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A two-parameter Gamma-Uniform distribution was recently introduced as a prominent alternative in modeling bounded phenomena. Unfortunately, however, its maximum likelihood estimators (MLEs) are fou...

It is well known that the maximum likelihood estimates (MLEs) have appealing statistical properties. Under fairly mild conditions their asymptotic distribution is normal, and no other estimator has a smaller asymptotic variance.However, in finite samples the maximum likelihood estimates are often biased estimates and the bias disappears as the sample size grows.Mazucheli, Menezes, and Ghitany (2018b) introduced a two-parameter unit-Weibull distribution which is useful for modeling data on the unit interval, however its MLEs are biased in finite samples.In this paper, we adopt three approaches for bias reduction of the MLEs of the parameters of unit-Weibull distribution.The first approach is the analytical methodology suggested by Cox and Snell (1968), the second is based on parametric bootstrap resampling method, and the third is the preventive approach introduced by Firth (1993).The results from Monte Carlo simulations revealed that the biases of the estimates should not be ignored and the bias reduction approaches are equally efficient. However, the first approach is easier to implement.Finally, applications to two real data sets are presented for illustrative purposes.

In this paper we introduce bivariate Weibull distributions derived from copula functions in presence of cure fraction, censored data and covariates. Two copula functions are explored: the FGM (Farlie- Gumbel Morgenstern) copula and the Gumbel copula. Inferences for the proposed models are obtained under the Bayesian approach, using standard MCMC (Markov Chain Monte Carlo) methods. An illustration of the proposed methodology is given considering a medical data set.

One of the key features in regression models consists in selecting appropriate characteristics that explain the behavior of the response variable, in which stepwise-based procedures occupy a prominent position. In this paper we performed several simulation studies to investigate whether a specific stepwise-based approach, namely Strategy A, properly selects authentic variables into the generalized additive models for location, scale and shape framework, considering Gaussian, zero inflated Poisson and Weibull distributions. Continuous (with linear and nonlinear relationships) and categorical explanatory variables are considered and they are selected through some goodness-of-fit statistics. Overall, we conclude that the Strategy A greatly performed.

A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.

Modal regression is an alternative approach for investigating the relationship between the most likely response and covariates and can hence reveal important structure missed by usual regression methods. This paper provides a collection of parametric mode regression models for bounded response variable by considering some recently introduced probability distributions with bounded support along with the well-established Beta and Kumaraswamy distribution. The main properties of the distributions are highlighted and compared. An empirical comparison between the considered modal regression is demonstrated through the analysis of three data sets from health and social science. For reproducible research, the proposed models are freely available to users as an R package unitModalReg.

In a note about the paper titled ‘On the one parameter unit-Lindley distribution and its associated regression model for proportion data’, Mazucheli et al. [J. Appl. Stat. 46 (2019), pp. 700–714] and Nadarajah and Chan [On moments of the unit Lindley distribution, J. Appl. Stat. (under review)] claim that ‘The expressions given for the moments and incomplete moments are not correct and not in closed form’. While we agree that they are not in closed form and observe a typo in the expressions for [Image: see text] and [Image: see text] , [Image: see text] , the expressions for [Image: see text] , [Image: see text] , [Image: see text] and [Image: see text] are, however, entirely correct.

Methods to obtain discrete analogs of continuous distributions have been widely applied in recent years. In general, the discretization process provides probability mass functions that can be competitive with traditional models used in the analysis of count data. The discretization procedure also avoids the use of continuous distribution to model strictly discrete data. In this paper, we propose two discrete analogs for the quasi xgamma distribution as alternatives to model under- and overdispersed datasets. The methods of infinite series and survival function have been considered to derive the models and, despite the difference between the methods, the resulting distributions are interchangeable. Several statistical properties of the proposed models have been derived. The maximum likelihood theory has been considered for estimation and asymptotic inference concerns. An intensive simulation study has been carried out in order to evaluate the main properties of the maximum likelihood estimators. The usefulness of the proposed models has been assessed by using two real datasets provided by literature. A general comparison of the proposed models with some well-known discrete distributions has been provided.

We propose a new semiparametric Weibull cure rate model for fitting nonlinear effects of explanatory variables on the mean, scale and cure rate parameters. The regression model is based on ...

