Properties of maximum likelihood estimators of variance components in the one-way classification,balanced data

We studied properties of maximum likelihood estimators (MLEs) of the variance components obtained from balanced data of the one-way classification. Exact and asymptotic expected values and variances of these MLEs were derived under the usual normality assumptions. Numerical studies illustrate these expected values and variances, and also illustrate the probability of obtaining a negative solution to the maximum likelihood (ML) equation for the between-class variance component. Simulations were used to study the robustness of the ML estimators under non-normal distributions.     

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

A comparison of LS/ML and GMM estimation in a simple AR(1) model

This paper compares least squares (LS)/maximum likelihood (ML) and generalised method of moments (GMM) estimation in a simple. Gaussian autoregressive of order one (AR(1)) model. First, we show that the usual LS/ML estimator is a corner solution to a general minimisation problem that involves two moment conditions, while the new GMM we devise is not. Secondly, we examine asymptotic and finite sample properties of the new GMM estimator in comparison to the usual LS/ML estimator in a simple AR(1) model. For both stable and unstable (unit root) specifications, we show asymptotic equivalence of the distributions of the two estimators. However, in finite samples, the new GMM estimator performs better.     

Componentwise B-spline estimation for varying coefficient models with longitudinal data

A componentwise B-spline method is proposed for estimating the unknown functions in the varying-coefficient models with longitudinal data. Different amounts of smoothing are used for different individual coefficient functions and the estimators of different coefficient functions are obtained by different minimization operations. The local asymptotic bias and variance of the estimators are derived. It is shown that our estimators achieve the local and global optimal convergence rates even if the coefficient functions belong to different smoothness families. The asymptotic distributions of the estimators are also established and are used to construct approximate pointwise confidence intervals for coefficient functions. Finite sample properties of our procedures are studied through Monte Carlo simulations.     

The Two-Piece Normal-Laplace Distribution

This article considers the two-piece normal-Laplace (TPNL) distribution, a split skew distribution consisting of a normal part, and a Laplace part. The distribution is indexed by three parameters, representing location, scale, and shape. As illustrated with several examples, the TPNL family of distributions provides a useful alternative to other families of asymmetric distributions on the real line. However, because the likelihood function is not well behaved, standard theory of maximum-likelihood (ML) estimation does not apply to the TPNL family. In particular, the likelihood function can have multiple local maxima. We provide a procedure for computing ML estimators, and prove consistency and asymptotic normality of ML estimators, using non standard methods.     

Self-consistent estimators of survival functions with doubly-censored data

The consistency and asymptotic normality of self-consistent estimators (SCE) of survival functions with doubly-censored data have been studied by many authors. However, to the best of our knowledge, expressions of the asymptotic variance of the SCE have not been derived in the literature. In this paper, under the assumption that the survival time and censoring time distributions are discrete with finitely many jump points, an expression and a consistent estimator of the asymptotic variance of the SCE are presented. A proof of the strong consistency of the SCE is also presented. Our simu¬lation studies indicate that the estimate of the asymptotic variance is very close to the true value even with moderate sample sizes and high censoring rates     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Estimation of parameters for the truncated exponential distribution

We consider the right truncated exponential distribution where the truncation point is unknown and show that the ML equation has a unique solution over an extended parameter space. In the case of the estimation of the truncation point T we show that the asymptotic distribution of the MLE is not centered at T. A modified MLE is introduced which outperforms all other considered estimators including the minimum variance unbiased estimator. Asymptotic as well as small sample properties of different estimators are investigated and compared. The truncated exponential distribution has an increasing failure rate, ideally suited for use as a survival distribution for biological and industrial data.     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

A note on umvu estimation in the transformed chi-square family

The transformed chi-square family includes many common one-parameter continuous distributions. In that family, we give conditions under which a given function of the mean admits a minimum variance unbiased estimator and an orthogonal expansion for this estimator in terms of the generalized Laguerre polynomials. We show that such expansion is useful for obtaining bounds for the variance and for the study of the asymptotic properties of the unbiased estimators.     

