Some characterization results for parallel systems with two heterogeneous exponential components

In this paper, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the likelihood ratio order (reversed hazard rate order) and the hazard rate order (stochastic order). We establish, among others, that the weakly majorization order between two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between lifetimes of two parallel systems, and that the p-larger order between two hazard rate vectors is equivalent to the hazard rate order (stochastic order) between lifetimes of two parallel systems. Moreover, we extend the results to the proportional hazard rate models. The results derived here strengthen and generalize some of the results known in the literature.     

A Bayesian approach to Weibull survival models—Application to a cancer clinical trial

In this paper we outline a class of fully parametric proportional hazards models, in which the baseline hazard is assumed to be a power transform of the time scale, corresponding to assuming that survival times follow a Weibull distribution. Such a class of models allows for the possibility of time varying hazard rates, but assumes a constant hazard ratio. We outline how Bayesian inference proceeds for such a class of models using asymptotic approximations which require only the ability to maximize the joint log posterior density. We apply these models to a clinical trial to assess the efficacy of neutron therapy compared to conventional treatment for patients with tumors of the pelvic region. In this trial there was prior information about the log hazard ratio both in terms of elicited clinical beliefs and the results of previous studies. Finally, we consider a number of extensions to this class of models, in particular the use of alternative baseline functions, and the extension to multi-state data.     

Stochastic properties of the mixed accelerated hazard models

The accelerated hazard model in survival analysis assumes that the covariate effect acts the time scale of the baseline hazard rate. In this paper, we study the stochastic properties of the mixed accelerated hazard model since the covariate is considered basically unobservable. We build dependence structure between the population variable and the covariate, and also present some preservation properties. Using some well-known stochastic orders, we compare two mixed accelerated hazards models arising out of different choices of distributions for unobservable covariates or different baseline hazard rate functions.     

Fitting and testing the Marshall–Olkin extended Weibull model with randomly censored data

The random censorship model (RCM) is commonly used in biomedical science for modeling life distributions. The popular non-parametric Kaplan–Meier estimator and some semiparametric models such as Cox proportional hazard models are extensively discussed in the literature. In this paper, we propose to fit the RCM with the assumption that the actual life distribution and the censoring distribution have a proportional odds relationship. The parametric model is defined using Marshall–Olkin's extended Weibull distribution. We utilize the maximum-likelihood procedure to estimate model parameters, the survival distribution, the mean residual life function, and the hazard rate as well. The proportional odds assumption is also justified by the newly proposed bootstrap Komogorov–Smirnov type goodness-of-fit test. A simulation study on the MLE of model parameters and the median survival time is carried out to assess the finite sample performance of the model. Finally, we implement the proposed model on two real-life data sets.     

Goodness-of-fit test for Gaussian regression with block correlated errors

We propose a procedure to test that the expectation of a Gaussian vector is linear against a nonparametric alternative. We consider the case where the covariance matrix of the observations has a block diagonal structure. This framework encompasses regression models with autocorrelated errors, heteroscedastic regression models, mixed-effects models and growth curves. Our procedure does not depend on any prior information about the alternative. We prove that the test is asymptotically of the nominal level and consistent. We characterize the set of vectors on which the test is powerful and prove the classical √log log (n)/n convergence rate over directional alternatives. We propose a bootstrap version of the test as an alternative to the initial one and provide a simulation study in order to evaluate both procedures for small sample sizes when the purpose is to test goodness of fit in a Gaussian mixed-effects model. Finally, we illustrate the procedures using a real data set.     

Marshall–Olkin q-Weibull distribution and max–min processes

In this paper, we introduce a new probability model known as Marshall–Olkin q-Weibull distribution. Various properties of the distribution and hazard rate functions are considered. The distribution is applied to model a biostatistical data. The corresponding time series models are developed to illustrate its application in times series modeling. We also develop different types of autoregressive processes with minification structure and max–min structure which can be applied to a rich variety of contexts in real life. Sample path properties are examined and generalization to higher orders are also made. The model is applied to a time series data on daily discharge of Neyyar river in Kerala, India.     

Generalised quasi‐symmetry models for ordinal contingency tables

This paper extends the ordinary quasi‐symmetry (QS) model for square contingency tables with commensurable classification variables. The proposed generalised QS model is defined in terms of odds ratios that apply to ordinal variables. In particular, we present QS models based on global, cumulative and continuation odds ratios and discuss their properties. Finally, the conditional generalised QS model is introduced for local and global odds ratios. These models are illustrated through the analysis of two data sets.     

