Estimation of order-restricted means of two normal populations under the LINEX loss function

In this paper, the estimation of order-restricted means of two normal distributions is studied under the LINEX loss function, when the variances are unknown and possibly unequal. Under certain sufficient conditions to be described in this paper, the proposed plug-in estimators uniformly perform better than the existing unrestricted maximum likelihood estimators. Further, the restricted maximum likelihood estimators are compared with the unrestricted maximum likelihood estimators under the Pitman nearness criterion. A simulation study is conducted and it is shown that our proposed plug-in estimators perform better than the unrestricted maximum likelihood estimators. An illustrative example of real data analysis is also given to compare the estimators.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

On universal admissibility of scale parameter estimators

A characterization result of Kushary (1998) regarding universal admissibility of equivariant estimators in the one parameter gamma distribution is generalized to a scale family of distributions with monotone likelihood ratio. New examples are given, among them the F-distribution with a scale parameter. In particular, universal admissibility is characterized within the class of location-scale equivariant estimators of the ratio of the variances of two normal distributions with unknown means. In this context the maximum likelihood estimator is shown to be universally inadmissible by virtue of a general sufficient condition for universal inadmissibility of a scale equivariant estimator. Received: January 2000     

Hybrid estimators for small diffusion processes based on reduced data

We deal with the Bayes type estimators and the maximum likelihood type estimators of both drift and volatility parameters for small diffusion processes defined by stochastic differential equations with small perturbations from high frequency data. From the viewpoint of numerical analysis, initial Bayes type estimators for both drift and volatility parameters based on reduced data are required, and adaptive maximum likelihood type estimators with the initial Bayes type estimators, which are called hybrid estimators, are proposed. The asymptotic properties of the initial Bayes type estimators based on reduced data are derived and it is shown that the hybrid estimators have asymptotic normality and convergence of moments. Furthermore, a concrete example and simulation results are given.     

Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.     

Restricted maximum likelihood estimation under Eisenhart model Ill

For a balanced two-way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non-negativity requirements of the variance components by L ee and K apadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non-negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.     

Nonparametric maximum likelihood estimation in a non locally compact setting

Maximum likelihood estimation is considered in the context of infinite dimensional parameter spaces. It is shown that in some locally convex parameter spaces sequential compactness of the bounded sets ensures the existence of minimizers of objective functions and the consistency of maximum likelihood estimators in an appropriate topology. The theory is applied to revisit some classical problems of nonparametric maximum likelihood estimation, to study location parameters in Banach spaces, and finally to obtain Varadarajan’s theorem on the convergence of empirical measures in the form of a consistency result for a sequence of maximum likelihood estimators. Several parameter spaces sharing the crucial compactness property are identified. This research was supported by grants from the National Sciences and Engineering Research Council of Canada and the Fonds FCAR de la Province de Québec.     

The Estimation of Pareto Distribution by a Weighted Least Square Method

The two-parameter Pareto distribution provides reasonably good fit to the distributions of income and property value, and explains many empirical phenomena. For the censored data, the two parameters are regularly estimated by the maximum likelihood estimator, which is complicated in computation process. This investigation proposes a weighted least square estimator to estimate the parameters. Such a method is comparatively concise and easy to perceive, and could be applied to either complete or truncated data. Simulation studies are conducted in this investigation to show the feasibility of the proposed method. This report will demonstrate that the weighted least square estimator gives better performance than unweighted least square estimators with simulation cases. We also illustrate that the weighted least square estimator is very close to maximum likelihood estimator with simulation studies.     

Estimation and testing for a Poisson autoregressive model

This article considers statistical inference for a Poisson autoregressive model. A condition for ergodicity and a necessary and sufficient condition for the existence of moments are given. Asymptotics for maximum likelihood estimator and weighted least squares estimators with estimated weights or known weights of the parameters are established. Testing conditional heteroscedasticity and testing the parameters under a simple ordered restriction are noted. A simulation study is also given.     

