Bias reduction for the maximum likelihood estimator of the doubly-truncated Poisson distribution

ABSTRACT

We derive an analytic expression for the bias of the maximum likelihood estimator of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. For smaller sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. Bias is of little concern in the positive Poisson distribution, the most common form of truncation in the applied literature. Bias appears to be the most severe in the doubly-truncated Poisson distribution, when the mean of the distribution is close to the right (upper) truncation.     

Extreme value index estimator using maximum likelihood and moment estimation

ABSTRACT

When a distribution function is in the max domain of attraction of an extreme value distribution, its tail can be well approximated by a generalized Pareto distribution. Based on this fact we use a moment estimation idea to propose an adapted maximum likelihood estimator for the extreme value index, which can be understood as a combination of the maximum likelihood estimation and moment estimation. Under certain regularity conditions, we derive the asymptotic normality of the new estimator and investigate its finite sample behavior by comparing with several classical or competitive estimators. A simulation study shows that the new estimator is competitive with other estimators in view of average bias, average MSE, and coefficient of variance of the new device for the optimal selection of the threshold.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Performance of nonparametric maximum likelihood estimator in a

When the probability of selecting an individual in a population is propor­tional to its lifelength, it is called length biased sampling. A nonparametric maximum likelihood estimator (NPMLE) of survival in a length biased sam­ple is given in Vardi (1982). In this study, we examine the performance of Vardi's NPMLE in estimating the true survival curve when observations are from a length biased sample. We also compute estimators based on a linear combination (LCE) of empirical distribution function (EDF) estimators and weighted estimators. In our simulations, we consider observations from a mix­ture of two different distributions, one from F and the other from G which is a length biased distribution of F. Through a series of simulations with vari­ous proportions of length biasing in a sample, we show that the NPMLE and the LCE closely approximate the true survival curve. Throughout the sur­vival curve, the EDF estimators overestimate the survival. We also consider a case where the observations are from three different weighted distributions, Again, both the NPMLE and the LCE closely approximate the true distribu­tion, indicating that the length biasedness is properly adjusted for. Finally, an efficiency study shows that Vardi's estimators are more efficient than the EDF estimators in the lower percentiles of the survival curves.     

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change-plane Cox model

The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. First, it shows that the convergence rate for the classification parameter can be arbitrarily close to $n^{-1}$ up to a logarithmic factor under a certain condition on covariates and the choice of tuning parameter. Given this convergence rate result, it also establishes the asymptotic normality for the regression parameter.     

Modified maximum likelihood estimator for the mixing distribution in a finite mixture

Using the likelihood equations, a method based on the generalized inverse of matrices is proposed to derive closed-form estimates for the mixing proportions in a finite mixture, when the component distributions are known.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

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.     

Towards a unified definition of maximum likelihood

A unified definition of maximum likelihood (ml) is given. It is based on a pairwise comparison of probability measures near the observed data point. This definition does not suffer from the usual inadequacies of earlier definitions, i.e., it does not depend on the choice of a density version in the dominated case. The definition covers the undominated case as well, i.e., it provides a consistent approach to nonparametric ml problems, which heretofore have been solved on a more less ad hoc basis. It is shown that the new ml definition is a true extension of the classical ml approach, as it is practiced in the dominated case. Hence the classical methodology can simply be subsumed. Parametric and nonparametric examples are discussed.     

Inference for the extreme value distribution under progressive Type-II censoring

The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. In this paper, we discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error. The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.     

On the minimaxiy of the maximum likelihood estimator in a multivariate problem

This study looks at the minimaxity of the maximum likelihood estimator (m.1.e), of the mean of a p-normal population, that has been given by Dahel, Giri and Lepage (1985). This estimator is computed on the basis of three independent samples: the first one is drawn from the whole vector of dimension p and the two others are based on the first p₁ and the last p₂ components respectively, such as p₁ +p₂=p.     

On the use of the selection matrix in the maximum likelihood estimation of normal distribution models with missing data

In this article, by using the constant and random selection matrices, several properties of the maximum likelihood (ML) estimates and the ML estimator of a normal distribution with missing data are derived. The constant selection matrix allows us to obtain an explicit form of the ML estimates and the exact relationship between the EM algorithm and the score function. The random selection matrix allows us to clarify how the missing-data mechanism works in the proof of the consistency of the ML estimator, to derive the asymptotic properties of the sequence by the EM algorithm, and to derive the information matrix.     

The maximum likelihood estimators of the parameters of the gamma distribution are always positively biased

Several authors have conjectured, on the basis of their numerical work, that the maximum likelihood estimators of the shape and scale parameters of the Gamma distribution are positively biased. It is proved that their conjecture is always true.     

Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution

Using the techniques developed by Subrahmaniam and Ching’anda (1978), we study the robustness to nonnormality of the linear discriminant functions. It is seen that the LDF procedure is quite robust against the likelihood ratio rule. The latter yields in all cases much smaller overall error rates; however, the disparity between the error rates of the LDF and LR procedures is not large enough to warrant the recommendation to use the more complicated LR procedure.     

A general algorithm for non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data

In this paper, we study an algorithm to compute the non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data. The algorithm, simply denoted by SQP (sequential quadratic programming), re-parameterizes the likelihood function to make the order constraints as a set of linear constraints, approximates the log-likelihood function as a quadratic function, and updates the estimate by solving a quadratic programming. We particularly consider two stochastic orderings, simple and uniform orderings, although the algorithm can also be applied to many other stochastic orderings. We illustrate the algorithm using the breast cancer data reported in Finkelstein and Wolfe (1985 Finkelstein, D. M., and R. A. Wolfe. 1985. A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41:933–45. [Google Scholar]).     

An iterative sparse algorithm for the penalized maximum likelihood estimator in mixed effects model

In this paper, we propose a new iterative sparse algorithm (ISA) to compute the maximum likelihood estimator (MLE) or penalized MLE of the mixed effects model. The sparse approximation based on the arrow-head (A-H) matrix is one solution which is popularly used in practice. The A-H method provides an easy computation of the inverse of the Hessian matrix and is computationally efficient. However, it often has non-negligible error in approximating the inverse of the Hessian matrix and in the estimation. Unlike the A-H method, in the ISA, the sparse approximation is applied “iteratively” to reduce the approximation error at each Newton Raphson step. The advantages of the ISA over the exact and A-H method are illustrated using several synthetic and real examples.     

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.