Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength

In this paper, we estimate multicomponent stress-strength reliability by assuming Burr-XII distribution. The research methodology adopted here is to estimate the parameter using maximum likelihood estimation. Reliability is estimated using the maximum likelihood method of estimation and results are compared using the Monte Carlo simulation for small samples. Using real data sets we illustrate the procedure clearly.     

Obtaining a Trapezoidal Distribution

Given a most believed value for a quantity together with upper and lower possible deviations from that value, a rectangular distribution might be used to represent state-of-knowledge about the quantity. If the deviations are themselves known by probability distributions, and the value conditioned on the deviations is rectangular, then the marginal distribution of the value is determined by the distributions of the deviations. Here we show under quite general conditions that conversely, given the marginal distribution, the distributions of the deviations are uniquely determined. The case in which the marginal distribution is trapezoidal is studied in some detail.     

Fisher Information Matrix for the Generalized Feller-Pareto Distribution

In this article, the exact form of Fisher information matrix for the generalized Feller-Pareto (GFP) distribution is determined. The GFP family is a general distribution which includes a variety of distributions as special cases. For example:

??generalized Singh-Maddala distribution which in turn includes Burr, Fisk, and Lomax distribution (see Kleiber and Kotz, 2003 Kleiber, C., Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken, NJ: John Wiley.[Crossref] , [Google Scholar]);

??a Pareto IV distribution which includes a hierarchy of Pareto models, omitted an additional location parameter (see Arnold, 1983 Arnold, B.C. (1983). Pareto Distributions. Fairland, MD: International Cooperative Publishing House. [Google Scholar], 2008 Arnold, B.C. (2008). Pareto and generalized pareto distributions. In: Modeling Income Distributions and Lorenz Curves, Economic Studies in Equality, Social Exclusion and Well-Being, Chotikapanich, D. (Ed.), New York: Springer. pp. 119–145.[Crossref] , [Google Scholar]); and

??beta Lomax distribution which includes, for example, beta II and Lomax distributions.

Application of these distributions covers a wide spectrum of areas ranging from actuarial science, economics, finance to bioscience, telecommunications, and medicine.     

On Wavelet Estimation of the Derivatives of a Density Based on Biased Data

Here, we consider wavelet based estimation of the derivatives of a probability density function under random sampling from a weighted distribution and extend the results regarding the asymptotic convergence rates under the i.i.d. setup studied in Prakasa Rao (1996 Rao, B. L.S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernat. 28:91–100. [Google Scholar]) to the biased-data setup. We compare the performance of the wavelet based estimator with that of the kernel based estimator obtained by differentiating the Efromovich (2004 Efromovich, S. (2004). Density estimation for biased data. Ann. Statist. 32:1137–1161.[Crossref], [Web of Science ®] , [Google Scholar]) kernel density estimator through a simulation study.     

A model with long-term survivors: negative binomial Birnbaum-Saunders

Abstract

We propose a cure rate survival model by assuming that the number of competing causes of the event of interest follows the negative binomial distribution and the time to the event of interest has the Birnbaum-Saunders distribution. Further, the new model includes as special cases some well-known cure rate models published recently. We consider a frequentist analysis for parameter estimation of the negative binomial Birnbaum-Saunders model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We illustrate the usefulness of the proposed model in the analysis of a real data set from the medical area.     

Bias reduction for the maximum likelihood estimator of the doubly-truncated Poisson distribution

ABSTRACT

We derive an analytic expression for the bias of the maximum likelihood estimator of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. For smaller sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. Bias is of little concern in the positive Poisson distribution, the most common form of truncation in the applied literature. Bias appears to be the most severe in the doubly-truncated Poisson distribution, when the mean of the distribution is close to the right (upper) truncation.     

Gamma shared frailty model based on reversed hazard rate

Abstract

Frailty models are used in survival analysis to account for unobserved heterogeneity in individual risks to disease and death. To analyze bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), shared frailty models were suggested. Shared frailty models are frequently used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor(frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, we introduce shared gamma frailty models with reversed hazard rate. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply the proposed model to the Australian twin data set.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

A unified approach to multilevel sample selection models

Abstract

We propose a unified approach for multilevel sample selection models using a generalized result on skew distributions arising from selection. If the underlying distributional assumption is normal, then the resulting density for the outcome is the continuous component of the sample selection density and has links with the closed skew-normal distribution (CSN). The CSN distribution provides a framework which simplifies the derivation of the conditional expectation of the observed data. This generalizes the Heckman’s two-step method to a multilevel sample selection model. Finite-sample performance of the maximum likelihood estimator of this model is studied through a Monte Carlo simulation.     

The Poisson-conjugate Lindley mixture distribution

ABSTRACT

A new discrete distribution that depends on two parameters is introduced in this article. From this new distribution the geometric distribution is obtained as a special case. After analyzing some of its properties such as moments and unimodality, recurrences for the probability mass function and differential equations for its probability generating function are derived. In addition to this, parameters are estimated by maximum likelihood estimation numerically maximizing the log-likelihood function. Expected frequencies are calculated for different sets of data to prove the versatility of this discrete model.