This paper addresses the different methods of estimation of the unknown parameters of a two-parameter unit-logistic distribution from the frequentist point of view. We briefly describe different approaches, namely, maximum likelihood estimators, percentile based estimators, least squares estimators, maximum product of spacings estimators, methods of minimum distances: Cramér-von Mises, AndersonDarling and four variants of Anderson-Darling. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The performances of the estimators have been compared in terms of their relative bias, root mean squared error, average absolute difference between the theoretical and empirical estimate of the distribution functions and the maximum absolute difference between the theoretical and empirical distribution functions using simulated samples. Also, for each method of estimation, we consider the interval estimation using the Bootstrap confidence interval and calculate the coverage probability and the average width of the Bootstrap confidence intervals. Finally, two real data sets have been analyzed for illustrative purposes.

A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with ot...

The methods to obtain discrete analogs of continuous distributions have been widely considered in recent years. In general, the discretization process provides probability mass functions that can be competitive with the tra ditional model used in the analysis of count data, the Poisson distribution. The discretization procedure also avoids the use of continuous distribution in the analysis of strictly discrete data. In this paper, we seek to introduce two discrete analogs for the Shanker distribution using the method of the in finite series and the method based on the survival function as alternatives to model overdispersed datasets. Despite the difference between discretization methods, the resulting distributions are interchangeable. However, the dis tribution generated by the method of the infinite series method has simpler mathematical expressions for the shape, the generating functions, and the central moments. The maximum likelihood theory is considered for estima tion and asymptotic inference concerns. A simulation study is carried out in order to evaluate some frequentist properties of the developed methodology. The usefulness of the proposed models is evaluated using real datasets pro vided by the literature. Resumen Los métodos para obtener análogos discretos de distribuciones continuas han sido ampliamente considerados en los últimos años. En general, el pro ceso de discretización proporciona funciones de probabilidad en masa que pueden ser competitivas con el modelo tradicional utilizado en el análisis de datos de conteo, la distribución de Poisson. El procedimiento de discretización también evita el uso de la distribución continua en el análisis de datos estrictamente discretos. En este artículo, intentamos introducir dos análogos discretos para la distribución de Shanker utilizando el método de la serie infinita y el método basado en la función de supervivencia como al ternativas para modelar conjuntos de datos sobre dispersados. A pesar de la diferencia entre los métodos de discretización, las distribuciones resultantes son intercambiables. Sin embargo, la distribución generada por el método de series infinitas tiene expresiones matemáticas más simples para la forma, las funciones de generación y los momentos centrales. La teoría de máxi ma verosimilitud se considera para la estimación y las preocupaciones de inferencia asintótica. Se lleva a cabo un estudio de simulación para evaluar algunas propiedades frecuentistas de la metodología desarrollada. La utili dad de los modelos propuestos se evalúa utilizando conjuntos de datos reales proporcionados por la literatura.

The Half-Normal distribution has been intensively extended in the recent years. A review of the literature showed that at least 10 extensions of the Half-Normal distribution were introduced between 2008 and 2016. These extensions generalized the behavior of the density and hazard functions, which are restricted to monotonous decreasing and monotonically increasing, respectively. In this paper we propose a new extension called the transmuted Half-Normal distribution using the quadratic rank transmutation map, introduced by Shaw & Buckley (2009). A comprehensive account of mathematical properties of the new distribution is presented. We provide explicit expressions for the moments, moment-generating function, Shannon’s entropy, mean deviations, Bonferroni and Lorenz curves, order statistics, and reliability. The estimation of the parameters is implemented by the maximum likelihood method. The bias and accuracy of the estimators are assayed by the Monte Carlo simulations. This proposed distribution allows us to incorporate covariates directly in the mean and consequently to quantify their influences on the average of the response variable. Experiment with two real data sets show usefulness and its value as a good alternative to several extensions of the Half-Normal distribution in data modeling with and without covariates.

In this article, considering the two-parameter Johnson SB distribution, bounded on the unit interval, we derived, for the first time, the analytical expressions for bias-reduction of maximum likeli...

In this paper considering the transformation $X=\frac{Y}{1+Y}$, where $Y \sim\text{Lindley}(\theta)$, we propose the unit-Lindley distribution and investigate some of its mathematical properties. A important fact associated with this new distribution is that is possible to obtain the analytical expression for bias correction of the maximum likelihood estimator. Moreover, it belongs to the exponential family. This distribution allows us to incorporate covariates directly in the mean and consequently to quantify the influence on the average of the response variable. Finally, a practical application is present and it is shown that our model fits much better than the Beta regression., Comment: 17 pages, 4 figures

Maximum likelihood estimators usually have biases of the order O(n-1) for large sample size n which are very often ignored because of the fact that they are small when compared to the stand...

