Satish K. Agarwal

A generalized gamma distribution and its application in reliabilty

A generalization of gamma distribution is defined by slightly modifying the form of Kobayashis generalized gamma function (1991). Its possible application in reliability theory, to study displacement phenomenon of the corrosion problem in a new machine or metal fatigue, is discussed. A few well known probability distributions are shown to be its particular cases. Hazard function, mean, variance etc are worked out for generalized gamma distribution.

Finite mixture of gamma distributions: A conjugate prior

A finite mixture of gamma distributions [Finite mixture of certain distributions. Comm. Statist. Theory Methods 31(12), 2123-2137] is used as a conjugate prior, which gives a nice form of posterior distribution. This class of conjugate priors offers a more flexible class of priors than the class of gamma prior distributions. The usefulness of a mixture gamma-type prior and the posterior of uncertain parameters @l for the Poisson distribution are illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach, on hierarchical models. Using the generalized hypergeometric function, the method to approximate maximum likelihood estimators for the parameters of Agarwal and Al-Saleh [Generalized gamma type distribution and its hazard rate function. Comm. Statist. Theory Methods 30(2), 309-318] generalized gamma-type distribution is also suggested.

GENERALIZED GAMMA TYPE DISTRIBUTION AND ITS HAZARD RATE FUNCTION

A new generalized gamma distribution is defined involving a parameter δ = λ − 1; λ ≥ 0 in the Kobayashis (1991) function Γλ(m,n). The parameter δ will relax the restriction on the parameter λ > 0 in all probability distributions using Kobayashis (1991) type functions. The hazard rate function of this distribution has the property of monotonicity and that of bathtub. The trend of density function and statistical properties are studied for generalized gamma distribution. With the help of a real life data set the fitting of the distribution is shown. A new Weibull type distribution is also considered and the hazard rate function of this distribution also has both the properties namely bathtub and monotonicity.

FINITE MIXTURE OF CERTAIN DISTRIBUTIONS

ABSTRACT A two parameter extended form of standard gamma function is suggested which provide extra flexibility to the density function over positive range. A finite mixture of beta distribution is defined by using the suggested extended form. The shape of density function of extended gamma distribution and also that of finite mixture of beta distribution for various values of the parameters are shown. Inverted distribution of extended gamma and that of finite mixture of beta distribution are given.

Alternative estimators for the population totals in multiple characteristic surveys

The problem of estimating the population totals of multiple characteristics using without replacement sampling design is addressed. The alternative estimators for the study characteristics which are unrelated or have the low correlations with the characteristic used in sample selection are suggested. The efficiency of the estimators under two super population models, approximating the situations in which both the characteristics are unrelated and also when they are related, is studed. A numerical investigation is also carried out to get insight into the performance of the estimators in real applications.

An estimator of population total in multiple characteristics and its robustness to product moment correlation coefficient

We developed an alternative estimator for the probability proportional to size with replacement sampling scheme when certain characteristics under study have low correlation with the size measured used for sample selection. The performance of the proposed estimator has been studied with other related alternative estimators by comparing biases and the variances of respective alternative estimators. Most of the alternative estimators assume the knowledge of the product moment correlation coefficient. Therefore an empirical study, with the help of wide variety of populations, has been carried out to study their respective efficiency when correlation coefficient is departed from its true value.

On the relative efficiency of estimators of population total in unequal probability sampling when study variable has weak relationship with size variable

The performance of alternative biased estimators of population total in probability proportional to size sampling schemes is studied when the study variable has weak relationship with the size variable. A variety of natural populations are considered to study the performance of these estimators empirically that present useful comparisons among the alternative estimators available at present. It may help the survey practitioners/users to choose the suitable estimator among the class of alternative estimators.

Distributions of the lifetimes of system components operating under an unknown common environment

Families of joint distributions for describing the lifetimes of a system of components that operate under an unknown environment, when the environment follows a Weibull distribution, are derived. The reliability function for this system is calculated and several properties of the aforementioned joint distributions are investigated.

Selecting between two concomitant variables in ratio method of estimation

The issues related to the choice of a concomitant variable when information is available on more than one concomitant variables, under the ratio method of estimation, are addressed. Several situations of preference are derived. The conditions thus obtained are tested on the wide variety of natural populations and the results indicate that the ratio estimator, based on the concomitant variable with the coefficient of variation closed as possible to the coefficient of variation of the study variable, may be preferred by efficiency arguments.

EuroSCORE overestimated cardiac surgery related mortality: Comparing EuroSCORE model and Bayesian approach using new generalized probabilistic model with new form of prior information

Objectives: An attempt is made to suggest an alternative approach using Bayesian analysis in predicting the mortality during cardiac surgery. The risk index predicted through Bayesian approach is compared with the traditional approach EuroSCORE Model for known data sets available in the internet [see [4]]. Methods: The overall mortality related to cardiac surgery was traced from 2002 to 2012.The data was extracted from the official blue book website of the Society for Cardiothoracic Surgery in Great Britain & Ireland. The cohort included patients who underwent cardiac surgery in NHS hospital and other private hospitals in U.K. During each year, the true mortality, predicted mortality by EuroSCORE, and estimated mortality through Bayesian approach were compared. Results: Overall mortality rates derived from EuroSCORE Model were much higher (more than double) of true reported mortality rate for the period 2006 to 2012, and almost 60% higher for the period 2003 to 2005, than the true mortality reported by the Society for Cardiothoracic Surgery in Great Britain & Ireland in its official website. While the Bayesian approach predicted mortality rates estimates that are more consistent, significantly lower than the one estimated by EuroSCORE, and much closer to the reported true mortality. Conclusions: Mortality rate post-surgery is considered a quality metric to hospitals. Bayesian approach provides greater advantage over EuroSCORE in predicting mortality post cardiac surgery. Both are noticed to overestimate the real surgical risk, being less with the Bayesians. Bayesian approach is an alternative predictor to estimate mortality post cardiac surgery and should be verified by further research.