Bias-corrected estimators of scalar skew normal

One problem of skew normal model is the difficulty in estimating the shape parameter, for which the maximum likelihood estimate may be infinite when sample size is moderate. The existing estimators suffer from large bias even for moderate size samples. In this article, we proposed five estimators of the shape parameter for a scalar skew normal model, either by bias correction method or by solving a modified score equation. Simulation studies show that except bootstrap estimator, the proposed estimators have smaller bias compared to those estimators in literature for small and moderate samples.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.     

Statistical inference of Marshall-Olkin bivariate Weibull distribution with three shocks based on progressive interval censored data

There are several failure modes may cause system failed in reliability and survival analysis. It is usually assumed that the causes of failure modes are independent each other, though this assumption does not always hold. Dependent competing risks modes from Marshall-Olkin bivariate Weibull distribution under Type-I progressive interval censoring scheme are considered in this paper. We derive the maximum likelihood function, the maximum likelihood estimates, the 95% Bootstrap confidence intervals and the 95% coverage percentages of the parameters when shape parameter is known, and EM algorithm is applied when shape parameter is unknown. The Monte-Carlo simulation is given to illustrate the theoretical analysis and the effects of parameters estimates under different sample sizes. Finally, a data set has been analyzed for illustrative purposes.     

Sample-based Maximum Likelihood Estimation of the Autologistic Model

New recursive algorithms for fast computation of the normalizing constant for the autologistic model on the lattice make feasible a sample-based maximum likelihood estimation (MLE) of the autologistic parameters. We demonstrate by sampling from 12 simulated 420×420 binary lattices with square lattice plots of size 4×4, …, 7×7 and sample sizes between 20 and 600. Sample-based results are compared with ‘benchmark’ MCMC estimates derived from all binary observations on a lattice. Sample-based estimates are, on average, biased systematically by 3%–7%, a bias that can be reduced by more than half by a set of calibrating equations. MLE estimates of sampling variances are large and usually conservative. The variance of the parameter of spatial association is about 2–10 times higher than the variance of the parameter of abundance. Sample distributions of estimates were mostly non-normal. We conclude that sample-based MLE estimation of the autologistic parameters with an appropriate sample size and post-estimation calibration will furnish fully acceptable estimates. Equations for predicting the expected sampling variance are given.     

Point and Interval Estimation for a Generalized Logistic Distribution Under Progressive Type II Censoring

The likelihood equations based on a progressively Type II censored sample from a Type I generalized logistic distribution do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. Therefore we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.     

Penalized likelihood approach for the four-parameter kappa distribution

The four-parameter kappa distribution (K4D) is a generalized form of some commonly used distributions such as generalized logistic, generalized Pareto, generalized Gumbel, and generalized extreme value (GEV) distributions. Owing to its flexibility, the K4D is widely applied in modeling in several fields such as hydrology and climatic change. For the estimation of the four parameters, the maximum likelihood approach and the method of L-moments are usually employed. The L-moment estimator (LME) method works well for some parameter spaces, with up to a moderate sample size, but it is sometimes not feasible in terms of computing the appropriate estimates. Meanwhile, using the maximum likelihood estimator (MLE) with small sample sizes shows substantially poor performance in terms of a large variance of the estimator. We therefore propose a maximum penalized likelihood estimation (MPLE) of K4D by adjusting the existing penalty functions that restrict the parameter space. Eighteen combinations of penalties for two shape parameters are considered and compared. The MPLE retains modeling flexibility and large sample optimality while also improving on small sample properties. The properties of the proposed estimator are verified through a Monte Carlo simulation, and an application case is demonstrated taking Thailand’s annual maximum temperature data.     

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.     

Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution

This article is concerned with modifications of both maximum likelihood and moment estimators for parameters of the three-parameter gamma distribution. Modifications employed here are essentially the same as those previously considered by the authors (1980, 1981) in connection with the lognormal distribution. Sampling behavior of the estimates is indicated by a Monte Carlo simulation. For certain combinations of parameter values, these new estimators appear better than both maximum likelihood and moment estimators with respect to bias, variance and/or ease of calculation.     