Cooray and Ananda introduced a two-parameter generalized Half-Normal distribution which is useful for modelling lifetime data, while its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters of the model. In this paper, we adopt two approaches for bias reduction of the MLEs of the parameters of generalized Half-Normal distribution. The first approach is the analytical methodology suggested by Cox and Snell and the second is based on parametric Bootstrap resampling method. Additionally, the method of moments (MMEs) is used for comparison purposes. The numerical evidence shows that the analytic bias-corrected estimators significantly outperform their bootstrapped-based counterpart for small and moderate samples as well as for MLEs and MMEs. Also, it is apparent from the results that bias- corrected estimates of shape parameter perform better than that of scale parameter. Further, the results show that b...

In this study, classical and Bayesian inference methods are introduced to analyze lifetime data sets in the presence of left censoring considering two generalizations of the Lindley distribution: a first generalization proposed by Ghitany et al. [Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), pp. 20–33], denoted as a power Lindley distribution and a second generalization proposed by Sharma et al. [The inverse Lindley distribution: A stress–strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng. 32 (2015), pp. 162–173], denoted as an inverse Lindley distribution. In our approach, we have used a distribution obtained from these two generalizations denoted as an inverse power Lindley distribution. A numerical illustration is presented considering a dataset of thyroglobulin levels present in a group of individuals with differentiated cancer of thyroid.

This paper addresses different methods of estimation of the unknown parameters of a two-parameter Kumaraswamy distribution from a frequentist point of view. We briefly describe ten different frequentist approaches, namely, maximum likelihood estimators, moments estimators, L-moments estimators, percentile based estimators, least squares estimators, weighted least squares estimators, maximum product of spacings estimators, Cramer–von-Mises estimators, Anderson–Darling estimators and right tailed Anderson–Darling estimators. Monte Carlo simulations and two real data applications are performed to compare the performances of the estimators for both small and large samples.

The Beta distribution is the standard model for quantifying the influence of covariates on the mean of a response variable on the unit interval. However, this well-known distribution is no longer useful when we are interested in quantifying the influence of such covariates on the quantiles of the response variable. Unlike Beta, the Kumaraswamy distribution has a closed-form expression for its quantile and can be useful for the modeling of quantiles in the absence/presence of covariates. As an alternative to the Kumaraswamy distribution for the modeling of quantiles, in this paper the unit-Weibull distribution was considered. This distribution was obtained by the transformation of a random variable with Weibull distribution. The same transformation applied to a random variable with Exponentiated Exponential distribution generates the Kumaraswamy distribution. The suitability of our proposal was demonstrated to model quantiles, conditional on covariates, with two simulated examples and three real applications with datasets from health, accounting and social science. For such data sets, the obtained fits of the proposed regression model were compared with those provided by the Beta and Kumaraswamy regression models.

Different cure fraction models have been used in the analysis of lifetime data in presence of cured patients. This paper considers mixture and nonmixture models based on discrete Weibull distribution to model recurrent event data in presence of a cure fraction. The novelty of this study is the use of a discrete lifetime distribution in place of usual existing continuous lifetime distributions for lifetime data in presence of cured fraction, censored data, and covariates. In the verification of the fit of the proposed model it is proposed the use of randomized quantile residuals. An extensive simulation study is considered to evaluate the properties of the estimates of the parameters related to the proposed model. As an illustration of the proposed methodology, it is considered an application considering a medical dataset related to lifetimes in a retrospective cohort study conducted by Puchner et al. (2017) that consists of 147 consecutive cases with surgical treatment of a sarcoma of the pelvis between the years of 1980 and 2012.

In this paper, it is introduced a new parametric distribution to be used in multivariate lifetime data analysis as an alternative for the use of some existing multivariate parametric models...

Several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. Recently, a new generalization of the Lindley distribution was proposed by Ghitany et al. (2013), called power Lindley distribution. Another generalization was proposed by Sharma et al. (2015a), known as inverse Lindley distribution. In this paper a distribution is obtained from these two generalizations and named as inverse power Lindley distribution. Some properties of this distribution and study of the behavior of maximum likelihood estimators are presented and discussed. It is also applied considering two real data sets and compared with the fits obtained for already-known distributions. When applied, the inverse power Lindley distribution was found to be a good alternative for modeling survival data.

