Pushpa L. Gupta

Modeling failure time data by lehman alternatives

The proportional hazards model has been extensively used in the literature to model failure time data. In this paper we propose to model failure time data by F*(f) = [F(t)]θ where F(t) is the baseline distribution function and θ is a positive real number. This model gives rise to monotonic as well as non-monotonic failure rates even though the baseline failure rate is monotonic. The monotonicity of the failure rates are studied, in general, and some order relations are examined. Some examples including exponentiated Weibull, exponential, gamma and Pareto distributions are investigated in detail.

Analysis of zero-adjusted count data

Abstract In this paper a zero adjusted discrete model is developed. Such a situation arises when the proportion of zeros in the data is higher (lower) than that predicted by the original model. The effect of such an adjustment is studied. The failure rates and the survival functions of the adjusted and the non-adjusted models are compared. The relative error incurred by ignoring the adjustment is studied and it is shown that the relative error is a decreasing function of the count. An adjusted generalized Poisson distribution is studied and the three parameters of this model are estimated by the maximum likelihood method. Finally, two examples are presented. In both the examples, it is shown that the zero adjusted generlized Poisson distribution fits very well and the estimates of the parameters are obtained by simple and straight-forward methods.

Score Test for Zero Inflated Generalized Poisson Regression Model

Abstract In certain applications involving count data, it is sometimes found that zeros are observed with a frequency significantly higher (lower) than predicted by the assumed model. Examples of such applications are cited in the literature from engineering, manufacturing, economics, public health, epidemiology, psychology, sociology, political science, agriculture, road safety, species abundance, use of recreational facilities, horticulture and criminology. In this article, a zero adjusted generalized Poisson distribution is studied and a score test is developed, with and without covariates, to determine whether such an adjustment is necessary. Examples, with and without covariates, are provided to illustrate the results.

On the monotonic properties of discrete failure rates

Abstract As is well known, the monotonicity of failure rate of a life distribution plays an important role in modeling failure time data. In this paper, we develop techniques for the determination of increasing failure rate (IFR) and decreasing failure rate (DFR) property for a wide class of discrete distributions. Instead of using the failure rate, we make use of the ratio of two consecutive probabilities. The method developed is applied to various well known families of discrete distributions which include the binomial, negative binomial and Poisson distributions as special cases. Finally, a formula is presented to determine explicitly the failure rate of the families considered. This formula is used to determine the failure rate of various classes of discrete distributions. These formulas are explicit but complicated and cannot normally be used to determine the monotonicity of the failure rates.

A class of Hurwitz–Lerch Zeta distributions and their applications in reliability

Abstract In this paper, we revisit the study of the Hurwitz–Lerch Zeta (HLZ) distribution by investigating its structural properties, reliability properties and statistical inference. More specifically, we explore the reliability properties of the HLZ distribution and investigate the monotonic structure of its failure rate, mean residual life function and the reversed hazard rate. It is shown that the HLZ distribution is log-convex and hence that it is infinitely divisible. Both the hazard rate and the reversed hazard rate are found to be decreasing. The maximum likelihood estimation of the parameters is discussed and an example is provided in which the HLZ distribution fits the data remarkably well.

Inflated modified power series distributions with applications

In certain applications involving discrete data, it is sometimes found that X = 0 is observed with a frequency significantly higher than predicted by the assumed model. Zero inflated Poisson, binomial and negative binomial models have been employed in some clinical trials and in some regression analysis problems. In this paper, we study the zero inflated modified power series distributions (IMPSD) which include among others the generalized Poisson and the generalized negative binomial distributions and hence the Poisson, binomial and negative binomial distributions. The structural properties along with the distribution of the sum of independent IMPSD variables are studied. The maximum likelihood estimation of the parameters of the model is examined and the variance-covariance matrix of the estimators is obtained. Finally, examples are presented for the generalized Poisson distribution to illustrate the results.

Ageing Characteristics of the Weibull Mixtures

It is well known that mixtures of decreasing failure rate (DFR) distributions have the DFR property. A similar result is, of course, not true for increasing failure rate (IFR) distributions. In a recent note, Gurland and Sethuraman (1994, Technometrks 36(4): 416–418) presented two examples where mixtures of IFR distributions show DFR property. In this paper, we present a general approach to study the mixtures of distributions and show that the failure rates of the unconditional and conditional distributions cross at most at one point. Mixtures of Weibull distribution with a shape parameter greater than 1 are examined in detail. This also enables us to study the monotonic properties of the mean residual life function of the mixture. Some examples are provided to illustrate the results.

Analysis op failure time data by burr distribution

Sometimes it is appropriate to model the survival and failure time data by a non-monotonic failure rate distribution. This may be desirable when the course of disease is such that mortality reaches a peak after some finite period and then slowly declines.In this paper we study Burr, type XII model whose failure rate exhibits the above behavior. The location of the critical points (at which the monotonicity changes) for both the failure rate and the mean residual life function (MRLF) are studied. A procedure is described for estimating these critical points. Necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are provided and it is shown that the conditions provided by Wingo (1993) are not sufficient. A data set pertaining to fibre failure strengths is analyzed and the maximum likelihood estimates of the critical points are obtained.

Effect of length-biased sampling on the modeling error

In statistical data analysis, the choice of an appropriate model is a very important factor. An inappropriate model leads to a different kind of error in the analysis. This error has been called by C. R. Rao as type III error or modeling error as opposed to type I and type II errors in statistical inference.In This paper we Study the relative errors in Incurred by Erroneously Assuming the Distribution of the Family Size N as P(n) While in fact it is the Length-biased (Weighted) Version of P(n).An Analytical Expression for the Relative Error,When the Distribution of N Belongs to the Class of Modified Power Series Distributions, is Derived. More Specifically, the Effect of length-biasing on the Relative Error is Investigated, When N Follows a Generalized Poisson Distribution. These Results are Compared With the Case When N Follows a Poisson Distribution.

Correction for bias introduced by truncation in pharmacokinetic studies of environmental contaminants

Pharmacokinetic studies of biomarkers for environmental contaminants in humans are generally restricted to a few measurements per subject taken after the initial exposure. Subjects are selected for inclusion in the study if their measured body burden is above a threshold determined by the distribution of the biomarker in a control population. Such selection procedures introduce bias in the ordinary weighted least squares estimate of the decay rate caused by the truncation. We show that if the data are conditioned to lie above a line with slope-λ on the log scale then the weighted least squares estimate of λ is unbiased. We give an iterative estimation algorithm that produces this unbiased estimate with commercially available software for fitting a repeated measures linear model. The estimate and its efficiency are discussed in the context of a pharmacokinetic study of 2,3,7,8-tetrachlorodibenzo-p-dioxin. Unbiasedness and efficiency are demonstrated with a simulation.