Best linear unbiased estimator of the Rayleigh scale parameterbased on fairly large censored samples

This paper tabulates: coefficients and relative s-efficiency of the best linear unbiased estimator (BLUE) of the scale parameter of the Rayleigh distribution for type II censored samples of size N=20(5)40, r=0(1)4 (number of observations censored from the left) and s=0(1)4 (number of observations censored from the right); and ranks, coefficients, variances, and relative s-efficiencies of the BLUE of a based on a selected few order statistics (k) for sample size N=20(1)40 and k=2(1)4. These estimators have the minimum variance among the BLUE of a based on the same number of order statistics. As compared to maximum likelihood estimators (MLE) and approximate MLE, the s-efficiency of BLUE of ρ is very high. When estimating the parameter using only a few observations, the k-optimum BLUE of ρ is the only choice, as the MLE of ρ is not available. Therefore, these tables for coefficients of BLUE of ρ based on censored samples and few observations for moderately large samples, have many applications     

Multiple Comparison for Weibull Parameters

Two problems are considered: 1) testing the hypothesis that the shape parameters of k 2-parameter Weibull populations are equal, given a sample of n observations censored (Type II) at r failures, from each population; and 2) Under the assumption of equal shape parameters, the problem of testing the equality of the p-th percentiles. Test statistics (for these hypotheses), which are simple functions of the maximum likelihood estimates, follow distributions that depend only upon r,n,k,p and not upon the Weibull parameters. Critical values of the test statistics found by Monte Carlo sampling are given for selected values of r,n,k,p. An expression is found and evaluated numerically for the exact distribution of the ratio of the largest to smallest maximum likelihood estimates of the Weibull shape parameter in k samples of size n, Type II censored at r = 2. The asymptotic behavior of this distribution for large n is also found.     

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.     

Best Linear Unbiased Estimator of the Parameter of the Rayleigh Distribution?Part I: Small Sample Theory for Censored Order Statistics

The best linear unbiased estimator of the parameter of the Rayleigh distribution using order statistics in a Type II censored sample from a potential sample of size N is considered. The coefficients for this estimator are tabled to five decimal places for N = 2(1)15 and censoring values of r1, (the number of observations censored from the left) and r2 (the number of observations censored from the right) such that r1 + r2 ? N - 2 for N = 2(1)10, r1 + r2 ? N - 3 for N = 11(1)15.     

Confidence interval for the mean of the exponential distribution,based on grouped data

When the available data from an exponential distribution are grouped, the maximum likelihood estimator (MLE) for the mean and several modified MLE have been discussed in literature. However, little work has been done on interval estimators based on such grouped data. This paper derives the asymptotic property of a statistic which is used to construct an approximate confidence interval for the mean. The width of this approximate confidence interval, based on grouped data, is compared with those based on complete samples, and samples with type-I and type-II censoring. The limits of the ratios of these widths are derived when the sample size approaches infinity. The approximate confidence interval from grouped data is wider than those from complete and censored samples. However, Monte Carlo simulation indicates that the proposed method based on grouped data is adequate, considering the restricted information in this case     

Estimation of Periodically Changing Failure Rate

This paper considers two life testing procedures (progressively censored samples and Bartholomew's experiment) under the assumption that the life of an item follows the exponential distribution. The failure rates are different under n different conditions of usage of the item at regular intervals of time. The maximum likelihood estimates of the n failure rates have been derived along with their asymptotic variances for both types of data (when failure times are recorded and when only the number of items failing in each interval are recorded). A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

Tables for Exact Lower Confidence Limits for Reliability and Quantiles, Based on Least-Squares Estimators of Weibull Parameters

For the 2-parameter Weibull distribution, this paper gives tables to obtain exact lower s-confidence limits for reliability on the basis of the least-squares method and median plotting positions. The tables use a method based on the ancillary property of these estimators. They apply to samples of size N = 3(1)13, censored after the first m observations, m = 3(1)N. The same tables enable one to obtain lower s-confidence limits for population quantiles. The use of the tables is illustrated with a numerical example.     

Point and Interval Estimation, From One-Order Statistic, of the Location Parameter of an Extreme-Value Distribution with Known Scale Parameter and of the Scale Parameter of a Weibull Distribution with Known Shape Parameter

This paper derives a one-order statistic estimator ?mn b for the location parameter of the (first) extreme-value distribution of smallest values with cumulative distribution function F(x;u,b) = 1 - exp {-exp[(x-u)/b]} using the minimum-variance unbiased one-order statistic estimator for the scale parameter of an exponential distribution, as was done in an earlier paper for the scale parameter of a Weibull distribution. It is shown that exact confidence bounds, based on one-order statistic, can be easily derived for the location parameter of the extreme-value distribution and for the scale parameter of the Weibull distribution, using exact confidence bounds for the scale parameter of the exponential distribution. The estimator for u is shown to be b ln cmn + xmn, where xmn is the mth order statistic from an ordered sample of size n from the extreme-value distribution with scale parameter b and Cmn is the coefficient for a one-order statistic estimator of the scale parameter of an exponential distribution. Values of the factor cmn, which have previously viously been tabulated for n = 1(1)20, are given for n = 21(1)40. The ratios of the mean-square-errors of the maximum-likelihood estimators based on m order statistics to those of the one-order statistic estimators for the location parameter of the extreme-value distribution and the scale parameter of the Weibull distribution are investigated by Monte Carlo methods. The use of the table and related tables is discussed and illustrated by numerical examples.     

