Influence diagnostics for polyhazard models in the presence of covariates

In this paper, we present various diagnostic methods for polyhazard models. Polyhazard models are a flexible family for fitting lifetime data. Their main advantage over the single hazard models, such as the Weibull and the log-logistic models, is to include a large amount of nonmonotone hazard shapes, as bathtub and multimodal curves. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration.     

Frailty regression models in mixture distributions

We propose frailty regression models in mixture distributions and assume the distribution of frailty as gamma or positive stable or power variance function distribution. We consider Weibull mixture as an example. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate to study this particular model.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

Mini-batch learning of exponential family finite mixture models

Mini-batch algorithms have become increasingly popular due to the requirement for solving optimization problems, based on large-scale data sets. Using an existing online expectation–maximization (EM) algorithm framework, we demonstrate how mini-batch (MB) algorithms may be constructed, and propose a scheme for the stochastic stabilization of the constructed mini-batch algorithms. Theoretical results regarding the convergence of the mini-batch EM algorithms are presented. We then demonstrate how the mini-batch framework may be applied to conduct maximum likelihood (ML) estimation of mixtures of exponential family distributions, with emphasis on ML estimation for mixtures of normal distributions. Via a simulation study, we demonstrate that the mini-batch algorithm for mixtures of normal distributions can outperform the standard EM algorithm. Further evidence of the performance of the mini-batch framework is provided via an application to the famous MNIST data set.     

A Bayesian analysis for the Block and Basu bivariate exponential distribution in the presence of covariates and censored data

In this paper, we introduce a Bayesian Analysis for the Block and Basu bivariate exponential distribution using Markov Chain Monte Carlo (MCMC) methods and considering lifetimes in presence of covariates and censored data. Posterior summaries of interest are obtained using the popular WinBUGS software. Numerical illustrations are introduced considering a medical data set related to the recurrence times of infection for kidney patients and a medical data set related to bone marrow transplantation for leukemia.     

On the consistency of MLE in finite mixture models of exponential families

Finite mixtures of densities from an exponential family are frequently used in the statistical analysis of data. Modelling by finite mixtures of densities from different exponential families provide more flexibility in the fittings, and get better results. However, in mixture problems, the log-likelihood function very often does not have an upper bound and therefore a global maximum does not always exist. Redner and Walker (1984. Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev. 26, 195–239) provide conditions to assure the existence, consistency and asymptotic normality of the maximum likelihood estimator.     

On entropy-based goodness-of-fit test for asymmetric Student-t and exponential power distributions

This paper examines the goodness-of-fit (GOF) test for a generalized asymmetric Student-t distribution (ASTD) and asymmetric exponential power distribution (AEPD). These distributions are known to include a broad class of distribution families and are quite suitable to modelling the innovations of financial time series. Despite their popularity, to our knowledge, no studies in the literature have so far investigated their affinity and differences in implementation. To fill this gap, we examine the empirical power behaviour of entropy-based GOF tests for hypotheses wherein the ASTD and AEPD play the role of null and alternative distributions. Our findings through a simulation study and real data analysis indicate that the two distributions are generally hard to distinguish and that the ASTD family accommodates AEPDs to a greater degree than the other way around for larger samples.     

Optimal spacing of quantiles for the estimation of the mixing parameters in a mixture of two exponential distributions

Methods for estimating the mixing parameters in a mixture of two exponential distributions are proposed. The estimators proposed are consistent and BAN(best asymptotically normal). The optimal spacings for estimating these mixture parameters are calculated.     

Nonlinear mixed-effects models with scale mixture of skew-normal distributions

Aiming to avoid the sensitivity in the parameters estimation due to atypical observations or skewness, we develop asymmetric nonlinear regression models with mixed-effects, which provide alternatives to the use of normal distribution and other symmetric distributions. Nonlinear models with mixed-effects are explored in several areas of knowledge, especially when data are correlated, such as longitudinal data, repeated measures and multilevel data, in particular, for their flexibility in dealing with measures of areas such as economics and pharmacokinetics. The random components of the present model are assumed to follow distributions that belong to scale mixtures of skew-normal (SMSN) distribution family, that encompasses distributions with light and heavy tails, such as skew-normal, skew-Student-t, skew-contaminated normal and skew-slash, as well as symmetrical versions of these distributions. For the parameters estimation we obtain a numerical solution via the EM algorithm and its extensions, and the Newton-Raphson algorithm. An application with pharmacokinetic data shows the superiority of the proposed models, for which the skew-contaminated normal distribution has shown to be the most adequate distribution. A brief simulation study points to good properties of the parameter vector estimators obtained by the maximum likelihood method.     