A new generalized weighted Weibull distribution with decreasing,increasing, upside-down bathtub,N-shape and M-shape hazard rate

Recently, Domma et al. [An extension of Azzalinis method, J. Comput. Appl. Math. 278 (2015), pp. 37–47] proposed an extension of Azzalini's method. This method can attract readers due to its flexibility and ease of applicability. Most of the weighted Weibull models that have been introduced are with monotonic hazard rate function. This fact limits their applicability. So, our aim is to build a new weighted Weibull distribution with monotonic and non-monotonic hazard rate function. A new weighted Weibull distribution, so-called generalized weighted Weibull (GWW) distribution, is introduced by a method exposed in Domma et al. [13]. GWW distribution possesses decreasing, increasing, upside-down bathtub, N-shape and M-shape hazard rate. Also, it is very easy to derive statistical properties of the GWW distribution. Finally, we consider application of the GWW model on a real data set, providing simulation study too.     

On the Mixture of Proportional Odds Models

This article studies the mixed proportional odds model. We build TP2 dependence between the overall population variable and the unobservable covariate and present some preservation properties. Relations on aging characteristics, such as odds function and (reversed) hazard rate, are discussed. Stochastic comparisons on overall population variables are conducted as well.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

Convergence Rate of Strong Consistency of the Maximum Likelihood Estimator in Exponential Family Nonlinear Models

This article proposes some regularity conditions. On the basis of the proposed regularity conditions, we show the strong consistency of the maximum likelihood estimator (MLE) in exponential family nonlinear models (EFNM) and give its convergence rate. In an important case, we obtain the convergence rate O(n ^?1/2(log log n)^1/2)—the rate as that in the Law of the Iterated Logarithm (LIL) for iid partial sums and thus cannot be improved anymore.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

Dynamic proportional hazard rate and reversed hazard rate models

Cox (1972) proportional hazard (PH) model has been used to model failure time data in Reliability and Survival Analysis. Recently, proportional reversed hazard model has been analyzed in the literature. Sometimes, the hazard rate (or the reversed hazard rate) may not be proportional over the whole time interval, but may be proportional differently in different intervals. In order to take care of this kind of problems, in this paper, we introduce the dynamic proportional hazard rate model, and the dynamic proportional reversed hazard rate model, and study their properties for different aging classes. The closure of the models under different stochastic orders has also been studied. Examples are presented to illustrate different properties of the models.     

Odd-Burr generalized family of distributions with some applications

We study a new family of continuous distributions with two extra shape parameters called the Burr generalized family of distributions. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for some of its mathematical quantities. The estimation of the model parameters is performed by maximum likelihood. We prove the flexibility of the new family by means of applications to two real data sets. Furthermore, we propose a new extended regression model based on the logarithm of the Burr generalized distribution. This model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.     

On additive-multiplicative hazards model

In survival analysis and reliability theory, a fundamental problem is the study of lifetime properties of a live organism or system. In this regard, there have been considered and studied several models based on different concepts of ageing such as hazard rate and mean residual life. In this paper, we consider an additive-multiplicative hazard model (AMHM) and study some reliability and ageing properties of the proposed model. We then specify the bivariate models whose conditionals satisfy AMHM. Several properties of the proposed bivariate model are investigated and adequacy of the model is evaluated based on a real data set.     

Defective models induced by gamma frailty term for survival data with cured fraction

In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, São Paulo, Brazil.     

Measure of departure from marginal homogeneity using marginal odds for multi-way tables with ordered categories

For square contingency tables with ordered categories, this paper proposes a measure to represent the degree of departure from the marginal homogeneity model. It is expressed as the weighted sum of the power-divergence or Patil–Taillie diversity index, and is a function of marginal log odds ratios. The measure represents the degree of departure from the equality of the log odds that the row variable is i or below instead of i+1 or above and the log odds that the column variable is i or below instead of i+1 or above for every i. The measure is also extended to multi-way tables. Examples are given.     

Estimation of the cumulative baseline hazard function for dependently right-censored failure time data

In this paper, we study the properties of a special class of frailty models when the frailty is common to several failure times. The models are closely linked to Archimedean copula models. We establish a useful formula for cumulative baseline hazard functions and develop a new estimator for cumulative baseline hazard functions in bivariate frailty regression models. Based on our proposed estimator, we present a graphical model checking procedure. We fit a leukemia data set using our model and end our paper with some discussions.     

MRL ordering of parallel systems with two heterogeneous components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the mean residual life order. We establish, among others, that the reciprocal majorization order between parameter vectors implies the mean residual life order between the lifetimes of two parallel systems. We then extend this result to the proportional hazard rate models.     

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.