This article deals with the Bayesian and non Bayesian estimation of multicomponent stress–strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with common and known shape parameter. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach and the results are compared using Markov Chain Monte Carlo (MCMC) technique for both small and large samples. Finally, two data sets are analyzed for illustrative purposes.

Using the number of thrips found in the maize seedlings from a longitudinal study, in this work a regression analysis was conducted through polynomial models and nonlinear logistic, under the assumption of homoscedasticity and multiplicative heteroscedasticity of variances. The response variable represents the number of thrips in adult phase observed in an entirely randomized trial involving 4 ground cover systems (tumbled oat, desiccated oat, mowed oat and incorporated oat), 8 replications with portions of 57.6 m2 and evaluated at 7, 11, 16, 23 and 31 days after the seedlings emergence. The parameters from the adopted models (homoscedastic and heteroscedastic) were estimated by maximizing the logarithm of the verisimilitude function. The adjusted model, via maximum verisimilitude, under the assumption of heteroscedasticity of the variances, did not over estimate the true asymptotic average number of thrips.

Made available in DSpace on 2018-11-26T16:27:47Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-01-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) The aim of this paper is to compare the parameters' estimations of the Marshall-Olkin extended Lindley distribution obtained by six estimation methods: maximum likelihood, ordinary least-squares, weighted least-squares, maximum product of spacings, Cramer-von Mises and Anderson-Darling. The bias, root mean-squared error, average absolute difference between the true and estimate distributions' functions and the maximum absolute difference between the true and estimate distributions' functions are used as comparison criteria. Although the maximum product of spacings method is not widely used, the simulation study concludes that it is highly competitive with the maximum likelihood method. Univ Estadual Paulista, Dept Estat, Presidente Prudente, SP, Brazil Univ Estadual Maringa, Dept Estat, Maringa, Parana, Brazil Univ Estadual Paulista, Dept Estat, Presidente Prudente, SP, Brazil

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a three-parameter Burr XII distribution, which includes the popularWeibull mixture model as a special case. Classical and Bayesian procedures are used to get point estimates and confidence intervals for the unknown parameters. We consider a general survival model where the scale and shape parameters of the Burr XII distribution are dependent of some covariates. To illustrate the proposed methodology, we consider an application considering a leukaemia data set where the proposed model gives better fit for the data when compared to other existing models.

Recently, Mazucheli (2017) uploaded the package mle.tools to CRAN. It can be used for bias corrections of maximum likelihood estimates through the methodology proposed by Cox and Snell (1968). The main function of the package, coxsnell.bc(), computes the bias corrected maximum likelihood estimates. Although in general, the bias corrected estimators may be expected to have better sampling properties than the uncorrected estimators, analytical expressions from the formula proposed by Cox and Snell (1968) are either tedious or impossible to obtain. The purpose of this paper is twofolded: to introduce the mle.tools package, especially the coxsnell.bc() function; secondly, to compare, for thirty one continuous distributions, the bias estimates from the coxsnell.bc() function and the bias estimates from analytical expressions available in the literature. We also compare, for five distributions, the observed and expected Fisher information. Our numerical experiments show that the functions are efficient to estimate the biases by the Cox-Snell formula and for calculating the observed and expected Fisher information.

Made available in DSpace on 2018-11-12T17:26:01Z (GMT). No. of bitstreams: 0 Previous issue date: 2016-12-01. Added 1 bitstream(s) on 2018-11-12T17:34:10Z : No. of bitstreams: 1 S0101-74382016000300547.pdf: 525570 bytes, checksum: 3d27195cf01df2f212a021687807bdb4 (MD5) ABSTRACT In this paper, a new compounding distribution, named zero truncated Lindley-Poisson distribution is introduced. The probability density function, cumulative distribution function, survival function, failure rate function and quantiles expressions of it are provided. The parameters estimatives were obtained by six methods: maximum likelihood (MLE), ordinary least-squares (OLS), weighted least-squares (WLS), maximum product of spacings (MPS), Crame´r-von-Mises (CM) and Anderson-Darling (AD), and intensive simulation studies are conducted to evaluate the performance of parameter estimation. Some generalizations are also proposed. Application in a real data set is given and shows that the composed zero truncated Lindley-Poisson distribution provides better fit than the Lindley distribution and three of its generalizations. The paper is motivated by application in real data set and we hope this model may be able to attract wider applicability in survival and reliability. Universidade Estadual Paulista Universidade Estadual de Maringá Universidade Estadual Paulista

