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.     

Prior Distributions Fitted to Observed Reliability Data

This paper describes methods of fitting prior distributions to equipment MTBF = ?, shows the priors fitted to different equipments, establishes data criteria for fitting prior distribution to ?, and presents the results of a robustness analysis performed on the fitted priors. Systematic procedures for fitting priors are shown for Type 1 data (number of failures in fixed time T) and Type 2 data (observed MTBF, number of failures not the same for all equipments), and specific data criteria, in the form of minimum values of n (number of equipments) and K (number of failures) are presented. The inverted-gamma prior-distributions were derived from operational failure data obtained from Tinker AFB. The equipments are primarily electronic, therefore, the time-to-failure distribution was assumed to be exponential; however, the methods are generally applicable whatever the form of the conditional distribution. The robustness analysis shows the effects of errors in estimating the parameters of the prior on the posterior distribution. In general, the effect of errors in estimating parameters of the prior was practically negligible for large values of K.     

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.     

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.     

Modified Goodness-of-Fit Tests for Gamma Distributions with Unknown Location and Scale Parameters

The common Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises goodness-of-fit tests require continuous underlying distributions with known parameters. This paper gives tables of critical values for these tests for gamma distributions with unknown location and scale parameters and known shape parameters. The powers of these tests are given for a number of alternative distributions. A relation between the critical values and the inverse square of the shape parameter is presented. For larger sample sizes, the modified CvM test is usually the most powerful of the three tests. One exception is for the alternative of a lognormal distribution where the modified AD test is most powerful. The equation, C = ao + a1(1/?2) describes the relation between critical value and shape parameter quite well.     

Minimum Expected Loss Estimators of Reliability and Parameters of Certain Lifetime Distributions

The estimators of reliability and parameters of certain lifetime distributions which are widely used in reliability, repairability, and maintainability are obtained by using a different form of loss function and minimizing the s-expected loss with respect to the posterior distribution. These estimators are called MELO estimators. The applications, and the comparison between MELO, Bayes, and Maximum likelihood estimators are discussed.     

Modified KS, AD, and C-vM tests for the Pareto distribution withunknown location and scale parameters

Standard goodness-of-fit tests based on the empirical CdF (Edf) require continuous underlying distributions with all parameters specified. Three modified Edf-type tests, the Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D), and Cramer-von Mises (C-vM), are developed for the Pareto distribution with unknown parameters of location and scale and known shape parameter. The unknown parameters are estimated using best linear unbiased estimators. For each test, Monte Carlo techniques are used to generate critical values for sample sizes 5(5)30 and Pareto shape parameters 0.5(0.5)4.0. The powers of the modified tests are investigated under eight alterative distributions. In most cases, the powers of the modified K-S, A-D, C-vM tests are considerably higher than the chi-square test. Finally, a functional relationship is identified between the modified K-S and C-vM test statistics and the Pareto shape parameter. Powerful goodness-of-fit tests that supplement the best linear unbiased estimates are provided     

Estimation of Constrained Parameters With Guaranteed MSE Improvement

We address the problem of estimating an unknown parameter vector x in a linear model y=Cx+v subject to the a priori information that the true parameter vector x belongs to a known convex polytope X. The proposed estimator has the parametrized structure of the maximum a posteriori probability (MAP) estimator with prior Gaussian distribution, whose mean and covariance parameters are suitably designed via a linear matrix inequality approach so as to guarantee, for any xisinX, an improvement of the mean-squared error (MSE) matrix over the least-squares (LS) estimator. It is shown that this approach outperforms existing "superefficient" estimators for constrained parameters based on different parametrized structures and/or shapes of the parameter membership region X     

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.     

Decomposition of nonlinear operators into `harmonic? components, with applications to audio signal processing

We show that time-invariant nonhysteretic nonlinear operators K may be decomposed into `nth-harmonic? components Kn (n=?,?2, ?1, 0, 1, 2, ?) consisting of frequencies n1 F1?+?+n1Fi, where F1,?, Fi are frequencies in the input signal and n1+?+ni=n. This is applied to the design of r.f. and a.f. audio signal processors.     

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.     

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.     

