Estimation of scale parameters for two exponential distributions[reliability theory]

The scale parameter of the exponential distribution is estimated using conditional specification. When there are two censored samples available for estimating the scale parameter, a preliminary test is usually used to determine whether to pool the samples or to use the individual minimum-variance unbiased estimator. This latter estimator (usual preliminary-test estimator) is studied. The optimum levels of significance and their corresponding critical values for the preliminary test are obtained on the basis of the minimax regret criterion. A preliminary-test shrinkage estimator is proposed, and the optimum values of its shrinkage estimator is proposed, and the optimum values of its shrinkage coefficients are obtained. For a mean-square-error criterion of goodness of estimation, the preliminary-test shrinkage estimator is better than the usual preliminary-test estimator     

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.     

The logarithmic series distribution as a failure model from the Bayesian point of view

In this presentation the logorithmic series is studied as a failure model from the Bayesian point of view. It is assumed that the location parameter behaves as a random variable with beta as its prior distribution. Based on this assumption Bayes estimators for the location parameter and reliability function are derived. By using computer simulation we compare the Bayes estimator for the parameter with the corresponding minimum variance unbiased estimator (MVUE) and the Bayes estimator for the reliability with a corresponding unbiased estimator derived from the MVUE of the probability function.     

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.     

Large-sample estimation of reliability under the exponentialdistribution in life testing

The exact formula for the variance of the uniformly minimum variance unbiased estimator (UMVUE) of the reliability function, R(t), for exponential life is quite cumbersome. A simple expression for the asymptotic variance of the UMVUE of R(t) is found. Interval estimates of R(t) are also constructed. By comparing the approximations to the exact values in several configurations of sample size and population parameter, the derived formula is shown to be useful for sample sizes larger than 15     

On the Joint Synchronization of Clock Offset and Skew in RBS-Protocol

Motivated by the necessity of having a good clock synchronization amongst the nodes of wireless ad-hoc sensor networks, the joint maximum likelihood (JML) estimator for clock phase offset and skew under exponential noise model for reference broadcast synchronization (RBS) protocol is formulated and found via a direct algorithm. The Gibbs sampler is also proposed for joint clock phase offset and skew estimation and shown to provide superior performance relative to JML- estimator. Lower and upper bounds for the mean-square errors (MSE) of JML-estimator and Gibbs Sampler are introduced in terms of the MSE of the uniform minimum variance unbiased (UMVU) estimator and the conventional best linear unbiased estimator (BLUE), respectively.     

A recursive algorithm for computing Cramer-Rao-type bounds onestimator covariance

We give a recursive algorithm to calculate submatrices of the Cramer-Rao (CR) matrix bound on the covariance of any unbiased estimator of a vector parameter &thetas;_. Our algorithm computes a sequence of lower bounds that converges monotonically to the CR bound with exponential speed of convergence. The recursive algorithm uses an invertible “splitting matrix” to successively approximate the inverse Fisher information matrix. We present a statistical approach to selecting the splitting matrix based on a “complete-data-incomplete-data” formulation similar to that of the well-known EM parameter estimation algorithm. As a concrete illustration we consider image reconstruction from projections for emission computed tomography     

Estimating the cumulative downtime distribution of a highlyreliable component

The distribution of the cumulative downtime for a highly reliable component over an interval of time is approximated using a compound Bernoulli process. Given a set of observed cumulative downtimes, the maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimator of the approximate cumulative downtime distribution are derived under the assumption of exponential repair times and are compared to the nonparametric estimator. The ML estimator is more efficient than the UMVU estimator which itself is more efficient than the nonparametric estimator     

A wavelet-based joint estimator of the parameters of long-rangedependence

A joint estimator is presented for the two parameters that define the long-range dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a wavelet-based estimator of the scaling parameter (Abry and Veitch 1998), as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well-justified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed-form expressions are given for the covariance matrix of the estimator as a function of data length, and are shown by simulation to be very accurate even when the technical idealizations are not satisfied. Comparisons are made against two maximum-likelihood estimators. In terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable. We apply the tool to the analysis of Ethernet teletraffic data, completing an earlier study on the scaling parameter alone     

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.     

Robust weighted likelihood estimation of exponential parameters

The problem of estimating the parameter of an exponential distribution when a proportion of the observations are outliers is quite important to reliability applications. The method of weighted likelihood is applied to this problem, and a robust estimator of the exponential parameter is proposed. Interestingly, the proposed estimator is an /spl alpha/-trimmed mean type estimator. The large-sample robustness properties of the new estimator are examined. Further, a Monte Carlo simulation study is conducted showing that the proposed estimator is, under a wide range of contaminated exponential models, more efficient than the usual maximum likelihood estimator in the sense of having a smaller risk, a measure combining bias & variability. An application of the method to a data set on the failure times of throttles is presented.     

Approximate MLEs for the location and scale parameters of thehalf-logistic distribution with type-II right-censoring

For the half-logistic distribution the maximum likelihood method does not provide an explicit estimator for the scale parameter based on either complete or right-censored samples. The authors provide a simple method of deriving an explicit estimator by approximating the likelihood function. The bias and variance of this estimator are studied, and it is shown that this estimator is as efficient as the best linear unbiased estimator. An example to illustrate the method is presented     

Goodness-of-fit tests based on Kullback-Leibler information

We evaluate the power of the sample entropy goodness-of-fit tests for s-normal, exponential, and uniform distributions. We compare them with the mainstream statistical tests, the W test based on the best linear unbiased estimator (BLUE) of the location parameter, the Z test based on the sample spacings, and the R test based on the correlation coefficient between the order statistics of the sample & the corresponding population quantiles. We show that the latter are more powerful overall. The mainstream statistical tests, particularly the Z test, readily extend to censored samples and to multi-sample situations.     

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     

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.     

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.     

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.     

Shrunken estimators for the scale parameter of classical Pareto distribution

The shrunken estimators for the scale parameter of classical Pareto distribution by shrinking the maximum likelihood estimator and the unbiased estimator towards the guess value are proposed. Comparisons with the usual estimators in terms of mean square error have been made. The proposed estimators are preferable in some regions of parametric space.     

Estimation of failure probability under a bivariate normal stress-strength distribution

The probability of a chance failure is a parameter of great importance for design reliability and is often examined through stress-strength models. The uniform minimum variance unbiased estimator for the same under a bivariate normal stress-strength distribution has been obtained.