Alex S. Papadopoulos

The Burr Distribution as a Failure Model from a Bayesian Approach

The 2-parameter Burr distribution is introduced as a failure model from a Bayesian approach. Bayesian estimators for the parameter p and reliability function are derived. Two priors are considered when the Bayesian approach is used: gamma and uniform. The Bayesian estimators of p and R(t) were derived in a closed form. Thus the Bayesian estimators are mathematically tractable and easy to use. A hypothetical example shows how the Burr distribution would be used as a failure model.

The Burr type XII distribution as a failure model under various loss functions

Bayesian estimates of the parameter p and the reliability function for the two-parameter Burr type XII failure model under three different loss functions, absolute difference, squared error and logarithmic are derived. It is assumed that the parameter p behaves as a random variable having (i) a gamma prior and (ii) a vague prior. Monte Carlo simulations are presented to compare the Bayesian estimators and the maximum likelihood estimators of the parameter p and the reliability function. The results show that the “popular” squared error loss function is not always the best, and that the other loss functions give comparable results.

On characterization and goodness-of-fit test of some discrete distribution families

Abstract In this article a non-negative integer-valued random variable which is distributed according to a power-series distribution law, is characterized as Poisson, binomial, or negative binomial by relating its first- and second-order moments. In addition, we show that the characterizations can be used to derive goodness-of-fit tests for Poisson, binomial, and negative binomial families. The resulting tests are closely related to those based on Fisher’s index of dispersion. Their performances compared with the classical Pearson–Fisher chi-squared tests are investigated in a Monte Carlo study.

A nonparametric method for estimating the joint probability density of BOD and DO

Abstract A nonparametric probability estimator, having a uniform biochemical-oxygen demand kernel, is utilized to obtain the joint probability density function of (BOD) and dissolved oxygen (DO) along the stretch of a stream when the pollutant is discharged over an interval and the velocity is distance-dependent. Furthermore, probabilistic confidence bounds for BOD and DO are derived. An example and computations are presented to illustrate the usefulness of the model.

Hierarchical Bayes estimation for the exponential-multinomial model in reliability and competing risks

The exponential-multinomial distribution arises from: (1) observing the system failure of a series system with p components having independent exponential lifetimes, or (2) a competing-risks model with p sources of failure, as well as (3) the Marshall-Olkin multivariate exponential distribution under a series sampling scheme. Hierarchical Bayes (HB) estimators of the component sub-survival function and the system reliability are obtained using the Gibbs sampler. A large-sample approximation of the posterior pdf is used to derive the HB estimators of the parameters of the model with respect to the quadratic loss function. The exact risk of the HE estimator is obtained and is compared with those corresponding to some other estimators such as Bayes, maximum likelihood, and minimum variance unbiased estimators.

Bootstrap procedures for time series analysis of BOD data

Abstract Recently, Hannan studied the resolution of closely adjacent spectral lines. In the paper, he discusses the problem of determining the parameters of a trigonometric polynomial when it is observed with stationary noise. He also obtains the asymptotic properties of the estimates of these parameters. The purpose of this paper is to fit a trigonometric polynomial to a Biochemical Oxygen Demand (BOD) data set and to examine the sampling properties of the estimates of the parameters of the model by using the empirical bootstrap procedure. In particular, 95% bootstrap confidence intervals of the estimates are obtained. We also employ the Bayesian bootstrap technique to obtain the interval estimates of the parameters. The problem of prediction using the empirical and Bayesian bootstrap procedures is also studied. This model, when compared with time series models of the form ARIMA ( p , d , q ), gives better fit in terms of a lower mean squared error ( MSE ).

ON BAYES ESTIMATION FOR MIXTURES OF TWO WEIBULL DISTRIBUTIONS UNDER TYPE I CENSORING

Abstract A failure population need not be homogeneous. It can be a mixture of two or more distinct subpopulations. In this paper a population which can be divided into two subpopulations each representing a different cause of failure, is considered. The subpopulations each follow a Weibull distribution. Maximum likelihood estimators, Bayes estimators with respect to proper priors, and with respect to improper priors for the population parameters and the reliability function are derived. All analysis is based on samples censored at a predetermined test termination time. A comparison of the Bayes estimation and the maximum likelihood estimation is made through Monte Carlo simulation.

Shortest bayes credibility intervals for the lognormal failure model

Abstract The lognormal distribution with density function f(x|λ,δ 2 )= (2π) 1 2 δx −1 exp − ( In x−λ) 2 2δ 2 , for x>0,δ>0,−∞ is considered as a failure model from the Bayesian point of view. The shortest Bayesian confidence intervals for the parameters and reliability function are obtained for two cases. First, it is assumed that σ2 is known and that γ has a normal prior distribution. Then, the case that γ is known and σ2 has an inverted gamma prior distribution is considered. Some computer simulations are given to demonstrate the advantage of the shortest Bayesian confidence intervals over the ordinary Bayesian confidence intervals in terms of their length ratios.

Bayesian approach for BOD and DO when the discharged pollutants are random

A stochastic model for biochemical oxygen demand (BOD) and dissolved oxygen (DO) at a distance t downstream, when pollutants are discharged over a continuous stretch, is a random differential equation of the form X(t)=AX(t) + Y(t), t⩾0, with the initial condition X0 = X(0). Assuming that X0 is a random vector having a bivariate normal distribution with the mean vector μ0 and the precision (the inverse of the variance-covariance) matrix Λ0, we provide the prediction equation X(t) at any point t a for μ0 keeping Λ0 fixed and known, and (ii) a normal-Wishart prior for (μ0, Λ0. The theory is supplemented by numerical studies.

A Stochastic Model for BOD and DO with Random Initial Conditions and Distance Dependent Velocity

Abstract A random differential equation arises in stream pollution models when the initial conditions are stochastic and the stream velocity is distance dependent. The mean squared solution gives the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) at any distance downstream from the pollution source. The probability density function and some of the moments of the BOD and DO are obtained, thus completely characterizing the BOD and DO process. An example illustrates the results and compares them with previous ones.