Random variate generation from D-distributions

Within the context of non-parametric Bayesian inference, Dykstra and Laud (1981) define an extended gamma (EG) process and use it as a prior on increasing hazard rates. The attractive features of the extended gamma (EG) process, among them its capability to index distribution functions that are absolutely continuous, are offset by the intractable nature of the computation that needs to be performed. Sampling based approaches such as the Gibbs Sampler can alleviate these difficulties but the EG processes then give rise to the problem of efficient random variate generation from a class of distributions called D-distributions. In this paper, we describe a novel technique for sampling from such distributions, thereby providing an efficient computation procedure for non-parametric Bayesian inference with a rich class of priors for hazard rates.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

Modelling the probability distribution of prize winnings in the UK National Lottery: consequences of conscious selection

Summary.  In the statistical and economics literature on lotteries, the problem of designing attractive games has been studied by using models in which sales are a function of the structure of prizes. Recently the prize structure has been proxied by using the moments of the prize distribution. Such modelling is a vital input into the process of designing appealing new lottery games that can generate large revenues for good causes. We show how conscious selection, the process by which lottery players choose numbers non-randomly, complicates the multivariate distribution of prize winners by introducing massive overdispersion of numbers of winners, and large correlations between the numbers of different types of prize winner. Although it is possible intuitively to reach a qualitative understanding of the data, an a priori model does not fit well. We therefore construct an empirical model of the joint distribution of prize winners and use it to calculate the moments of ticket value as a function of sales. The new model gives much higher estimates of ticket value moments, particularly skewness, than previously obtained. Our results will have consequences for policy decisions regarding game design. A spin-off result is that, on the basis of the results of model fitting, lottery players may increase the expected value of their ticket by strategically choosing numbers which are less popular with other lottery players.     

A bivariate generalized linear model with an application to meteorological data analysis

A bivariate generalized linear model is developed as a mixture distribution with one component of the mixture being discrete with probability mass only at the origin. The use of the proposed model is illustrated by analyzing local area meteorological measurements with constant correlation structure that incorporates predictor variables. The Monte Carlo study is performed to evaluate the inferential efficiency of model parameters for two types of true models. These results suggest that the estimates of regression parameters are consistent and the efficiency of the inference increases for the proposed model for ρ≥0.50 especially in larger samples. As an illustration of a bivariate generalized linear model, we analyze a precipitation monitoring data of adjacent local stations for Tokyo and Yokohama.     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

The Conway–Maxwell–Poisson-generalized gamma regression model with long-term survivors

In this paper, we proposed a flexible cure rate survival model by assuming the number of competing causes of the event of interest following the Conway–Maxwell distribution and the time for the event to follow the generalized gamma distribution. This distribution can be used to model survival data when the hazard rate function is increasing, decreasing, bathtub and unimodal-shaped including some distributions commonly used in lifetime analysis as particular cases. Some appropriate matrices are derived in order to evaluate local influence on the estimates of the parameters by considering different perturbations, and some global influence measurements are also investigated. Finally, data set from the medical area is analysed.     

The gamma-linear failure rate distribution: theory and applications

For any continuous baseline G distribution, Zografos and Balakrishnan [On families of beta- and generalized gamma-generated distributions and associated inference. Statist Methodol. 2009;6:344–362] introduced the generalized gamma-generated distribution with an extra positive parameter. A new three-parameter continuous model called the gamma-linear failure rate (LFR) distribution, which extends the LFR model, is proposed and studied. Various structural properties of the new distribution are derived, including some explicit expressions for ordinary and incomplete moments, generating function, probability-weighted moments, mean deviations and Rényi and Shannon entropies. We estimate the model parameters by maximum likelihood and obtain the observed information matrix. The new model is modified to cope with possible long-term survivors in lifetime data. We illustrate the usefulness of the proposed model by means of two applications to real data.     

Small Sample Bias in the Gamma Frailty Model for Univariate Survival

The gamma frailty model is a natural extension of the Cox proportional hazards model in survival analysis. Because the frailties are unobserved, an E-M approach is often used for estimation. Such an approach is shown to lead to finite sample underestimation of the frailty variance, with the corresponding regression parameters also being underestimated as a result. For the univariate case, we investigate the source of the bias with simulation studies and a complete enumeration. The rank-based E-M approach, we note, only identifies frailty through the order in which failures occur; additional frailty which is evident in the survival times is ignored, and as a result the frailty variance is underestimated. An adaption of the standard E-M approach is suggested, whereby the non-parametric Breslow estimate is replaced by a local likelihood formulation for the baseline hazard which allows the survival times themselves to enter the model. Simulations demonstrate that this approach substantially reduces the bias, even at small sample sizes. The method developed is applied to survival data from the North West Regional Leukaemia Register.     

x~2分布对Gamma分布的近似

对寿命数据进行统计分析是工程界和医学界专业人士常常遇到的问题,寿命数据常有一个被称为截尾的特性,这种特性对我们的统计分析有着巨大的影响。而指数分布在寿命分布中占有重要位置,根据实际工作经验,我们常常假定一些电子产品的寿命数据服从指数分布。本文主要考虑了简单Ⅰ型截尾数据在指数分布下的估计和显著性检验。     

Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.