Partial maximum likelihood estimation of spatial probit models

This paper analyzes spatial Probit models for cross sectional dependent data in a binary choice context. Observations are divided by pairwise groups and bivariate normal distributions are specified within each group. Partial maximum likelihood estimators are introduced and they are shown to be consistent and asymptotically normal under some regularity conditions. Consistent covariance matrix estimators are also provided. Estimates of average partial effects can also be obtained once we characterize the conditional distribution of the latent error. Finally, a simulation study shows the advantages of our new estimation procedure in this setting. Our proposed partial maximum likelihood estimators are shown to be more efficient than the generalized method of moments counterparts.     

逐次定数截尾下Weibull分布参数的逆矩估计

基于逐次定数截尾样本下,讨论了两参数Weibull分布的参数估计,得到了两参数的逆矩估计.并利用模拟方法与极大似然估计作比较,模拟结果表明逆矩估计优于极大似然估计。     

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.     

Optimal allocation of measurements in a linear calibration process

A problem of allocation of measurements for a linear calibration process is considered in this article. It is assumed that a total of N measurements are made some of which may be measurements on two distinct standards, while the remaining measurements are on m different unknown specimens. We discuss allocation of the N measurements for the two standards and m unknown specimens based on A-optimality criterion, which is applied to asymptotic variances of maximum likelihood estimators for the true values of unknown specimens. It can be shown that the optimal allocation depends on the true values of unknown specimens. Hence, the user may resort to locally or Bayesian A-optimal measurement designs. Some practical solution is presented. Furthermore, the impact of prior on the allocation is also discussed.     

On Maximum Likelihood Estimation of Dynamic Panel Data Models

We analyse the finite sample properties of maximum likelihood estimators for dynamic panel data models. In particular, we consider transformed maximum likelihood (TML) and random effects maximum likelihood (RML) estimation. We show that TML and RML estimators are solutions to a cubic first‐order condition in the autoregressive parameter. Furthermore, in finite samples both likelihood estimators might lead to a negative estimate of the variance of the individual‐specific effects. We consider different approaches taking into account the non‐negativity restriction for the variance. We show that these approaches may lead to a solution different from the unique global unconstrained maximum. In an extensive Monte Carlo study we find that this issue is non‐negligible for small values of T and that different approaches might lead to different finite sample properties. Furthermore, we find that the Likelihood Ratio statistic provides size control in small samples, albeit with low power due to the flatness of the log‐likelihood function. We illustrate these issues modelling US state level unemployment dynamics.     

The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.     

Right censoring in a discrete life model

Although there are many sophisticated models for estimation of failure rate based on censored data in continuous distributions, not much work has been done in the discrete case. We introduce a discrete model for life lengths and consider its properties. For this model, we derive the corresponding maximum likelihood estimators of the parameters under Type I and Type II right-censoring. Received May 2000     

Some identification and estimation results for regression models with stochastically varying coefficients

Although various theoretical and applied papers have appeared in recent years concerned with the estimation and use of regression models with stochastically varying coefficients, little is available in the literature on the properties of the proposed estimators or the identifiability of the parameters of such models. The present paper derives sufficient conditions under which the maximum likelihood estimator is consistent and asymptotically normal and also provides sufficient conditions for the estimation of regression models with stationary stochastically varying coefficients. In many instances these requirements are found to have simple, intuitively appealing interpretations. Consistency and asymptotic normality is also proven for a two-step estimator and a method suggested by Rosenberg for generating initial estimates.     

Regression analysis with dichotomous regressors and misclassification

We discuss a regression model in which the regressors are dummy variables. The basic idea is that the observation units can be assigned to some well-defined combination of treatments, corresponding to the dummy variables. This assignment can not be done without some error, i.e. misclassification can play a role. This situation is analogous to regression with errors in variables. It is well-known that in these situations identification of the parameters is a prominent problem. We will first show that, in our case, the parameters are not identified by the first two moments but can be identified by the likelihood. Then we analyze two estimators. The first is a moment estimator involving moments up to the third order, and the second is a maximum likelihood estimator calculated with the help of the EM algorithm. Both estimators are evaluated on the basis of a small Monte Carlo experiment.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999