given for the estimation of a cumulative distribution. Performance measures are given based on samples from a Weibull distribution. Due to small size and modest space demands, these subroutines are easily implemented on most small computers.     

Shrinkage and modification techniques in estimation of variance and the related problems: A review

One of the surprising decision-theoretic results Charles Stein discovered is the inadmissibility of the uniformly minimum variance unbiased estirnator(UMVUE) of the variance of a normal distribution with an unknown mean. Some methods for deriving estimators better than the UMVUE were given by Stein. Brown, Brewster and Zidek. Recently Kubokawa established a novel approach, called the IERD method, by use of which one gets a unified class of improved estimators including their previous procedures. This paper gives a review for a series of these decision-theoretical developments as well as surveys the study of the variance-estimation problem from various aspects. Related to this issue, the paper enumerates several topics with the situations where the usual plain estimators are required to be shrunken or modified, and gives reasonable procedures improving the usual ones through the IERD method.     

On estimation of the PDF and CDF of the Lindley distribution

This article addresses two methods of estimation of the probability density function (PDF) and cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE). Since the Lindley distribution is more flexible than the exponential distribution, the same estimators have been found out for the exponential distribution and compared. Monte Carlo simulations and a real data analysis are performed to compare the performances of the proposed methods of estimation.     

Bayesian estimation of mean and square of mean of normal distribution using linex loss function

In this paper, the Bayes estimators for mean and square of mean ol a normal distribution with mean μ and vaiiance σ r2 (known), relative to LINEX loss function are obtained Comparisons in terms of risk functions and Bayes risks of those under LINEX loss and squared error loss functions with their respective alternative estimators viz, UMVUE and Bayes estimators relative to squared error loss function, are made. It is found that Bayes estimators relative to LINEX loss function dominate the alternative estimators m terms of risk function snd Bayes risk. It is also found that if t2 is unknown the Bayes estimators are still preferable over alternative estimators.     

On the pade tablé and its relationship to the r and sarrays and arm a modeling

The Box-Jenkins method is a popular and important technique for modeling and forecasting of time series. Unfortunately the problem of determining the appropriate ARMA forecasting model (or indeed if an ARMA model holds) is a major drawback to the use of the Box-Jenkins methodology. Gray et al. (1978) and Woodward and Gray (1979) have proposed methods of estimating p and qin ARMA modeling based on the R and Sarrays that circumvent some of these modeling difficulties.

In this paper we generalize the R and S arrays by showing a relationship to Padé approximunts and then show that these arrays have a much wider application than in just determining model order. Particular non-ARMA models can be identified as well. This includes certain processes that consist of deterministic functions plus ARMA noise, indeed we believe that the combined R and S arrays are the best overall tool so fur developed for the identification of general 2nd order (not just stationary) time scries models.     

An identity for exponential distributions with the common location parameter and its applications

An identity for exponential distributions with an unknown common location parameter and unknown and possibly unequal scale parameters is established.Through use of the identity the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) of a quantile of an exponential population are compared under the squared error loss.A class of estimators dominating both MLE and UMVUE is obtained by using the identity.