One-Order-Statistic Conditional Estimators of Shape Parameters of Limited and Pareto Distributions and Scale Parameters of Type II Asymptotic Distributions of Smallest and Largest Values

One-order-statistic estimators are derived for the shape parameter K of the limited distribution function F1(x, ?, K) = 1 - (? - x)K and the Pareto distribution function F2(y, ?, K) = 1 - (y - ?)-K, given the location parameters ? and ?, respectively. Similar estimators are derived for the scale parameters v1 and Vn, of the Type II asymptotic distributions of smallest and largest values, F3(w, v1, K) = 1 - exp[-(w/v1)-K] and F4(z, vn K) = exp [-(z/vn)-K], given the shape parameter K and assuming the location parameter is zero. The one-order-statistic estimators are K?|? = -1/Cmn 1n(? - xmn) for the limited distribution, K?|? = 1/Cmn 1n(ymn - ?) for the Pareto distribution, ?1|K = Cmn-1/K Wmn and ?n|K = Cmn-1/K Zn-m+1,n for the Type II distributions of smallest and largest values, where Xmn, Ymn, Wmn, Zmn are the mth order statistics of samples of size n from the respective distributions and Cmn is the coefficient for a one-order-statistic estimator of the scale parameter of an exponential distribution, which has been tabled in an earlier paper. It is shown that exact confidence bounds can be easily derived for these parameters using exact confidence bounds for the scale parameter of the exponential distribution. Use of the estimators is illustrated by numerical examples.     

On Shrinkage Estimation of the Exponential Scale Parameter

This paper shows some simple shrunken estimators for the scale parameter of an exponential distribution and compares them with minimum MSE estimator and the estimator proposed by Pandey We have also obtained a Bayes estimator, which is a shrinkage estimator and has smaller MSE than the estimator (sample mean) n/(n + 1) if sample size, n, is small and other restrictions apply.     

One-Order-Statistic Estimation of the Scale Parameters of Weibull Populations

This paper calculates the minimum-variance unbiased one-order-statistic estimator of the parameter of a one-parameter exponential population. The estimator is given for N = 2(1)20 along with its efficiency with respect to an unbiased M-order-statistic estimator for a sample of N items which is truncated after M items have failed. Furthermore, it is shown that by using the estimator for exponential populations one can obtain a consistent estimator for the scale parameter of Weibull populations with any known shape parameter and with ## location parameter zero. A section on the use of the tabled data and a numerical example are included.     

Linear Estimation of the Scale Parameter of the First Asymptotic Distribution of Extreme Values

A Lagrange multiplier technique is used to obtain linear, minimum-variance, unbiased estimators for the scale parameters of the first asymptotic distributions of smallest and largest values with known mode. Coefficients for multiplying ordered observations are computed for complete and censored samples of size n = 1(1) 15. Each sample of size n is censored from above and all m-order-statistic estimators are obtained where m ? n. Then the smallest subset of # order statistics from the set of m available order statistics is found which yields a 99% efficiency relative to the m-order-statistic estimator. The Cramér-Rao lower bound for the variances of the estimators for complete samples is derived and tabled for n = 1(1) 15. For censored samples the asymptotic variances of the maximum-likelihood m-order-statistic estimators are presented for comparative purposes.     

Monte Carlo Comparisons of Parameter Estimators of the 2-parameter Weibull Distribution

A Monte Carlo Simulation was carried out in order to compare three different estimators of the 2-parameter Weibull distribution. The estimators were the ML (maximum likelihood) estimators and two other estimator pairs suggested by Bain & Antle. The Bain-Antle estimators are better than the ML estimator for small samples (in that their bias, standard deviation, and rms error are smaller), whereas the ML estimator is superior in large samples.     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.     

Uniqueness of maximum likelihood estimators of the 2-parameterWeibull distribution

We present a simple statistic, calculated from either complete failure data or from right-censored data of type-I or -II. It is useful for understanding the behavior of the parameter maximum likelihood estimates (MLE) of a 2-parameter Weibull distribution. The statistic is based on the logarithms of the failure data and can be interpreted as a measure of variation in the data. This statistic provides: (a) simple lower bounds on the parameter MLE, and (b) a quick approximation for parameter estimates that can serve as starting points for iterative MLE routines; it can be used to show that the MLE for the 2-parameter Weibull distribution are unique     

On estimating the shape parameter of the Weibull distribution by shrinkage towards an interval

This paper proposes some shrunken estimators for the shape parameter of the Weibull distribution under censored sampling when some apriori or guessed interval containing the parameter β is available. The extensions of the work done in Pandey and Singh (1984) have been considered. Comparisons of the proposed estimators with the usual unbiased estimator, in terms of mean squared error are made. It is found that the proposed estimators are preferable to the usual estimator in some guessed interval of the parameter space of β.     

Unbiased Maximum-Likelihood Estimation of a Weibull Percentile when the Shape Parameter is Known

The maximum-likelihood (ML) estimator for a percentile of a Weibull distribution with a known shape parameter is considered. Multiplicative correction factors are listed for rendering the ML estimator mean or median unbiased in the cases where the samples are type II censored with or without replacement. The correction factors depend upon the number of failures and the shape parameter but are independent of the sample size and the percentile being estimated.     