Local power of some tests in exponential family nonlinear models

In this paper we obtain asymptotic expansions up to order n^−1/2 for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in exponential family nonlinear models (Cordeiro and Paula, 1989), under a sequence of Pitman alternatives. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the dispersion parameter, thus generalising the results given in Cordeiro et al. (1994) and Ferrari et al. (1997). We also present Monte Carlo simulations in order to compare the finite-sample performance of these tests.     

Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Bayesian inference for truncated exponential distributions

Truncated normal distributions are considered as prior distributions for the truncation parameter in truncated exponential models. Posterior istributions re obtained, and inferenceforthe truncation parameter and for the reliability function is discussed. One and two parameter models are considered.     

Adjusting ROC curves for covariates in the presence of verification bias

The ROC (receiver operating characteristic) curve is frequently used for describing effectiveness of a diagnostic marker or test. Classical estimation of the ROC curve uses independent identically distributed samples taken randomly from the healthy and diseased populations. Frequently not all subjects undergo a definitive gold standard assessment of disease status (verification). Estimation of the ROC curve based on data only from subjects with verified disease status may be badly biased (verification bias). In this work we investigate the properties of the doubly robust (DR) method for estimating the ROC curve adjusted for covariates (ROC regression) under verification bias. We develop the estimator's asymptotic distribution and examine its finite sample size properties via a simulation study. We apply this procedure to fingerstick postprandial blood glucose measurement data adjusting for age.     

Uneven two-sided power distributions with applications in econometric models

In this paper we propose and analyze a bounded density function with a jump discontinuity at a threshold. Its properties are presented and a maximum likelihood estimation (MLE) procedure for the threshold location and jump size is developed. The distribution seems be appropriate in the context of financial engineering, production analysis, standard auction models and the equilibrium job search problem. An example of the MLE procedure is given utilizing an i.i.d. sample of standardized log differences of bi-monthly US Certificate Deposit interest rates for the period from 1966-2002. The corresponding time series was constructed using an Auto-Regressive Conditional Heteroscedastic (ARCH) model.     

Estimation in two-parameter exponential distributions

Exponential distributions are used extensively in the field of life-testing. Estimation of parameters is revisited in two-parameter exponential distributions. A comparison study between the maximum likelihood method, the unbiased estimates which are linear functions of the maximum likelihood method, the method of product spacings, and the method of quantile estimates are presented. Finally, a simulation study is given to demonstrate the small sample properties     

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.     

Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data

In this paper, the identifiability of a finite mixture of generalized exponential distributions (GE(τ, α)) is proved and the maximum likelihood estimates (MLE’s) of the parameters are obtained using EM algorithm based on a general form of right-censored failure times. The results are specialized to type-I and type-II censored samples. A real data set is introduced and analyzed using a mixture of two GE(τ, α) distributions and also using a mixture of two Weibull(α, β) distributions. A comparison is carried out between the mentioned mixtures based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the GE(τ, α) mixture model fits the data better than the other mixture model.     

Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models

This paper considers the estimation of the stress–strength reliability of a multi-state component or of a multi-state system where its states depend on the ratio of the strength and stress variables through a kernel function. The article presents a Bayesian approach assuming the stress and strength as exponentially distributed with a common location parameter but different scale parameters. We show that the limits of the Bayes estimators of both location and scale parameters under suitable priors are the maximum likelihood estimators as given by Ghosh and Razmpour [15 M. Ghosh and A. Razmpour, Estimation of the common location parameter of several exponentials, Sankhyā, Ser. A 46 (1984), pp. 383–394. [Google Scholar]]. We use the Bayes estimators to determine the multi-state stress–strength reliability of a system having states between 0 and 1. We derive the uniformly minimum variance unbiased estimators of the reliability function. Interval estimation using the bootstrap method is also considered. Under the squared error loss function and linex loss function, risk comparison of the reliability estimators is carried out using extensive simulations.     

Stationary mixture transition distribution (MTD) models via predictive distributions

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist.29, 657–663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy. Statist. Soc. B.47, 528–539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series.