In general, data that are obtained by counting processes, strictly discrete or discretized (from truncations and/or rounding), are analyzed, without exhaustion, by the Geometric, Logarithmic, Poisson and Negative Binomial distributions. In recent years a large number of discrete distributions have been proposed in the literature from the discretization of continuous random variables. Many of the discretization methods preserve one or more characteristics of the continuous version, with the proposal of Nakagawa e Osaki (1975) being the most used. In this paper, from this methodology, which makes use of the survival function, we propose the discrete version of the continuous generalized Half-Normal distribution, introduced in the literature by Cooray e Ananda (2008). Some of its properties are discussed and Monte Carlo simulations evaluate the bias and accuracy of the estimates obtained by the maximum likelihood method and method of moments. Some discrete data sets found in the literature are considered to illustrate the applicability of the proposed distribution.

The aim of this study was to model the individual height and volume of eucalyptus wood in two integrated crop-livestock-forest systems (ICLF1 and ICLF2) in Campo Grande, a city in the state of Mato Grosso do Sul, Brazil. Classic nonlinear growth models were adjusted for height (Logistic, Gompertz, Richards, Weibull, Van Bertalanffy, Brody, Mitscherlich, and Chapman and Richards) and volume (Shumacher-hall nonlinear, Takata, Honner, Logistic, Gompertz, and Weibull) in two structural arrangements: ICLF1, with a spacing of 14 x 2 m and density of 357 trees ha-1, and ICLF2, with a spacing of 22 x 2 m and density of 227 trees ha-1. Diameter at Breast Height (DBH) measurements were performed in 100% of trees, with measurements of the total height of some individuals and a rigorous scaling procedure in diameter classes. According to the calculated value of Student's t-test, there was no significant evidence that DBH and the average height of the trees were different between ICLF1 and ICLF2. Based on the Akaike information criterion (AIC), the corrected Akaike information criterion (AICC) and the Bayesian information criterion (BIC), the Richards model was selected to estimate heights and the Takata model was selected to estimate the volume.

This paper presents estimates for the parameters included in the Block and Basu bivariate lifetime distributions in the presence of covariates and cure fraction, applied to analyze survival data when some individuals may never experience the event of interest and two lifetimes are associated with each unit. A Bayesian procedure is used to get point and confidence intervals for the unknown parameters. Posterior summaries of interest are obtained using standard Markov Chain Monte Carlo methods in rjags package for R software. An illustration of the proposed methodology is given for a Diabetic Retinopathy Study data set.

The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary least-squares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results.

We introduce the Weibull distributions in presence of cure fraction, censored data and covariates. Two models are explored in this paper: mixture and non-mixture models. Inferences for the proposed models are obtained under the Bayesian approach, using standard MCMC (Markov Chain Monte Carlo) methods. An illustration of the proposed methodology is given considering a life- time data set.

The identification of outliers plays an important role in the statistical analysis, since such observations may contain important information regarding the hypotheses of the study. If classical statistical models are blindly applied to data containing atypical values, the results may be misleading and mistaken decisions can be made. Moreover, in practical situations, the outliers themselves are often the special points of interest and their identification may be the main objective of the investigation. In this way, it was proposed to propose a technique of detection of multivariate outliers, based on cluster analysis and to compare this technique with the method of identification of outliers via Mahalanobis Distance. For data generation, Monte Carlo method simulation and the mixed multivariate normal distribution technique were used. The results presented in the simulations showed that the proposed method was superior to the Mahalanobis method for both sensitivity and specificity, that is, it presented greater ability to correctly diagnose outliers and non-outliers individuals. In addition, the proposed methodology was illustrated with an application in real data from the health area.