Loss probability behavior of Pareto/M/1/K queue

In this letter, we investigate the queue behavior with Pareto interarrival and exponential service time distribution. By numerical analysis and simulations, we analyze the asymptotic and the exact loss probabilities of GI/M/1/K to show the big discrepancy between the asymptotic and the actual loss probability and propose a model for the loss probability of Pareto/M/1/K as a function of the buffer size K and the geometric parameter.     

The Gaussian Transform of Distributions: Definition, Computation and Application

This paper introduces the general-purpose Gaussian transform of distributions, which aims at representing a generic symmetric distribution as an infinite mixture of Gaussian distributions. We start by the mathematical formulation of the problem and continue with the investigation of the conditions of existence of such a transform. Our analysis leads to the derivation of analytical and numerical tools for the computation of the Gaussian transform, mainly based on the Laplace and Fourier transforms, as well as of the afferent properties set (e.g., the transform of sums of independent variables). The Gaussian transform of distributions is then analytically derived for the Gaussian and Laplacian distributions, and obtained numerically for the generalized Gaussian and the generalized Cauchy distribution families. In order to illustrate the usage of the proposed transform we further show how an infinite mixture of Gaussians model can be used to estimate/denoise non-Gaussian data with linear estimators based on the Wiener filter. The decomposition of the data into Gaussian components is straightforwardly computed with the Gaussian transform, previously derived. The estimation is then based on a two-step procedure: the first step consists of variance estimation, and the second step consists of data estimation through Wiener filtering. To this purpose, we propose new generic variance estimators based on the infinite mixture of Gaussians prior. It is shown that the proposed estimators compare favorably in terms of distortion with the shrinkage denoising technique and that the distortion lower bound under this framework is lower than the classical minimum mean-square error bound.     

Best Estimates of Functions of the Parameters of the Gaussian and Gamma Distributions

General formulae for minimum variance unbiased estimators of a broad class of functions of the parameters of the Gaussian and gamma distributions are given. The formulae are easy to employ and are applied here to several examples familiar to the reliability engineer.     

Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.     

Effect of Mean Pressure Levels on the Dynamic Response Characteristics of the Carotid Sinus Baroceptor Process in Dog

The purpose of this investigation was to delineate the effect of imposed mean pressure levels on the open-loop dynamic response characteristics of the carotid sinus baroceptors in dogs. The experimental design consisted of measuring the intrasinus pressure and the gross baroceptor nerve activity while forcing the isolated sinus with sinusoidal pulse pressures, with peak-to-peak amplitude of 50 mm Hg, superimposed on mean pressures of 75, 125, 175, and 225 mm Hg at frequencies of 0.5, 1, 2, 3, 4, 5, 7, 10, 15 and 20 Hz. With this forcing protocol, we were able to divide the traditional sigmoidal pressurenerve activity relationship into three piecewise linear segments whose input-output (transfer) functions could then be determined by conventional linear system analysis. We found that (a) at each mean pressure level, the transfer function relating nerve activity, N(s), to forcing pressure, P(s), was second order and of the form, N(s)/P(s) = ?(1 + ?s + ?s2), and (b) the coefficients ?, ?, and ? were all quadratic functions of the mean pressure level, P? Incorporating the equations for each coefficient as a function of mean pressure into the transfer function yielded the nonlinear differential equation, N(t) = k?(P?) [(P(t) - P?) + ?(P?) (dp(t)/dt) + ?(P?) (d2P(t)/dt2)], which describes the dynamic response of the carotid sinus baroceptor nerve, N(t), over the entire pressure range, P(t), studied.     

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.     

Extreme Value Characteristics of Distributions of Cumulative Processes

The paper introduces the concept of a cumulative stochastic process and derives the general mathematical expression of the distribution corresponding to such processes when they can be assumed to be Markovian. The behaviour of such a distribution in correspondence to accumulation functions of the type u(t) = atb and u(t) = l ln(l + t) is explored. It is shown how the exponential, Weibull, gamma, normal and lognormal distributions are particular cases of the general distribution. Next, the characteristics of the extreme values of n independent observations coming from such a general distribution are investigated. The central characteristics of the extreme values distributions are related to the hazard rate of the initial distribution. In particular, a simple method for relating the modal smallest value and the modal largest value to the sample size using the asymptotic expression of the hazard rate is given. The tail characteristics of the extreme values distributions are investigated numerically or analytically. The mathematical findings are applied to the volume effect on the failure probability of materials.