Complete sufficiency and maximum likelihood estimation for the two-parameter negative binomial distribution

The difficult estimation problem associated with the two-parameter negative binomial distribution is discussed. The order statistic is shown to be minimal sufficient but not complete. It is proven that there is at least one maximum likelihood estimator of the parameterk when the second sample moment is greater than the sample mean. Contours and three-dimensional graphs of the natural logarithm of the likelihood function provide further insight into the estimation problem.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.     

The Estimation of Pareto Distribution by a Weighted Least Square Method

The two-parameter Pareto distribution provides reasonably good fit to the distributions of income and property value, and explains many empirical phenomena. For the censored data, the two parameters are regularly estimated by the maximum likelihood estimator, which is complicated in computation process. This investigation proposes a weighted least square estimator to estimate the parameters. Such a method is comparatively concise and easy to perceive, and could be applied to either complete or truncated data. Simulation studies are conducted in this investigation to show the feasibility of the proposed method. This report will demonstrate that the weighted least square estimator gives better performance than unweighted least square estimators with simulation cases. We also illustrate that the weighted least square estimator is very close to maximum likelihood estimator with simulation studies.     

On the inefficiency of the restricted maximum likelihood

The restricted maximum likelihood is preferred by many to the full maximum likelihood for estimation with variance component and other random coefficient models, because the variance estimator is unbiased. It is shown that this unbiasedness is accompanied in some balanced designs by an inflation of the mean squared error. An estimator of the cluster‐level variance that is uniformly more efficient than the full maximum likelihood is derived. Estimators of the variance ratio are also studied.     

ON THE FIRST–ORDER EFFICIENCY AND ASYMPTOTIC NORMALITY OF MAXIMUM LIKELIHOOD ESTIMATORS OBTAINED FROM DEPENDENT OBSERVATIONS

Abstract. In this paper we study the first–order efficiency and asymptotic normality of the maximum likelihood estimator obtained from dependent observations. Our conditions are weaker than usual, in that we do not require convergences in probability to be uniform or third–order derivatives to exist.
The paper builds on Witting and Nolle's result concerning the asymptotic normality of the maximum likelihood estimator obtained from independent and identically distributed observations, and on a martingale theorem by McLeish.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

Strong Consistency of the Maximum Likelihood Estimator in Generalized Linear and Nonlinear Mixed-Effects Models

Generalized linear and nonlinear mixed-effects models are used extensively in biomedical, social, and agricultural sciences. The statistical analysis of these models is based on the asymptotic properties of the maximum likelihood estimator. However, it is usually assumed that the maximum likelihood estimator is consistent, without providing a proof. A rigorous proof of the consistency by verifying conditions from existing results can be very difficult due to the integrated likelihood. In this paper, we present some easily verifiable conditions for the strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Based on this result, we prove that the maximum likelihood estimator is consistent for some frequently used models such as mixed-effects logistic regression models and growth curve models.     

Likelihood estimation and inference in threshold regression

This paper studies likelihood-based estimation and inference in parametric discontinuous threshold regression models with i.i.d. data. The setup allows heteroskedasticity and threshold effects in both mean and variance. By interpreting the threshold point as a “middle” boundary of the threshold variable, we find that the Bayes estimator is asymptotically efficient among all estimators in the locally asymptotically minimax sense. In particular, the Bayes estimator of the threshold point is asymptotically strictly more efficient than the left-endpoint maximum likelihood estimator and the newly proposed middle-point maximum likelihood estimator. Algorithms are developed to calculate asymptotic distributions and risk for the estimators of the threshold point. The posterior interval is proved to be an asymptotically valid confidence interval and is attractive in both length and coverage in finite samples.     

The conditional maximum likelihood estimator of the shape parameter in the gamma distribution

Summary The conditional maximum likelihood estimator of the shape parameter in the gamma distribution is studied for a finite sample size in comparison with the (unconditional) maximum likelihood estimator. The former estimator is concluded to be strictly superior to the latter. The reasons for the conclusion include the undesirable behavior of the residual likelihood, the consistency and relatively less bias of the conditional maximum likelihood estimator. Simulation studies for risk comparisons also support the conclusion.     

Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

Estimation of dynamic models with nonparametric simulated maximum likelihood

We propose an easy-to-implement simulated maximum likelihood estimator for dynamic models where no closed-form representation of the likelihood function is available. Our method can handle any simulable model without latent dynamics. Using simulated observations, we nonparametrically estimate the unknown density by kernel methods, and then construct a likelihood function that can be maximized. We prove that this nonparametric simulated maximum likelihood (NPSML) estimator is consistent and asymptotically efficient. The higher-order impact of simulations and kernel smoothing on the resulting estimator is also analyzed; in particular, it is shown that the NPSML does not suffer from the usual curse of dimensionality associated with kernel estimators. A simulation study shows good performance of the method when employed in the estimation of jump-diffusion models.     

Finite sample properties of maximum likelihood estimator in spatial models

We investigate the finite sample properties of the maximum likelihood estimator for the spatial autoregressive model. A stochastic expansion of the score function is used to develop the second-order bias and mean squared error of the maximum likelihood estimator. We show that the results can be expressed in terms of the expectations of cross products of quadratic forms, or ratios of quadratic forms in a normal vector which can be evaluated using the top order invariant polynomial. Our numerical calculations demonstrate that the second-order behaviors of the maximum likelihood estimator depend on the degree of sparseness of the weights matrix.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

Estimation of the trace of the scaled covariance matrix of a multivariate t-model using a known information

The trace of the scaled covariance matrix of the multivariate t-distribution is considered for estimation using a power transformation. The proposed estimator always dominates the usual maximum likelihood estimator in the sense of having smaller risk under a quadratic loss function. The dominance behaviour is proved analytically as well as computationally by using Monte-Carlo simulation.     

Comovements and asymmetric tail dependence in state housing prices in the USA: A nonparametric approach

We reexamine the methods used in estimating comovements among US regional home prices and find that there are insufficient moments to ensure a normal limit necessary for employing the quasi‐maximum likelihood estimator. Hence we propose applying the self‐weighted quasi‐maximum exponential likelihood estimator and a bootstrap method to test and account for the asymmetry of comovements as well as different magnitudes across state pairs. Our results reveal interstate asymmetric tail dependence based on observed house price indices rather than residuals from fitting autoregressive–generalized autoregressive conditional heteroskedasticity (AR‐GARCH) models.     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

A note on the maximum probability property of MLE's in multiparameter exponential families

Summary For multiparameter exponential models a short and direct proof is given that the maximum likelihood estimator is a maximum probability estimator with respect to a certain sequence of convex and bounded sets inR _(k) that are symmetric about the origin; asymptotically these sets are allowed to be unbounded.     

Estimating and testing linkage disequilibrium from sib data

We consider estimation and testing of linkage equilibrium from genotypic data on a random sample of sibs, such as monozygotic and dizygotic twins. We compute the maximum likelihood estimator with an EM‐algorithm and a likelihood ratio statistic that takes the family structure into account. As we are interested in applying this to twin data we also allow observations on single children, so that monozygotic twins can be included. We allow non‐zero recombination fraction between the loci of interest, so that linkage disequilibrium between both linked and unlinked loci can be tested. The EM‐algorithm for computing the maximum likelihood estimator of the haplotype frequencies and the likelihood ratio test‐statistic, are described in detail. It is shown that the usual estimators of haplotype frequencies based on ignoring that the sibs are related are inefficient, and the likelihood ratio test for testing that the loci are in linkage disequilibrium.