In animal production, the models that mimicry the biological reality are of great importance for optimization and sustainability of the productive system. The continuous Burr XII distribution is widely used in survival data analysis, however, the same does not occur with its discrete version, recently proposed in the literature. The purpose of this work is to use the discrete Burr XII distribution, obtained by the discretization method proposed by Nakagawa and Osaki (1975), in the analysis of data related to animal production. The data analyzed describe the time, in days, from birth to first laying of yellow quail (Coturnix coturnix japonica) submitted to two diets. For this purpose the discretized versions of five distributions were used: the discrete Burr XII, the discrete Weibull, the discrete gamma, the discrete inverse-Gaussian and the discrete log-normal. For all distributions, the parameter estimates were obtained by the maximum likelihood method. Despite the similarity between the estimates it is natural to choose the discrete given the nature of the data and assuming the discrete distribution, it could be calculated exactly, for example, the probability of the time to the first posture, which is not possible if a continuous distribution is assumed. Thus, among the discrete distributions, the chi-square goodness-of-fit test showed that the Burr XII distribution was the only one indicated to describe the behavior of the data considered.

In this paper we present a hierarchical Bayesian analysis for a predator–prey model applied to ecology considering the use of Markov Chain Monte Carlo methods. We consider the introduction of a random effect in the model and the presence of a covariate vector. An application to ecology is considered using a data set related to the plankton dynamics of lake Geneva for the year 1990. We also discuss some aspects of discrimination of the proposed models.

The methods of generate a probability function from a probability density function has long been used in recent years. In general, the discretization process produces probability functions that can be rivals to traditional distributions used in the analysis of count data as the geometric, the Poisson and negative binomial distributions. In this paper, by the method based on an infinite series, we studied an alternative discrete Lindley distribution to those study in Gomez (2011) and Bakouch (2014). For both distributions, a simulation study is carried out to examine the bias and mean squared error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence intervals. For the discrete Lindley distribution obtained by infinite series method we present the analytical expression for bias reduction of the maximum likelihood estimator. Some examples using real data from the literature show the potential of these distributions.

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a three-parameter Burr XII distribution, which includes the popular Weibull mixture model as a special case.

Bioequivalence studies have been generally used to compare a test formulation with a reference, in order to validate the interchangeability between them. Some pharmacokinetic (PK) parameters are compared in this type of study, typically using a model which assumes independence among PK parameters, the same variance for the different formulations, logarithmic transformation for the data and normal distribution for the residuals. We propose an alternative model based on a generalized gamma distribution, which permits the presence of positive asymmetry for the data and possible differences in the variances for the different formulations which could have more flexibility in this case. For the multivariate structure, we use a Gaussian copula function to capture the possible dependence between the PK parameters. We use Bayesian inference methods to obtain the results of interest. We also introduce a real data example from where we observe a good fit of the proposed model for the dataset. From this study, we conclude that the proposed model could be a good alternative in some applications where the distribution of the bioequivalence data presents a positive asymmetric distribution.

Summary In many applications of lifetime data analysis, it is important to perform inferences about the mode of the hazard function in situations of lifetime data modeling with unimodal hazard functions. For lifetime distributions where the mode of the hazard function can be analytically calculated, its maximum likelihood estimator is easily obtained from the invariance properties of the maximum likelihood estimators. From the asymptotical normality of the maximum likelihood estimators, confidence intervals can be obtained. However, these results might not be very accurate for small sample sizes and/or large proportion of censored observations. Considering the log-logistic distribution for the lifetime data with shape parameter β > 1 , we present and compare the accuracy of asymptotical confidence intervals with two confidence intervals based on bootstrap simulation. The alternative methodology of confidence intervals for the mode of the log-logistic hazard function are illustrated in three numerical examples.

Binary responses are correlated when the sampling units are clustered or when repeated binary responses are taken on the same experiment unit. In this paper we present a Bayesian analysis of logistic regression models for correlated binary data with random effects. We assume that the random effects, namely αi, i = 1, …, n are draw from a mixture of normal distributions. This assumption gives a great flexibility of fit by correlated binary data. Considering Gibbs sampling with Metropolis-Hastings algorithms, we obtain Monte Carlo estimates for the posterior quantities of interest

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. The main goal of this paper is to present a Bayesian inference procedure for the polyhazard models in the presence of covariates, generalizing the Bayesian analysis presented in Berger and Sun (J. Amer. Statist. Assoc. 88 (1993) 1412), Basu et al. (J. Statist. Plan Inference 78 (1999) 255) and Kuo and Yang (Statist. Probab. Lett. 47 (2000) 229). The two most important particular polyhazard models, namely poly-Weibull, poly-log-logistic and a combination of both are studied in detail. The methodology is illustrated in two real medical datasets.