Empirical likelihood ratio under infinite second moment

In this article, we show that the log empirical likelihood ratio statistic for the population mean converges in distribution to χ²₍₁₎ as n → ∞ when the population is in the domain of attraction of normal law but has infinite variance. The simulation results show that the empirical likelihood ratio method is applicable under the infinite second moment condition.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

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.     

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.     

Local maximum likelihood estimation and inference

Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.     

Asymptotic variances of maximum likelihood estimator for the correlation coefficient from a BVN distribution with one variable subject to censoring

This paper deals with the problem of estimating the Pearson correlation coefficient when one variable is subject to left or right censoring. In parallel to the classical results on the Pearson correlation coefficient, we derive a workable formula, through tedious computation and intensive simplification, of the asymptotic variances of the maximum likelihood estimators in two cases: (1) known means and variances and (2) unknown means and variances. We illustrate the usefulness of the asymptotic results in experimental designs.     

Penalized maximum likelihood estimation in the modified extended Weibull distribution

We address the issue of performing inference on the parameters that index the modified extended Weibull (MEW) distribution. We show that numerical maximization of the MEW log-likelihood function can be problematic. It is even possible to encounter maximum likelihood estimates that are not finite, that is, it is possible to encounter monotonic likelihood functions. We consider different penalization schemes to improve maximum likelihood point estimation. A penalization scheme based on the Jeffreys’ invariant prior is shown to be particularly useful. Simulation results on point estimation, interval estimation, and hypothesis testing inference are presented. Two empirical applications are presented and discussed.     

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.     

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.     

Pseudo maximum likelihood estimation of the univariate GARCH (2,2) and asymptotic normality under dependent innovations

In this paper, we first consider the pseudo maximum likelihood estimation of the univariate GARCH (2,2) model and derive the underlying estimator. Then, we make use of the technique of martingales to establish the asymptotic normality of the pseudo-maximum likelihood estimator (PMLE) of the univariate GARCH (2,2) model. Contrary to previous approaches encountered in the statistical literature, the pseudo-likelihood function uses the general form of the density laws of the quadratic exponential family.     

A novel weighted likelihood estimation with empirical Bayes flavor

We propose a novel approach to estimation, where a set of estimators of a parameter is combined into a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood evaluated at the estimators. We investigate the method for a set of estimators obtained by using the maximum likelihood principle applied to each individual observation. The method can be viewed as a Bayesian approach with a data-driven prior distribution. We provide several examples illustrating the new method and argue for its consistency, asymptotic normality, and efficiency. We also conduct simulation studies to assess the performance of the estimators. This straightforward methodology produces consistent estimators comparable with those obtained by the maximum likelihood method. The method also approximates the distribution of the estimator through the “posterior” distribution.     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

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.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

Bias-Corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution

The two-parameter weighted Lindley distribution is useful for modeling survival data, whereas its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters. We adopt a “corrective” approach to derive modified MLEs that are bias-free to second order. We also consider an alternative bias-correction mechanism based on Efron’s bootstrap resampling. Monte Carlo simulations are conducted to compare the performance between the proposed and two previous methods in the literature. The numerical evidence shows that the bias-corrected estimators are extremely accurate even for very small sample sizes and are superior than the previous estimators in terms of biases and root mean squared errors. Finally, applications to two real datasets are presented for illustrative purposes.     

Modified skew-slash distribution

Abstract

In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.     

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.