Unimodality of the pareto distribution likelihood function for multicensored samples and implications for estimation

This paper discusses maximum likelihood parameter estimation in the Pareto distribution for multicensored samples. In particu-

lar, the modality of the associated conditional log-likelihood function is investigated in order to resolve questions concerninc

the existence and uniqurneas of the lnarimum likelihood estimates.For the cases with one parameter known, the maximum likelihood

estimates of the remaining unknown parameters are shown to exist and to be unique. When both parameters are unknown, the maximum likelihood estimates may or may not exist and be unique. That is, their existence and uniqueness would seem to depend solely upon the information inherent in the sample data. In viav of the possible nonexistence and/or non-uniqueness of the maximum likelihood estimates when both parameters are unknown, alternatives to standard iterative numerical methods are explored.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.     

Pseudo-likelihood inference for clustered binary data

The maximum likelihood procedure to estimate paraneters of a model has scveral attractive properties including the existence of the covariance matrix which yield asymptotic covariances: for a sample size N the asymptotics are in general of order 1/N. Here we give an asymptotic for the skewness of the distribution of the maximum likelihood estimator of a parameter; this is of order 1/ n2 and this expression is new. Applications relate to the parameters of (i) the Poisson, binomial, and normal density. (ii) the gamna density and (iii) the Beta debsity. Other application are being considered. The expression for the asymptotic skowness at one phase of the study tured out to be unusually complicated involving the asymptotic expressions for variance and bias. When these were identified a much simpler compact expression appeared which we now describe. The work is a much improved treatment of the subject described in Shenton and Bowman (Mariunm likelihood estimation in small samples, Griffin. 1977).     

Editorial collaborators

A simulation study of the binomial-logit model with correlated random effects is carried out based on the generalized linear mixed model (GLMM) methodology. Simulated data with various numbers of regression parameters and different values of the variance component are considered. The performance of approximate maximum likelihood (ML) and residual maximum likelihood (REML) estimators is evaluated. For a range of true parameter values, we report the average biases of estimators, the standard error of the average bias and the standard error of estimates over the simulations. In general, in terms of bias, the two methods do not show significant differences in estimating regression parameters. The REML estimation method is slightly better in reducing the bias of variance component estimates.     

Computation of probability and non-centrality parameter of a non-central f-distribution

The small sample properties of the score function approximation to the maximum likelihood estimator for the three-parameter lognormal distribution using an alternative parameterization are considered. The new set of parameters is a continuous function of the usual parameters. However, unlike with the usual parameterization, the score function technique for this parameterization is extremely insensitive to starting values. Further, it is shown that whenever the sample third moment is less than zero, a local maximum to the likelihood function exists at a boundary point. For the usual parameterization, this point is unattainable. However, the alternative parameter space can be expanded to include these boundary points. This procedure results in good estimates of the expected value, variance, extreme percentiles and other parameters of the distribution even in samples where, with the typical parameterization, the estimation procedure fails to converge.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

New modified moment estimators for the two-parameter weighted Lindley distribution

We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.     

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS OF THE TRUNCATED CAUCHY DISTRIBUTION

Truncated Cauchy distribution with four unknown parameters is considered and derivation and existence of the maximum likelihood estimates is investigated here. We provide a sufficient condition for the maximum likelihood estimate of the scale parameter to be finite, and also show that the condition is necessary for sufficiently large samples. Note that all the moments of the truncated Cauchy distribution exist which makes it much more attractive as a model when compared to the regular Cauchy. We also study, using simulations, the small sample properties of the maximum likelihood estimates.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

Estimation for the generalized pareto distribution with censored data

We present a methodology for computing the point and interval maximum likelihood parameter estimation for the two-parameter generalized Pareto distribution (GPD) with censored data. The basic idea underlying our method is a reduction of the two-dimensional numerical search for the zeros of the GPD log-likelihood gradient vector to a one-dimensional numerical search. We describe a computationally efficient algorithm which implement this approach. Two illustrative examples are presented. Simulation results indicate that the estimates derived by maximum likelihood estimation are more reliable against those of method of moments. An evaluation of the practical sample size requirements for the asymptotic normality is also included.     

The Size and Power of Bootstrap and Bartlett-Corrected Tests of Hypotheses on the Cointegrating Vectors

In this paper we compare Bartlett-corrected, bootstrap, and fast double bootstrap tests on maximum likelihood estimates of cointegration parameters. The key result is that both the bootstrap and the Bartlett-corrected tests must be based on the unrestricted estimates of the cointegrating vectors: procedures based on the restricted estimates have almost no power. The small sample size bias of the asymptotic test appears so severe as to advise strongly against its use with the sample sizes commonly available; the fast double bootstrap test minimizes size bias, while the Bartlett-corrected test is somehow more powerful.