Estimation of p{y p> max(y1, y2, …, yp-1)}in theexponential case

This paper deals with the derivation of (i) the MLE (ii) the MVUE (iii) a Bayes estimator of the probability in the title, for the case p = 2. Simulation studies are carried out to compare these estimators. The results suggest that the MLE and the Bayes estimator are biased and the Bayes estimator have the smallest MSE. In the general case, explicit expression for the probability in the title is derived and the MLE and Bayes estimator are obtained. A general method of deriving the MVUE is pointed out. Because of the simulation studies for p = 2 it is recommended that the Bayes or predictive estimator should be used.     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82–86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.     

Estimation in the presence of incidental parameters

This paper considers the estimation of “structural” parameters when the number of unknown parameters increases with the sample size. Neyman and Scott (1948) had demonstrated that maximum likelihood estimators (MLE) of structural parameters may be inconsistent in this case. Patefield (1977) further observed that the asymptotic covariance matrix of the MLE is not equal to the inverse of the information matrix. In this paper we establish asymptotic properties of estimators (which include in particular the MLE) obtained via the usual likelihood approach when the incidental parameters are first replaced by their estimates (which are allowed to depend on the structural parameters). Conditions for consistency and asymptotic normality together with a proper formula for the asymptotic covariance matrix are given. The results are illustrated and applied to the problem of estimating linear functional relationships, and mild conditions on the incidental parameters for the MLE (or an adjusted MLE) to be consistent and asymptotically normal are obtained. These conditions are weaker than those imposed by previous authors.     

New Shrinkage Parameters for the Liu-type Logistic Estimators

The binary logistic regression is a widely used statistical method when the dependent variable has two categories. In most of the situations of logistic regression, independent variables are collinear which is called the multicollinearity problem. It is known that multicollinearity affects the variance of maximum likelihood estimator (MLE) negatively. Therefore, this article introduces new shrinkage parameters for the Liu-type estimators in the Liu (2003) in the logistic regression model defined by Huang (2012) in order to decrease the variance and overcome the problem of multicollinearity. A Monte Carlo study is designed to show the goodness of the proposed estimators over MLE in the sense of mean squared error (MSE) and mean absolute error (MAE). Moreover, a real data case is given to demonstrate the advantages of the new shrinkage parameters.     

New perspective on the benefits of the gene–environment independence in case–control studies

We study the benefit of exploiting the gene–environment independence (GEI) assumption for inferring the joint effect of genotype and environmental exposure on disease risk in a case–control study. By transforming the problem into a constrained maximum likelihood estimation problem we derive the asymptotic distribution of the maximum likelihood estimator (MLE) under the GEI assumption (MLE‐GEI) in a closed form. Our approach uncovers a transparent explanation of the efficiency gained by exploiting the GEI assumption in more general settings, thus bridging an important gap in the existing literature. Moreover, we propose an easy‐to‐implement numerical algorithm for estimating the model parameters in practice. Finally, we conduct simulation studies to compare the proposed method with the traditional prospective logistic regression method and the case‐only estimator. The Canadian Journal of Statistics 47: 473–486; 2019 © 2019 Statistical Society of Canada     

Maximum likelihood estimation in a model with interval data: a comment and extension

The purpose of this article is, first, to extend Poon et al. 's (1993) maximum likelihood estimation (MLE) of the correlation coefficient based on interval data to the regression case. Secondly, this paper shows how the traditional method of collecting interval data with the intervals chosen by the researcher can be easily modified to avoid the problems discussed by Poon et al. (1993). The MLE for this modification to the regression problem is presented. Finally, all the methods discussed in this paper are used to estimate the effects of grade point average and gender on student perceptions of the percentage of their classmates who have cheated on at least one exam in college.     

Variance-stabilizing and Confidence-stabilizing Transformations for the Normal Correlation Coefficient with Known Variances

Fosdick and Raftery (2012) recently encountered the problem of inference for a bivariate normal correlation coefficient ρ with known variances. We derive a variance-stabilizing transformation y(ρ) analogous to Fisher’s classical z-transformation for the unknown-variance case. Adjusting y for the sample size n produces an improved “confidence-stabilizing” transformation y_n(ρ) that provides more accurate interval estimates for ρ than the known-variance MLE. Interestingly, the z transformation applied to the unknown-but-equal-variance MLE performs well in the known-variance case for smaller values of |ρ|. Both methods are useful for comparing two or more correlation coefficients in the known-variance case.     

