On Maximum Likelihood Estimation for the Two Parameter Burr XII Distribution

This article considers three related aspects of maximum likelihood estimation of parameters in the two-parameter Burr XII distribution. Specifically, we first provide further clarification to some limiting results in Wingo (1993 Wingo , D. R. ( 1993 ). Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring . Microelectron. Reliab. 33 : 1251 – 1257 .[Crossref], [Web of Science ®] , [Google Scholar]). We then focus on details in a proof of the uniqueness of the maximum likelihood estimators. Finally, we consider using the likelihood approach for data which does not satisfy Wingo's criterion, and show that this results in fitting either a Pareto distribution or an intuitively sensible degenerate distribution to the data. The discussion here is completely general, and not restricted to data obtained under Type II censoring.     

Bayesian P-Values for Testing Independence in 2 × 2 Contingency Tables

This article considers Bayesian p-values for testing independence in 2 × 2 contingency tables with cell counts observed from the two independent binomial sampling scheme and the multinomial sampling scheme. From the frequentist perspective, Fisher's p-value (p _F ) is the most commonly used p-value but it can be conservative for small to moderate sample sizes. On the other hand, from the Bayesian perspective, Bayarri and Berger (2000 Bayarri , M. J. , Berger , J. O. ( 2000 ). P-values for composite null models (with discussion) . J. Amer. Statist. Assoc. 95 : 1127 – 1170 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first proposed the partial posterior predictive p-value (p _PPOST ), which can avoid the double use of the data that occurs in another Bayesian p-value proposed by Guttman (1967 Guttman , I. ( 1967 ). The use of the concept of a future observation in goodness-of-fit problems . J. Roy. Statist. Soc. Ser. B 29 : 83 – 100 . [Google Scholar]) and Rubin (1984 Rubin , D. B. ( 1984 ). Bayesianly justifiable and relevant frequency calculations for the applied statistician . Ann. Statist. 12 : 1151 – 1172 .[Crossref], [Web of Science ®] , [Google Scholar]), called the posterior predictive p-value (p _POST ). The subjective and objective Bayesian p-values in terms of p _POST and p _PPOST are derived under the beta prior and the (noninformative) Jeffreys prior, respectively. Numerical comparisons among p _F , p _POST , and p _PPOST reveal that p _PPOST performs much better than p _F and p _POST for small to moderate sample sizes from the frequentist perspective.     

On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood

One approach to handling incomplete data occasionally encountered in the literature is to treat the missing data as parameters and to maximize the complete-data likelihood over the missing data and parameters. This article points out that although this approach can be useful in particular problems, it is not a generally reliable approach to the analysis of incomplete data. In particular, it does not share the optimal properties of maximum likelihood estimation, except under the trivial asymptotics in which the proportion of missing data goes to zero as the sample size increases.     

Asymptotic Distribution of the Maximum Likelihood Estimators of the Parameters of the Right-Truncated Dagum Distribution

In this work we have determined the asymptotic distribution of the maximum likelihood estimators of the parameters β, λ, and δ for the right-truncated Dagum model. Some numerical comparisons show that, for each combination of the parameters and for each sample size, the variance of maximum likelihood estimators increases as the truncation point decreases, i.e., with the increase in the cut of the right tail of distribution.     

End-Point Estimation for Decreasing Densities: Asymptotic Behaviour of the Penalized Likelihood Ratio

Abstract.  We consider the problem of estimating the modal value of a decreasing density on the positive real line. This has application in several interesting phenomena arising, for example, in renewal theory, and in biased and distance samplings. We use a penalized likelihood ratio-based approach for inference and derive the scale-free universal large sample null distribution of the log-likelihood ratio, using a suitably chosen penalty parameter. We present simulation results and a real data analysis to corroborate our findings, and compare the performance of the confidence sets with the existing results.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

Estimation of parameters of Misclassified Size Biased Discrete Lindley Distribution

In this paper, a two-parameter discrete distribution named Misclassified Size Biased Discrete Lindley distribution is defined under the situation of misclassification where some of the observations corresponding to x = c + 1 are reported as x = c with misclassification errorα. Different estimation methods like maximum likelihood estimation, moment estimation, and Bayes Estimation are considered to estimate the parameters of Misclassified Size Biased Discrete Lindley distribution. These methods are compared by using mean square error through simulation study with varying sample sizes. Further general form of factorial moment is also obtained for Misclassified Size Biased Discrete Lindley distribution. Real life data set is used to fit Misclassified Size Biased Discrete Lindley distribution.     

