Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

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.     

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.     

Linear approximate ML estimation in scaled Type I generalized logistic distribution based on Type-II censored samples

The scaled (two-parameter) Type I generalized logistic distribution (GLD) is considered with the known shape parameter. The ML method does not yield an explicit estimator for the scale parameter even in complete samples. In this article, we therefore construct a new linear estimator for scale parameter, based on complete and doubly Type-II censored samples, by making linear approximations to the intractable terms of the likelihood equation using least-squares (LS) method, a new approach of linearization. We call this as linear approximate maximum likelihood estimator (LAMLE). We also construct LAMLE based on Taylor series method of linear approximation and found that this estimator is slightly biased than that based on the LS method. A Monte Carlo simulation is used to investigate the performance of LAMLE and found that it is almost as efficient as MLE, though biased than MLE. We also compare unbiased LAMLE with BLUE based on the exact variances of the estimators and interestingly this new unbiased LAMLE is found just as efficient as the BLUE in both complete and Type-II censored samples. Since MLE is known as asymptotically unbiased, in large samples we compare unbiased LAMLE with MLE and found that this estimator is almost as efficient as MLE. We have also discussed interval estimation of the scale parameter from complete and Type-II censored samples. Finally, we present some numerical examples to illustrate the construction of the new estimators developed here.     

A note on the common location parameter of several exponential populations

The problem of estimation of an unknown common location parameter of several exponential populations with unknown and possibly unequal scale parameters is considered. A wide class of estimators, including both a modified maximum likelihood estimator (MLE), and the uniformly minimum variance unbiased estimator (Umvue) proposed by ghosh and razmpour(1984), is obtained under a class of convex loss functions.     

ON THE COMMON SCALE PARAMETER OF SEVERAL PARETO POPULATIONS IN CENSORED SAMPLES

The problem of estimation of an unknown common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters in censored samples is considered. A new class of estimators which includes both the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) is proposed and examined under a squared error loss.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized half-normal distribution

Cooray and Ananda introduced a two-parameter generalized Half-Normal distribution which is useful for modelling lifetime data, while its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters of the model. In this paper, we adopt two approaches for bias reduction of the MLEs of the parameters of generalized Half-Normal distribution. The first approach is the analytical methodology suggested by Cox and Snell and the second is based on parametric Bootstrap resampling method. Additionally, the method of moments (MMEs) is used for comparison purposes. The numerical evidence shows that the analytic bias-corrected estimators significantly outperform their bootstrapped-based counterpart for small and moderate samples as well as for MLEs and MMEs. Also, it is apparent from the results that bias- corrected estimates of shape parameter perform better than that of scale parameter. Further, the results show that bias-correction scheme yields nearly unbiased estimates. Finally, six fracture toughness real data sets illustrate the application of our methods.     

Exponential Parameter Estimation for Data Sets Containing Left and Right Censored Observations

Lifetime data sets may contain both left and right censored observations. Regression, maximum likelihood, and the best iinear unbiased parameter estimates are given for the exponential distribution, along with a comparison of the estimators.     

On the estimation of the parameters of the von mises distribution

The usual maximum likelihood estimators of the parameters of the von Mises distribution are shown to perform badly in small samples. In view of this and the fact that these estimators require a large amount of computation, alternative, simpler estimators are proposed. It is shown that these estimators are at least comparable to the traditional estimators and are, in many cases, superior to them. We also apply the procedure of jackknifing to the maximum likelihood estimator of the concentration parameter of the von Mises distribution and compare the properties of the jackknifed estimator with the other estimators considered in this paper.     

Non-minimaxity of linear combinations of restricted location estimators and related problems

The estimation of a linear combination of several restricted location parameters is addressed from a decision-theoretic point of view. Although the corresponding linear combination of the unbiased estimators is minimax under the restricted problem, it has a drawback of taking values outside the restricted parameter space. Thus, it is reasonable to use the linear combination of the restricted estimators such as maximum likelihood or truncated estimators. In this paper, a necessary and sufficient condition for such restricted estimators to be minimax is derived, and it is shown that the restricted estimators are not minimax when the number of the location parameters is large. The condition for minimaxity is examined for some specific distributions. Finally, similar problems of estimating the product and sum of the restricted scale parameters are studied, and it is shown that analogous non-dominance properties appear when the number of the scale parameters is large.     

An asymptotic equivalence of the cross-data and predictive estimators

Abstract

Estimators using multiplicative tuning parameters for maximum likelihood estimators in cross-validation are called cross-data estimators in this paper. Single-sample versions of the cross-data estimators have been called predictive estimators in literatures, which are given by maximizing the expected log-likelihood, where the two-fold expectations are taken over the distributions of future and current data using maximum likelihood estimators based on current data. An asymptotic equivalence of the cross-data and predictive estimators is shown, which guarantees an optimality of the predictive estimator when an unknown population parameter vector is replaced by the sample counterpart. Examples using typical statistical distributions are shown.     

Properties of estimators for the gamma distribution

Accurate moments of maximum likelihood and moment estimators for the scale and shape parameters of a two parameter gamma density are given, the former being tabulated over a segment of the parameter space. In addition, joint acceptance regions are given for a particular case. The three parameter model is also considered and comments made on second order asymptotics for the maximum likelihood estimators     

Inference and prediction for the burr type x model based on records

Based on record values, the maximum likelihood, minimum variance unbiased and Bayes estimators of the one parameter of the Burr type X distribution are computed and compared. The Bayesian and non-Bayesian confidence intervals for this parameter are also presented. A Bayesian prediction interval for the sth future record is obtained in a closed form. Based on simulated record values, numerical computations and comparisons between the different estimators are given     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.     

Estimation of Pr( x<y ) in the exponential case with common location parameter

This paper considers the problem of estimating the probability P = Pr(X < Y) when X and Y are independent exponential random variables with unequal scale parameters and a common location parameter. Uniformly minimum variance unbiased estimator of P is obtained. The asymptotic distribution of the maximum likelihood estimator is obtained and then the asymptotic equivalence of the two estimators is established. Performance of the two estimators for moderate sample sizes is studied by Monte Carlo simulation. An approximate interval estimator is also obtained.     

Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type ii censored samples

It is known that the maximum likelihood methods does not provide explicit estimators for the mean and standard deviation of the normal distribution based on Type II censored samples. In this paper we present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We obtain the variances and covariance of these estimators. We also show that these estimators are almost as eficient as the maximum likelihood (ML) estimators and just as eficient as the best linear unbiased (BLU), and the modified maximum likelihood (MML) estimators. Finally, we illustrate this method of estimation by applying it to Gupta's and Darwin's data.     

The combined dynamically weighted modified maximum likelihood estimators of the location and scale parameters

In this study, two new types of estimators of the location and scale parameters are proposed having high efficiency and robustness; the dynamically weighted modified maximum likelihood (DWMML) and the combined dynamically weighted modified maximum likelihood (CDWMML) estimators. Three pairs of the DWMML and two pairs of the CDWMML estimators of the location and scale parameters are produced, namely, the DWMML1, the DWMML2 and the DWMML3, and the CDWMML1 and the CDWMML2 estimators, respectively. Based on the simulation results, the DWMML1 estimators of the location and scale parameters are almost fully efficient (under normality) and robust at the same time. The DWMML3 estimators are asymptotically fully efficient and more robust than the M-estimators. The DWMML2 estimators are a compromise between efficiency and robustness. The CDWMML1 and CDWMML2 estimators are jointly very efficient and robust. Particularly, the CDWMML1 and CDWMML2 estimators of the scale parameter are superior compared to the other estimators of the scale parameter.