Inconsistency of the MLE for the Joint Distribution of Interval-Censored Survival Times and Continuous Marks

Abstract.  This paper considers the non-parametric maximum likelihood estimator (MLE) for the joint distribution function of an interval-censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark-specific cumulative hazard function of Huang & Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE, which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our theoretical results are supported by simulations.     

Bias reduction of maximum likelihood estimates for a modified skew-normal distribution

ABSTRACT

This paper presents a modified skew-normal (SN) model that contains the normal model as a special case. Unlike the usual SN model, the Fisher information matrix of the proposed model is always non-singular. Despite of this desirable property for the regular asymptotic inference, as with the SN model, in the considered model the divergence of the maximum likelihood estimator (MLE) of the skewness parameter may occur with positive probability in samples with moderate sizes. As a solution to this problem, a modified score function is used for the estimation of the skewness parameter. It is proved that the modified MLE is always finite. The quasi-likelihood approach is considered to build confidence intervals. When the model includes location and scale parameters, the proposed method is combined with the unmodified maximum likelihood estimates of these parameters.     

Estimation of system reliability for exponential distributions based on L ranked set sampling

Abstract

In the case where strength and stress both follow exponential distributions, this paper considers the maximum likelihood estimator (MLE) of the system reliability based on L ranked set sampling (LRSS). The proposed MLE is shown to have existence, uniqueness and asymptotic normality, and its asymptotic variance is obtained by the Fisher information matrix of LRSS. The values of asymptotic relative efficiencies show that the proposed MLE is always more efficient than the MLE using simple random sampling (SRS). However, the MLE using LRSS cannot be written in closed form. Therefore, the modified MLE is proposed using the technique replaced some terms in the maximum likelihood equations by their expectations. The newly modified MLE using LRSS is shown to be superior to the MLE using SRS. Finally, the proposed method is applied to a real data set on metastatic renal carcinoma study.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

On second-order admissibilities of the estimators of gamma shape vector

Simultaneous estimation problem of gamma shape vector is considered.First, it is shown that the maximum likelihood estimator (MLE), the bias corrected MLE, and the conditional MLE of shape vector are second-order inadmissible. Second, these estimators are improved up to the second order. Finally, we identify whether these improved estimators are second-order admissible or not. Simulation studies are also given.     

Estimation of common location parameter of several heterogeneous exponential populations based on generalized order statistics

In this article, several independent populations following exponential distribution with common location parameter and unknown and unequal scale parameters are considered. From these populations, several independent samples of generalized order statistics (gos) are drawn. Under the setup of gos, the problem of estimation of common location parameter is discussed and various estimators of common location parameter are derived. The authors obtained maximum likelihood estimator (MLE), modified MLE and uniformly minimum variance unbiased estimator of common location parameter. Furthermore, under scaled-squared error loss function, a general inadmissibility result of invariant estimator is proposed. The derived results are further reduced for upper record values which is a special case of gos. Finally, simulation study and real life example are reported to show the performances of various competing estimators in terms of percentage risk improvement.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

AN ALTERNATIVE PARAMETRIC APPROACH FOR DISCRETE MISSING DATA PROBLEMS

We propose an iterative method of estimation for discrete missing data problems that is conceptually different from the Expectation–Maximization (EM) algorithm and that does not in general yield the observed data maximum likelihood estimate (MLE). The proposed approach is based conceptually upon weighting the set of possible complete-data MLEs. Its implementation avoids the expectation step of EM, which can sometimes be problematic. In the simple case of Bernoulli trials missing completely at random, the iterations of the proposed algorithm are equivalent to the EM iterations. For a familiar genetics-oriented multinomial problem with missing count data and for the motivating example with epidemiologic applications that involves a mixture of a left censored normal distribution with a point mass at zero, we investigate the finite sample performance of the proposed estimator and find it to be competitive with that of the MLE. We give some intuitive justification for the method, and we explore an interesting connection between our algorithm and multiple imputation in order to suggest an approach for estimating standard errors.     

Pitman closeness of the class of isotonic estimators for ordered scale parameters of two Gamma distributions

The problem of estimating order-restricted scale parameters of two Gamma distributions is considered under the Pitman closeness criterion. A class of isotonic estimators including the MLE is proposed. Some properties of this class of isotonic estimators is given under the Pitman closeness criterion. In particular, some properties of the MLE naturally follow. We use simulation comparisons to examine the results.