On the Estimation of Parameters of Kumaraswamy-G Distributions

Cordeiro and de Castro proposed a new family of generalized distributions based on the Kumaraswamy distribution (denoted as Kw-G). Nadarajah et al. showed that the density function of the new family of distributions can be expressed as a linear combination of the density of exponentiated family of distributions. They derived some properties of Kw-G distributions and discussed estimation of parameters using the maximum likelihood (ML) method. Cheng and Amin and Ranneby introduced a new method of estimating parameters based on Kullback–Leibler divergence (the maximum spacing (MSP) method). In this article, the estimates of parameters of Kw-G distributions are obtained using the MSP method. For some special Kw-G distributions, the new estimators are compared with ML estimators. It is shown by simulations and a real data application that MSP estimators have better properties than ML estimators.     

Variance estimation of the Buckley–James estimator under discrete assumptions

The Buckley–James estimator (BJE) is a widely recognized approach in dealing with right-censored linear regression models. There have been a lot of discussions in the literature on the estimation of the BJE as well as its asymptotic distribution. So far, no simulation has been done to directly estimate the asymptotic variance of the BJE. Kong and Yu [Asymptotic distributions of the Buckley–James estimator under nonstandard conditions, Statist. Sinica 17 (2007), pp. 341–360] studied the asymptotic distribution under discontinuous assumptions. Based on their methodology, we recalculate and correct some missing terms in the expression of the asymptotic variance in Theorem 2 of their work. We propose an estimator of the standard deviation of the BJE by using plug-in estimators. The estimator is shown to be consistent. The performance of the estimator is accessed through simulation studies under discrete underline distributions. We further extend our studies to several continuous underline distributions through simulation. The estimator is also applied to a real medical data set. The simulation results suggest that our estimation is a good approximation to the true standard deviation with reference to the empirical standard deviation.     

Generalized Pickands estimators for the extreme value index

The Pickands estimator for the extreme value index is generalized in a way that includes all of its previously known variants. A detailed study of the asymptotic behavior of the estimators in the family serves to determine its optimally performing members. These are given by simple, explicit formulas, have the same asymptotic variance as the maximum likelihood estimator in the generalized Pareto model, and are robust to departures from the limiting generalized Pareto model in case the convergence of the excess distribution to its limit is slow. A simulation study involving a wide range of distributions shows the new estimators to compare favorably with the maximum likelihood estimator.     

Improved inference for MCP-Mod approach using time-to-event endpoints with small sample sizes

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.     

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.     

Estimation of the SURE model

The paper presents the essentials of the SURE model and the estimation of its parameters β and ω. Two alternative compact representations of the model are being used. The parameter β is estimated by least squares (LS), generalized least squares (GLS) and maximum likelihood (ML) (under normality). For ω two estimators are being considered, viz an LS-related estimator and a maximum likelihood estimator (under normality). Attention is being given to the study of asymptotic properties of all estimators examined. It turns out that the LS-related and ML estimators of ω follow the same asymptotic (normal) distribution. Efficiency comparisons for the various estimators of β conclude the paper.     

Bootstrapping data with multiple levels of variation

The authors consider general estimators for the mean and variance parameters in the random effect model and in the transformation model for data with multiple levels of variation. They show that these estimators have different distributions under the two models unless all the variables have Gaussian distributions. They investigate the asymptotic properties of bootstrap procedures designed for the two models. They also report simulation results and illustrate the bootstraps using data on red spruce trees.     

A new linearized ridge Poisson estimator in the presence of multicollinearity

Poisson regression is a very commonly used technique for modeling the count data in applied sciences, in which the model parameters are usually estimated by the maximum likelihood method. However, the presence of multicollinearity inflates the variance of maximum likelihood (ML) estimator and the estimated parameters give unstable results. In this article, a new linearized ridge Poisson estimator is introduced to deal with the problem of multicollinearity. Based on the asymptotic properties of ML estimator, the bias, covariance and mean squared error of the proposed estimator are obtained and the optimal choice of shrinkage parameter is derived. The performance of the existing estimators and proposed estimator is evaluated through Monte Carlo simulations and two real data applications. The results clearly reveal that the proposed estimator outperforms the existing estimators in the mean squared error sense.KEYWORDS: Poisson regression, multicollinearity, ridge Poisson estimator, linearized ridge regression estimator, mean squared errorMathematics Subject Classifications: 62J07, 62F10