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.     

On the Empirical Likelihood Ratio for Smooth Functions of M-functionals

It is known that the empirical likelihood ratio can be used to construct confidence regions for smooth functions of the mean, Fréchet differentiable statistical functionals and for a class of M-functionals. In this paper, we argue that this use can be extended to the class of functionals which are smooth functions of M-functionals. In particular, we find the conditions under which the empirical log-likelihood ratio for this kind of functionals admits a χ² approxima tion. Furthermore, we investigate, by simulation methods, the related approximation error in some contexts of practical interest.     

An Extension of the Epsilon-Skew-Normal Distribution

This article is related with the probabilistic and statistical properties of an parametric extension of the so-called epsilon-skew-normal (ESN) distribution introduced by Mudholkar and Hutson (2000 Mudholkar , G. S. , Hutson , A. D. ( 2000 ). The epsilon-skew-normal distribution for analyzing near-normal data . J. Statist. Plann. Infer. 83 : 291 – 309 .[Crossref], [Web of Science ®] , [Google Scholar]), which considers an additional shape parameter in order to increase the flexibility of the ESN distribution. Also, this article concerns likelihood inference about the parameters of the new class. In particular, the information matrix of the maximum likelihood estimators is obtained, showing that it is non singular in the special normal case. Finally, the statistical methods are illustrated with two examples based on real datasets.     

Tests and Confidence Intervals for an Extended Variance Component Using the Modified Likelihood Ratio Statistic

Abstract.  The large deviation modified likelihood ratio statistic is studied for testing a variance component equal to a specified value. Formulas are presented in the general balanced case, whereas in the unbalanced case only the one-way random effects model is studied. Simulation studies are presented, showing that the normal approximation to the large deviation modified likelihood ratio statistic gives confidence intervals for variance components with coverage probabilities very close to the nominal confidence coefficient.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.     

Estimating Lorenz Curves Using a Dirichlet Distribution

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified, it is typically estimated by linear or nonlinear least squares estimation techniques that have good properties when the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This article proposes and applies a new methodology that recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution. Five Lorenz curve specifications are used to demonstrate the technique. Maximum likelihood estimates under the Dirichlet distribution assumption provide better fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.     

Partial Partial Likelihood

The maximum likelihood and maximum partial likelihood approaches to the proportional hazards model are unified. The purpose is to give a general approach to the analysis of the proportional hazards model, whether the baseline distribution is absolutely continuous, discrete, or a mixture. The advantage is that heavily tied data will be analyzed with a discrete time model, while data with no ties is analyzed with ordinary Cox regression. Data sets in between are treated by a compromise between the discrete time model and Efron's approach to tied data in survival analysis, and the transitions between modes are automatic. A simulation study is conducted comparing the proposed approach to standard methods of handling ties. A recent suggestion, that revives Breslow's approach to tied data, is finally discussed.     

On the Additive Weibull Distribution

We derive explicit expressions for the moments, incomplete moments, quantile function and generating function of the additive Weibull model pioneered by Xie and Lai (1995 Xie, M., Lai, C.D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safety 52:87–93.[Crossref], [Web of Science ®] , [Google Scholar]), which is a quite flexible distribution for fitting lifetime data with bathtub-shaped failure rate function. In addition, we estimate the model parameters by maximum likelihood and determine the observed information matrix. The flexibility of the additive Weibull distribution is illustrated by means of one application to real data.     

Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit

In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small simulation. From the simulation, the performance of the new estimator is roughly comparable with maximum likelihood for positive values of the shape parameter and often much better than maximum likelihood for negative values.     

Asymptotic posterior normality for multiparameter problems

For asymptotic posterior normality in the one-parameter cases, Weng [2003. On Stein's identity for posterior normality. Statist. Sinica 13, 495–506] proposed to use a version of Stein's Identity to write the posterior expectations for functions of a normalized quantity in a form that is more transparent and can be easily analyzed. In the present paper we extend this approach to the multi-parameter cases and compare our conditions with earlier work. Three examples are used to illustrate the application of this method.