Asymptotic Distribution of Bayes Estimators in Censored Type-I Samples From A Mixed Exponential Population

Bayes estimators in censored type-I samples, from a mixed exponential population are considered. Their large sample properties are examined by simulation. A log-normal distribution can explain adequately the asymptotic behaviour of Bayes estimators of the parameters.     

Inference for a Multivariate Exponential Distribution with a Censored Sample

Maximum likelihood estimators for the parameters of a multivariate exponential Cdf are easily obtained from partial information about a random sample, censored or not. The partial information consists of the minimum from each multivariate observation and the counts of how often each r.v. was equal to the minimum in an observation. The censoring might cause only the smallest r out of n minima to be observed along with the counts. The estimators depend on the total time-on-test statistic familiar in univariate exponential life testing. A likelihood ratio test for s-independence is derived which has s-significance ? = 0 and easily calculated power function.     

Point and interval estimation for Gaussian distribution, based on progressively Type-II censored samples

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.     

Entropy expressions and their estimators for multivariatedistributions

Entropy expressions for several continuous multivariate distributions are derived. Point estimation of entropy for the multinormal distribution and for the distribution of order statistics from D.G. Weinman's (Ph.D dissertation, Ariz. State Univ., Tempe, AZ, 1966) exponential distribution is considered. The asymptotic distribution of the uniformly minimum variance unbiased estimator for multinormal entropy is obtained. Simulation results on convergence of the means and variances of these estimators are provided     

Extended tables for the moments of gamma-distribution orderstatistics

The authors tabulate expectations, variances, and covariances of order statistics from a sample of size n from a standard gamma distribution with shape parameter r. The expected values are n=1(1)10(5)40 and r =5(1)8; the covariances are n =15(5)25 and r=2(1)5     

On estimating spectral moments in the presence of colored noise

Let{q^(1) (t)}, the signal, be a complex Gaussian process corrupted by additive Gaussian noise{q^(2) (t) }. Observations onp(t)q(t)andp(t) q^(2) (t)are assumed to be available wherep(t)is a smooth weighting function andq = q^(1) + q^(2). Using the Fourier transform of the samples ofp(t)q(t)andp(t) q^(2) (t), estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.     

Bayes parameter estimation for the bivariate Weibull model ofMarshall-Olkin for censored data [reliability theory]

Industrial applications frequently require statistical procedures to analyze life-test data from various sources, particularly in situations where observations are limited. The author develops a Bayes parameter estimation method for bivariate censored data collected from both system and component levels. The bivariate Weibull distribution of Marshall & Olkin (1967) is used to model lifetimes of components in a two-component system. Closed-form Bayes estimators of the model parameters are proposed along with their variances and high-probability density intervals     

Approximate MLE of the scale parameter of the Rayleigh distributionwith censoring

For the Rayleigh distribution, the maximum-likelihood method does not provide an explicit estimator for the scale parameter for left-censored samples; however, such censored samples arise quite often in practice. The author provides a simple method of deriving explicit estimators by approximating the likelihood function and obtains the asymptotic variance of this estimator. He shows that this estimator is as efficient as the best linear unbiased estimator. An example is given to illustrate this method     

Reliability sampling plans for lognormal distribution, based onprogressively-censored samples

This paper presents reliability sampling plans for the lognormal distribution based on progressively censored samples. In constructing these sampling plans, large-sample approximations to the best linear unbiased estimators of the location and scale parameters are used. For some selected progressive censoring schemes, reliability sampling plans are tabulated for p_α and p_β to match MIL-STD-105. While in general, variable-sampling plans require smaller sample size when compared with attribute-sampling plans, the ordinary complete and right-censored life test experiments are special cases of the progressively censored experiment. Hence, the progressively censored reliability sampling plans in this paper are widely applicable. General application of the procedure is discussed, and two examples are provided     

A New Method for Goodness-of-Fit Testing Based on Type-II Right Censored Samples

We present a simple method for testing goodness-of-fit based on Type-II right censored samples. Applying the property of order statistics due to Malmquist, we can transform any conventional Type-II right censored sample of size $r$ out of $n$ from a uniform distribution to a complete sample of size $r$ from a uniform distribution. This result is used to develop the proposed goodness-of-fit test procedure. The simulation studies reveal that the proposed approach provides as good or better overall power than the method of Michael & Schucany.      

Estimators for type-II censored (log)normal samples

Three common estimators for the parameters of the log-normal distribution are evaluated for censored samples. Correction factors which eliminate essentially all the bias, and formulas for the standard deviations of the estimators, are presented. It is reported that the Persson-Rootzen estimators are about as good as the maximum-likelihood estimators, without the penalty of requiring iterative (computer) optimization. Also, the estimators resulting from (least squares) fitting a line to the plot of log lifetimes on normal (Gaussian) probability paper are reasonably good. Formulas are given for obtaining these latter estimators without actually plotting the points. The author simulated 5 k to 30 k samples (more samples for smaller N for each case) and calculated the following: the means, standard deviations, and third moments of each estimator; correlations between the two members of each pair; comparisons between the estimators; and simple corrections to improve the performance of the estimators     

Quantile Estimates in Complete and Censored Samples From Extreme-Value and Weibull Distributions

This paper deals with the problem of estimating the quantiles (?) = 0.01, 0.05(0.05)0.95, 0.99 of the largest extreme-value distribution by using order statistics in small samples. Linear unbiased estimators, with minimum variance, based on ordered observations are constructed for sample sizes 5(5)20 for both complete and right-censored samples.