Admissible, Minimax and Equivariant Estimators of Life Distributions from Type II Censored Samples

In life testing, the unique minimum variance unbiased estimator (MVUE) ? is often used when it exists. However it has been shown for certain distributions that an estimator of the form k? with uniformly smaller mean square error exists. Such extimators are derived here for a class of life distributions and are shown to be admissible, minimax, and (in most cases) equivariant. The underlying distribution from which the samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein & Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are also given for the Type II asymptotic distribution of largest values, Pareto, and limited distributions. In addition, admissible linear estimators of the form a? + b are obtained and it is shown that they are a form of locally best estimators for some portion of the parameter space. Both k? and a? + b could be used in nonrepetitive estimation problems where bias causes no difficulty.     

Simultaneous Estimation of Location and Scale Parameters of the Weibull Distribution

This paper deals with the simultaneous estimation of the location parameter ? and the scale parameter ? of the Weibull distribution when both are unknown and the shape parameter ? is known. The best linear unbiased estimate (BLUE) (?, ?) based on a subset of k optimum ordered observations selected from the whole sample is compared with 1) Ogawa's asymptotically best linear estimate (ABLE) (?*, ?*) based on k ordered observations whose ranks are approximated by an asymptotic optimum selection, and 2) the BLUE based on the ranks in 1). Tables facilitating the computation of (?, ?) based on k = 3, 4 optimum ordered observations are provided.     

Tables to Facilitate Calculation of an Asymptotically Optimal t Test for Equality of Location Parameters of a Certain Extreme-Value Distribution

A t test, proposed by Ogawa and based on the use of a few sample quantiles selected from large samples, is considered for testing the hypothesis H0: ?1 = ?2 against the hypothesis H1: ?1 ? ?2 concerning the location parameters ?1 and ?2 of two extreme-value distributions with common unknown scale parameter ?. Tables that simplify the calculation of the test statistic and an example illustrating their use are provided.     

Shrinkage estimation of scale parameter of the extreme-valuedistribution

The author studies the usual preliminary test estimator of the scale parameter of the extreme-value distribution in censored samples. The optimum levels of significance and their corresponding critical values for the preliminary test are obtained based on the minimax regret criterion. A preliminary test shrinkage estimator that is smoother than the usual preliminary test estimator is proposed as well. The optimum values of shrinkage coefficients for the preliminary test shrinkage estimator are obtained, and are also based on the minimax regret criterion. Comparison of these two estimators shows that if the mean square error is a criterion of goodness of estimation then the preliminary test shrinkage estimator is better than the usual preliminary test estimator     

Life Tests with Periodic Change in the Scale Parameter of a Weibull Distribution

Two life testing procedures, namely, the progressively censored samples and Bartholomew's experiment are discussed under the assumption that the life of an item follows a specialized Weibull distribution. The scale parameter is different under two different conditions of usage of the item at regular intervals of time, the shape parameter remains unchanged throughout the experiment. The maximum likelihood estimates of the two scale parameters have been derived along with their variances. A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

A class of shrinkage estimators for the scale parameter of theexponential distribution

A class of shrinkage estimators for the scale parameter of the exponential distribution is suggested. It includes some previously published estimators as special cases. An analogous estimator based on censored samples is considered. An example is given     

Estimation in the 2-parameter exponential distribution with priorinformation

This paper proposes a class of estimators for the scale parameter and for the mean of a 2-parameter exponential distribution, which is important in life testing and reliability theory, given a prior estimate of the scale parameter. The class of estimators for the scale parameter is motivated by the work of Jani (1991). These estimators have smaller mean square error than the classical estimators for all values of the location parameter, and for values of the scale parameter in a neighborhood of the prior estimate. Numerical computations indicate that certain of these estimators substantially improve the classical estimators for values of the scale parameter near the prior estimate, especially for small sample sizes     

Robust Estimators of the 3-Parameter Weibull Distribution

Robust estimators of the location, scale, and shape parameters of the Weibull distribution are proposed. The estimators are easy to calculate and have few of the disadvantages associated with the maximum likelihood estimators. Their rms-errors are considerably smaller than those of the estimators available in the literature.     

Bayesian Analysis of the Weibull Process with Unknown Scale Parameter and Its Application to Acceptance Sampling

The Weibull process with unknown scale parameter is taken as a model for Bayesian decision making. The family of natural conjugate prior distributions for the scale parameter is exhibited and used in prior and posterior analysis. Preposterior analysis and several sampling schemes are then discussed. Preposterior analysis is given for an acceptance sampling problem with utility linear in the unknown mean of the Weibull process, in which the sampling scheme yields the first r failures in a life test of n items. An example is included.     

Properties of Moment Estimators For the 3-Parameter Weibull Distribution

This paper gives a method of constructing moment estimators for the shape, scale, and location parameters of the 3-parameter Weibull distribution. These estimators are asymptotically s-normally distributed around the population values; the asymptotic covariance matrix is obtained. The estimators and their estimated covariance matrix can be constructed by using tables in the paper; the covariance matrix can then be used to construct s-confidence regions for the estimators.     

Linear Estimation of the Parameters of the Extreme-Value Distribution Based on Suitably Chosen Order Statistics

Best linear unbiased estimates (BLUEs) based on a few order statistics are found for the location and scale parameters of the extreme-value distribution (Type-I asymptotic distribution of smallest values), when one or both parameters are unknown, such that the estimates have maximum efficiencies among the BLUEs based on the same number of order statistics. These estimates are then compared with the BLUEs and asymptotically best linear estimates (ABLEs) based on a few order statistics whose ranks were determined from the spacings that maximize the asymptotic efficiencies of the ABLEs. An application to the Weibull distribution is given.