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.     

Modified quasi-profile likelihoods from estimating functions

We discuss higher-order adjustments for a quasi-profile likelihood for a scalar parameter of interest, in order to alleviate some of the problems inherent to the presence of nuisance parameters, such as bias and inconsistency. Indeed, quasi-profile score functions for the parameter of interest have bias of order O(1)$\mathrm{O}(1)$, and such bias can lead to poor inference on the parameter of interest. The higher-order adjustments are obtained so that the adjusted quasi-profile score estimating function is unbiased and its variance is the negative expected derivative matrix of the adjusted profile estimating equation. The modified quasi-profile likelihood is then obtained as the integral of the adjusted profile estimating function. We discuss two methods for the computation of the modified quasi-profile likelihoods: a bootstrap simulation method and a first-order asymptotic expression, which can be simplified under an orthogonality assumption. Examples in the context of generalized linear models and of robust inference are provided, showing that the use of a modified quasi-profile likelihood ratio statistic may lead to coverage probabilities more accurate than those pertaining to first-order Wald-type confidence intervals.     

Semi-parametric estimation of the autoregressive parameter in non-Gaussian Ornstein–Uhlenbeck processes

ABSTRACT

This paper considers the problem of estimating the autoregressive parameter in discretely observed Ornstein–Uhlenbeck processes. Two consistent estimators are proposed: one obtained by maximizing a kernel-based likelihood function, and another by minimizing a Kolmogorov-type distance from independence. After establishing the consistency of these estimators, their finite-sample performance and possible normality in large samples, is investigated by means of extensive simulations. An illustrative example to credit rating is discussed.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

Bayesian,MLE, and GMM Estimation of a Spot Rate Model

ABSTRACT

We develop Markov chain Monte Carlo algorithms for estimating the parameters of the short-term interest rate model. Using Monte Carlo experiments we compare the Bayes estimators with the maximum likelihood and generalized method of moments estimators. We estimate the model using the Japanese overnight call rate data.     

Semiparametric estimation for the dispersion parameter in the analysis of over- or underdispersed count data

This paper investigates several semiparametric estimators of the dispersion parameter in the analysis of over- or underdispersed count data when there is no likelihood available. In the context of estimating the dispersion parameter, we consider the double-extended quasi-likelihood (DEQL), the pseudo-likelihood and the optimal quadratic estimating (OQE) equations method and compare them with the maximum likelihood method, the method of moments and the extended quasi-likelihood through simulation study. The simulation study shows that the estimator based on the DEQL has superior bias and efficiency property for moderate and large sample size, and for small sample size the estimator based on the OQE equations outperforms the other estimators. Three real-life data sets arising in biostatistical practices are analyzed, and the findings from these analyses are quite similar to what are found from the simulation study.     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

Design considerations for small experiments and simple logistic regression

Inference for a generalized linear model is generally performed using asymptotic approximations for the bias and the covariance matrix of the parameter estimators. For small experiments, these approximations can be poor and result in estimators with considerable bias. We investigate the properties of designs for small experiments when the response is described by a simple logistic regression model and parameter estimators are to be obtained by the maximum penalized likelihood method of Firth [Firth, D., 1993, Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38]. Although this method achieves a reduction in bias, we illustrate that the remaining bias may be substantial for small experiments, and propose minimization of the integrated mean square error, based on Firth's estimates, as a suitable criterion for design selection. This approach is used to find locally optimal designs for two support points.     

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.     

Estimation of odds ratios under order restrictions

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.     

A Robust and Almost Fully Efficient M-Estimator

Simultaneous robust estimates of location and scale parameters are derived from a class of M-estimating equations. A coefficient p ( p > 0), which plays a role similar to that of a tuning constant in the theory of M-estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi-variate normal and have positive breakdown for some choices of p . When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p , when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p . A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber's minimax and bisquare M-estimators.     

Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions

The skew normal model is a class of distributions that extends the Gaussian family by including a shape parameter. Despite its nice properties, this model presents some problems with the estimation of the shape parameter. In particular, for moderate sample sizes, the maximum likelihood estimator is infinite with positive probability. As a solution, we use a modified score function as an estimating equation for the shape parameter. It is proved that the resulting modified maximum likelihood estimator is always finite. For confidence intervals a quasi-likelihood approach is considered. When regression and scale parameters are present, the method is combined with maximum likelihood estimators for these parameters. Finally, also the skew t distribution is considered, which may be viewed as an extension of the skew normal. The same method is applied to this model, considering the degrees of freedom as known.     

Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Nonparametric maximum likelihood estimators for ar and ma time series

The problem of estimating the parameters of moving average or autoregressive time series is studied when the error distribution is completely unknown. Four nonparametric maximum likelihood estimators (NPMLE) are presented for this purpose. These estimators are compared with the classical moment and least squares estimators in a simulation study. The behavior of these NPMLEs is much better than the classical ones, suggesting that they should be used extensively when no parametric information is known in advance about the error distribution. An application of these estimators to coal mining accidents data is also included.     

On the integrated maximum likelihood estimators for a closed population capture–recapture model with unequal capture probabilities

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study.     

Estimates of the parameters of type-II extreme value distribution from censored samples

Maximum likelihood estimators of a Type-II extreme value distribution are derived from doubly censored samples. The asymptotic variances and covariances of the maximum likelihood estimators are discussed and these are numerically evaluated for different censoring proportions q₁ = 0.0(0. l) (0.9) from below and q₂ = 0.0 (0. l) (0.9- q₁) from above. The asymptotic relative efficiencies of the parameter estimates revealed that lower order statistics are more important for estimating the parameters of Type-II extreme value distribution as compared to higher order statistics.     

Estimation of the parameters of the gamma distribution by means of a maximal invariant

The use of a scale invariance criterion allows estimation of the shape parameter of the two parameter gamma distribution without estimating the scale parameter. Simulation experiments are used to show that the resulting estimators of both parameters are better than the usual maximum likelihood estimators in terms of both bias and mean square error. Approximately unbiased versions of the maximal invariant based estimators are derived and are shown to be as good as approximately unbiased versions of the usual maximum likelihood estimators