An estimator for the scale parameter of the two parameter weibull distribution for type i singly right-censored data

For type I censoring, in addition to the failure times, the number failures is also observed as part of the data. Using this feature of type I singly right-censored data a simple estimator is obtained for the scale parameter of the two parameter Weibull distribution. The exact mean and variance of the estimator are derived and computed for finite sample sizes. Its limiting properties such as asymptotic normality and asymptotic relative efficiency are obtained. The estimator has high efficiency for moderate and heavy censoring. Its use is illustrated by means of an example.     

Testing the two parameter weibull distribution

We investigate the power properties of Tiku’s 1980) goodness-of-fit statistics (defined in terms of the sample spacings) for testing the important Weibull distribution and show that these statistics are, on the whole, more powerful than their prominent competitors.     

Shrinkage testimators for the shape parameter of weibull distributin under type ii censoring

In this paper we propose some shrinkage testimators for the shape parameter of the Weibull distribution when censored samples are available and study their properties. Comparison of the testimators with Singh and Bhatkulikar (1978) and with the usual estimator, interms of mean squared error are made. It is shown that the proposed testimators     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

Approximate prediction limit for weibull failure based on preliminary test estimator

In this paper we have considered type II censored sample from a two parameter Weibull distribution with the known scale parameter. Using the preliminary test estimator of the unknown shape parameter (3 proposed by Pandey (1983), the paper derives a method of finding the approximate prediction limit for the minimum or, more generally,the j^th smallest of a set of future observations from the Weibull or even extreme-value distribution     

A comparison of robust regression techniques for the estimation of weibull parameters

In this paper several alternative robust reqression techniques are compared for estimating parameters of a Weibull distribution . In addition to the usual least squares (L₂) and least absolute deviation (L₁) methods, a number of one-step reweighting schemes based on the L₁residuals are considered. The results of an extensive series of Monte Carlo simulation experiments demonstrate that the Anscmbe reweighting scheme generally produces the best Weibull estimates over the range of sample sizes and parameter values studied.     

Statistical inference about the shape parameter of the Weibull distribution by upper record values

Summary In this paper, we provide some pivotal quantities to test and establish confidence interval of the shape parameter on the basis of the firstn observed upper record values. Finally, we give some examples and the Monte Carlo simulation to assess the behaviors (including higher power and more shorter length of confidence interval) of these pivotal quantities for testing null hypotheses and establishing confidence interval concerning the shape parameter under the given significance level and the given confidence coefficient, respectively.     

A family of shrinkage estimators for Weibull shape parameter in censored sampling

In this paper a new class of shrinkage estimators has been introduced for the shape parameter in an independently identically distributed two-parameterWeibull model under censored sampling. The main idea is to incorporate the prior guessed value by correcting the standard estimator, which is essentially an unbiased estimator, with optimally weighted ratios of the guessed value and the standard estimator, instead of considering a convex combination of the standard estimator and the difference of the guessed value and the standard estimator. The resulting estimator dominates the standard estimator in a surprisingly large neighborhood of the guessed value. The suggested estimator has also been compared with the minimum mean squared error estimator and a class of estimators suggested by Singh and Shukla in IAPQR Trans 25(2), 107–118, 2000. It is found that the suggested class of estimators has lesser bias as well as lesser mean squared error than its competitors subject to certain conditions.      

Least Squares estimation for the multivariate weibull model of bougaard based on accelerated life test of system and component

Accelerated life testing of products quickly yields information on life. In this article, we present a simple method to incorporate the information collected from accelerated life tests of both components and (series) systems. The multivariate Weibull distribution of Hougaard is applied to model lifetimes of components. Least-squares (LS) estimators of the model parameters and their joint asymptotic distribution are derived. The effects of the dependence parameter and the proportion of the system-data to the asymptotic relative efficiencies of the LS estimators are investigated.     

On selecting populations better than a control: weibull populations case

Consider k( k ≥ 1) independent Weibull populations and a control population which is also Weibull. The problem of identifying which of these k populations are better than the control using shape parameter as a criterion is considered. We allow the possibility of making at most m(0 ≤ m < k) incorrect identifications of better populations. This allowance results in significant savings in sample size. Procedures based on simple linear unbiased estimators of the reciprocal of the shape parameters of these populations are proposed. These procedures can be used for both complete and Type II-censored samples. A related problem of confidence intervals for the ratio of ordered shape parameters is also considered. Monte Carlo simulations as well as both chi-square and normal approximations to the solutions are obtained.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

Pooled estimators for the shape parameter of the three parameter gamma distribution

The aim of the paper is to study the pooled estimator of the shape parameter of the three parameter gamma distribution when k independent samples are available. Sufficient conditions for the existence of the pooled estimator are given and the small as well as the large sample properties are studied. The harmonic mean of the k estimators of the independent samples is proposed in the place of the pooled estimator, in the case in which the latter does not exist.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

Assessing shape differences in populations of shapes using the complex watson shape distribution

This paper presents a novel Bayesian method based on the complex Watson shape distribution that is used in detecting shape differences between the second thoracic vertebrae for two groups of mice, small and large, categorized according to their body weight. Considering the data provided in Johnson et al. (1988), we provide Bayesian methods of estimation as well as highest posterior density (HPD) estimates for modal vertebrae shapes within each group. Finally, we present a classification procedure that can be used in any shape classification experiment, and apply it for categorizing new vertebrae shapes in small or large groups.     

On the construction of bayesian prediction limits for the weibull distribution

This paper is concerned with a BAYESian construction of the prediction limits for the Weibull distribution as an example of extreme value distributions. Thus, considering Weibull and Uniform distributions for the parameters, the predictive functions, which may lead to approximative evaluation of the prediction limits, is determined by using simulation methods     

Estimation of weibull location

A simulation study was carried out to compare the performances of two different simple estimators of the location parameter for a three-parameter Weibull distribution Both Estimators have been suggested by recent paper in the literature. Bras and mean square error are examined for many different sample-size and shape-parameter-value combinations. Strong evidence of the domination of one estimator over the other is found.     

Confidence intervals for the threshold parameter. ii: unknown shape parameter

Confidence intervals based on the lower extreme values of a large sample are constructed for the threshold parameter (unknown minimum-life) of a life distribution. The confidence level is exact for a certain class of beta distributions and asymptotically exact for a wide class of distributions often used in life testing.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.