Anwar M. Hossain

Comparison of estimation methods for weibull parameters: Complete and censored samples

This paper compares several methods for estimating the parameters of the two-parameter Weibull distribution with complete, multiply time censored, and type II censored samples. An extensive simulation study compares the performance of these estimators.

Bayesian survival estimation of Pareto distribution of the second kind based on failure-censored data

This paper presents Bayesian estimation of the survival function of the Pareto distribution of the second kind using the methods of Lindley (1980) and Tierney and Kadane (1986). A numerical example is given to illustrate the results derived. Based on a Monte Carlo simulation study, comparisons are made between these two methods, as well as, their competitor, the maximum likelihood method, by considering different censored samples and several values of the true shape and scale parameters.

Detection of influential observations in multivariate regression

Several methods have been suggested, in the literature, to detect influential observations from the data fitting usual linear model y=X∗∗∗+∗∗∗, ∗∗∗∽N(0, ∗∗∗2I). Recently, Chatterjee & Hadi (1986) have reviewed most of these available methods and described the inter-relationships between them. In this article, we extend some of these methods to the case of multivariate regression data. We consider several data sets to illustrate the methods.

On Bayesian estimation and prediction from Rayleigh based on type II censored data

This paper presents Bayes estimators and highest posterior density intervals for the Rayleigh parameter and its reliability based on the first r lifetimes in a group of n components under test. Based on the same censored data, Bayes predictive estimators and highest posterior density prediction intervals for a future observation as well as for the remaining (n-r) order statistics are derived. A numerical example is included.

Monotone Log-Odds Rate Distributions in Reliability Analysis

Abstract Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLEs are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters. Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.

Estimation of parameters in the presence of outliers for a burr XII distribution

This paper deals with an unweighted least squares (LS) estimation of the parameters of a two-parameter Burr XII distribution and compares the results with the maximum likelihood (ML) and maximum product of spacings (MPS) methods. The performance of these estimators is examined with and without outliers through simulation studies. Also, we obtain approximate confidence intervals for the parameters c and k. We propose that the LS method when data, yi are generated by using log log [n/(n — i -f 0.5)) for small n be preferred over the ML and MPS methods, while for large samples the ML method has a slight edge over the LS method in terms of root-mean squared error (RMSE). However, the amount of computer time required by the ML method in solving the simultaneous equations negates this advantage.

Unweighted least squares estimation of weibull parameters

This paper considers several unweighted least squares estimation of the parameters of a two-parameter Weibull distribution and compares the results with the maximum likelihood estimator. The methods are illustrated with a numerical example. Within various parameters and sample sizes explored, in the simulation study, the least squares estimation procedure for small n over maximum likelihood is recommended for estimating the Weibull parameters. The unweighted least squares method is especially recomended for small samples and when the v-data are generated by using ln

Conditional tests for homogeneity of zero-inflated Poisson and Poisson-hurdle distributions

We develop two conditional tests for homogeneity of zero-inflated Poisson (ZIP) and Poisson-hurdle distributions. A Monte Carlo method is proposed for approximating the reference distributions of these tests. The techniques are applied to two examples.

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferronis inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welschs statistic and the Cooks distance are appropriate for detection of influential observations.