Another Approach to Incomplete Integrals

One usually writes the incomplete integrals as the sums of discrete probabilities by repeating the procedure of integration by parts. This work provides another approach by employing the binomial expansion.     

On a Class of Random Probability Measures with General Predictive Structure

Abstract. In this study, we investigate a recently introduced class of non‐parametric priors, termed generalized Dirichlet process priors. Such priors induce (exchangeable random) partitions that are characterized by a more elaborate clustering structure than those arising from other widely used priors. A natural area of application of these random probability measures is represented by species sampling problems and, in particular, prediction problems in genomics. To this end, we study both the distribution of the number of distinct species present in a sample and the distribution of the number of new species conditionally on an observed sample. We also provide the Bayesian Non‐parametric estimator for the number of new species in an additional sample of given size and for the discovery probability as function of the size of the additional sample. Finally, the study of its conditional structure is completed by the determination of the posterior distribution.     

Residual life estimation based on bivariate non-stationary gamma degradation process

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method.     

Best linear unbiased estimation of location and scale parameters of the log-logistic distribution

If an assumption, such as homoscedasticity, or some other aspect of an inference problem, such as the number of cases, is altered, our conclusions may change and different parts of the conclusions can be affected in different ways. Most diagnostic procedures measure the influence on one particular aspect of the conclusion - such as model fit or change in parameter estimates. The effect on all aspects of the conclusions can be described by the difference in two log likelihood functions and when the log likelihood functions come from an exponential family or are quasi-likelihoods, this difference can be factored into three terms: one depending only on the alteration, another depending only on the aspects of the conclusions to be considered, and a third term depending on both. The third term is interesting because it shows which aspects of the conclusions are relatively insensitive even to large alterations.     

Properties and applications of the sarmanov family of bivariate distributions

We discuss properties of the bivariate family of distributions introduced by Sarmanov (1966). It is shown that correlation coefficients of this family of distributions have wider range than those of the Farlie-Gumbel-Morgenstern distributins. Possible applications of this family of bivariate distributions as prior distributins in Bayesian inference are discussed. The density of the bivariate Sarmanov distributions with beta marginals can be expressed as a linear combination of products of independent beta densities. This pseudoconjugate property greatly reduces the complexity of posterior computations when this bivariate beta distribution is used as a prior. Multivariate extensions are derived.     

The cut-off point based on underlying distribution and cost function

Cut-off sampling has been widely used for business survey which has the right-skewed population with a long tail. Several methods are suggested to obtain the optimal cut-off point. The LH algorithm suggested by Lavallee and Hidiroglou [6] is commonly used to get the optimum boundaries by minimizing the total sample size with a given precision. In this paper, we suggest a new cut-off point determination method which minimizes a cost function. And that leads to reducing the size of take-all stratum. Also we investigate an optimal cut-off point using a typical parametric estimation method under the assumptions of underlying distributions. Small Monte-Carlo simulation studies are performed in order to compare the new cut-off point method to the LH algorithm. The Korea Transportation Origin – Destination data are used for real data analysis.     

The Additive Genetic Gamma Frailty Model

In this paper the additive genetic gamma frailty model is defined. Individual frailties are correlated as a result of an additive genetic model. An algorithm to construct additive genetic gamma frailties for any pedigree is given so that the variance–covariance structure among individual frailties equals the numerator relationship matrix times a variance. The EM algorithm can be used to estimate the parameters in the model. Calculations are similar using the EM algorithm in the shared frailty model, however the E step is not correspondingly simple. This is illustrated re-analysing data, analysed by the shared frailty model in Nielsen et al . (1992), from the Danish adoptive register. Goodness of fit of the additive genetic gamma frailty model can be tested after analysing data with the correlated frailty model. Doing so, a "defect" in the often used and otherwise well behaving likelihood was found     

Comparisons between parallel systems with exponentiated generalized gamma components

Consider two parallel systems with their independent components’ lifetimes following heterogeneous exponentiated generalized gamma distributions, where the heterogeneity is in both shape and scale parameters. We then obtain the usual stochastic (reversed hazard rate) order between the lifetimes of two systems by using the weak submajorization order between the vectors of shape parameters and the p-larger (weak supermajorization) order between the vectors of scale parameters, under some restrictions on the involved parameters. Further, by reducing the heterogeneity of parameters in each system, the usual stochastic (reversed hazard rate) order mentioned above is strengthened to the hazard rate (likelihood ratio) order. Finally, two characterization results concerning the comparisons of two parallel systems, one with independent heterogeneous generalized exponential components and another with independent homogeneous generalized exponential components, are derived. These characterization results enable us to find some lower and upper bounds for the hazard rate and reversed hazard rate functions of a parallel system consisting of independent heterogeneous generalized exponential components. The results established here generalize some of the known results in the literature, concerning the comparisons of parallel systems under generalized exponential and exponentiated Weibull models.     

Variable selection in gamma regression models via artificial bee colony algorithm

Variable selection is an important task in regression analysis. Performance of the statistical model highly depends on the determination of the subset of predictors. There are several methods to select most relevant variables to construct a good model. However in practice, the dependent variable may have positive continuous values and not normally distributed. In such situations, gamma distribution is more suitable than normal for building a regression model. This paper introduces an heuristic approach to perform variable selection using artificial bee colony optimization for gamma regression models. We evaluated the proposed method against with classical selection methods such as backward and stepwise. Both simulation studies and real data set examples proved the accuracy of our selection procedure.     

Lower bounds on Bayes risks for estimating a normal variance: With applications

Brown and Gajek (1990) gave useful lower bounds on Bayes risks, which improve on earlier bounds by various authors. Many of these use the information inequality. For estimating a normal variance using the invariant quadratic loss and any arbitrary prior on the reciprocal of the variance that is a mixture of Gamma distributions, we obtain lower bounds on Bayes risks that are different from Borovkov-Sakhanienko bounds. The main tool is convexity of appropriate functionals as opposed to the information inequality. The bounds are then applied to many specific examples, including the multi-Bayesian setup (Zidek and his coauthors). Subsequent use of moment theory and geometry gives a number of new results on efficiency of estimates which are linear in the sufficient statistic. These results complement earlier results of Donoho, Liu and MacGibbon (1990), Johnstone and MacGibbon (1992) and Vidakovic and DasGupta (1994) for the location case.