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     

Quadratic statistics for the goodness-of-fit test of the inverseGaussian distribution

The problem of using a quadratic test to examine the goodness-of-fit of an inverse Gaussian distribution with unknown parameters is discussed. Tables of approximate critical values of Anderson-Darling, Cramer-von Mises, and Watson test statistics are presented in a format requiring only the sample size and the estimated value of the shape parameter. A relationship is found between the sample size and critical values of these test statistics, thus eliminating a need to interpolate among sample sizes given in the table. A power study showed that the proposed modified goodness-of-fit procedures have reasonably good power     

A Class of Distributions Useful in Life Testing and Reliability

A new dass of distributions having a finite range that will be useful in life testing and reliability is proposed. Methods of estimating the unknown parameters are studied. Explicit expressions for lower moments of order statistics in complete random samples of any size are given. A uniformly most powerful test is derived for the `shape' parameter assuming the location and scale parameters known. An asymptotically optimal test procedure is also suggested when the location and scale parameters are unknown.     

On Bayes Estimation of Reliability for the Birnbaum-Saunders Fatigue Life Model

Estimation of reliability for the Birnbaum-Saunders fatigue life distribution is considered. The scale parameter is also the median lifetime, and assuming that the scale parameter is known, Bayes estimators of the reliability function are obtained for a family of proper conjugate priors as well as for Jeffreys' vague prior for the shape parameter. When both parameters are unknown, a modified Bayes estimator of reliability is proposed using a moment estimator of the scale parameter. In addition to being computationally simpler than the MLE of reliability, Monte Carlo simulations for small samples show that the modified Bayes estimator is better than the MME for all values of the shape parameter and as good as the MLE for small values of the shape parameter in the sense of root mean squared errors.     

Reliability tests for Weibull distribution with varyingshape-parameter, based on complete data

The Weibull distribution indexed by scale and shape parameters is generally used as a distribution of lifetime. In determining whether or not a production lot is accepted, one wants the most effective sample size and the acceptance criterion for the specified producer and consumer risks. (μ₀ ≡ acceptable MTTF; μ₁ ≡ rejectable MTTF). Decide on the most effective reliability test satisfying both constraints: Pr{reject a lot | MTTF = μ₀} ⩽ α, Pr{accept a lot | MTTF = μ₁ } ⩽ β. α, β are the specified producer, consumer risks. Most reliability tests for assuring MTTF in the Weibull distribution assume that the shape parameter is a known constant. Thus such a reliability test for assuring MTTF in Weibull distribution is concerned only with the scale parameter. However, this paper assumes that there can be a difference between the shape parameter in the acceptable distribution and in the rejectable distribution, and that both the shape parameters are respectively specified as interval estimates. This paper proposes a procedure for designing the most effective reliability test, considering the specified producer and consumer risks for assuring MTTF when the shape parameters do not necessarily coincide with the acceptable distribution and the rejectable distribution, and are specified with the range. This paper assumes that α < 0.5 and β < 0.5. This paper confirms that the procedure for designing the reliability test proposed here applies is practical     

Optimal simple step-stress plan for Khamis-Higgins model

Optimal times of changing stress-level for the simple step-stress plans under the Khamis-Higgins model (an alternative to the Weibull step-stress model) are determined for a wide range of values of the model parameters. The applicability of the "optimal time of changing stress" formula obtained by Khamis-Higgins for a known shape parameter to the case of unknown shape parameter is examined. Their formula provides a reasonable approximation to the actual optimal times of changing stress levels within a certain range of values of the stress levels and model parameters but a poor approximation outside of that range     

An Evaluation of Exponential and Weibull Test Plans

MIL-STD-781B gives sampling plans (sequential and fixed length) for reliability tests under the assumption of a constant failure rate. Using Monte Carlo techniques, the authors compare s-expected time to a decision and producer and consumer risks for some of these plans. It is shown that plans which assume an exponential distribution are not robust to departures from that assumption. A simple modification of these plans for use when life has a Weibull distribution with known shape parameter not equal to one, and an adaptive test procedure for use when life has a Weibull distribution with unknown shape parameter are proposed. The modified plans for a Weibull distribution with known shape parameter have the same designated producer and consumer risks, but different s-expected time to a decision than the corresponding exponential plans. Using Monte Carlo techniques, the authors determine s-expected time to a decision and producer and consumer risks for various forms of the adaptive procedure.     

Starting & Limiting Values for Reliability Growth

Modifications of the Duane model for reliability growth which permit formulating test plans prior to test-data availability are presented herein. The modified model yields finite and nonzero MTBFs at the start and end of development testing. The unmodified Duane model is inadequate for formulating the Development Test plan for a Reliability Program because the MTBF is zero at the start of testing and is unbounded for long tests. The parameters of the modified model; limiting MTBF, initial MTBF, and Duane shape parameter (logarithmic growth rate) are estimatable from handbooks and previous experience prior to the start of development tests. An implicit expression for the Duane-model scale-parameter as a function of the limiting MTBF, initial MTBF, and shape parameter is given. The time required and/or the reliability improvement potential of development tests are obtained from the model to plan the reliability program. When the development tests are underway interval estimates for the modified Duane model parameters, needed to monitor the progress of the reliability program, can be computed.     

Inference on Weibull Percentiles and Shape Parameter from Maximum Likelihood Estimates

Four functions of the maximum likelihood estimates of the Weibull shape parameter and any Weibull percentile are found. The sampling distributions are independent of the population parameters and depend only upon sample size and the degree of (Type II) censoring. These distributions, once determined by Monte Carlo methods, permit the testing of the following hypotheses: 1) that the Weibull shape parameter is equal to a specified value; 2) that a Weibull percentile is equal to a specified value; 3) that the shape parameters of two Weibull populations are equal; and 4) that a specified percentile of two Weibull populations are equal given that the shape parameters are. The OC curves of the various tests are shown to be readily computed. A by-product of the determination of the distribution of the four functions are the factors required for median unbiased estimation of 1) the Weibull shape parameter, 2) a Weibull percentile, 3) the ratio of shape parameters of two Weibull distributions, and 4) the ratio of a specified percentile of two Weibull distributions.     

Testing that a distribution is new better than used of age to

Hollander and Proschan (1972) gave a test of NBU-ness of a distribution F, but it was pointed out by Green and Tiwari (1989) that the test more appropriately related to much wider null and alternative hypotheses than those stated, and provided suitable critical values. Similarly, Hollander, Park and Proschan (1986) later provided a test that a distribution is NBU-t_o and a modified test was given by Ebrahimi and Habibullah (1990). However, a similar criticism may be levelled against these tests and a modified test is here provided for appropriate, more general, hypotheses.     

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.     

Selecting, under type-II censoring, Weibull populations that aremore reliable

It is pointed out that the problem of selecting Weibull populations that are more reliable is complex; the main result is that there is no simple selection rule. Under type-II censoring, the use of a locally optimal selection rule when the shape parameters are known and the use of a modified selection rule when the unknown shape parameters have some prior distributions are proposed. The performance of this modified rule was tested extensively by simulation; this rule was shown to be quite robust for a variety of beta prior distributions     

The stability test for symmetric alpha-stable distributions

Symmetric alpha-stable distributions are a popular statistical model for heavy-tailed phenomena encountered in communications, radar, biomedicine, and econometrics. The use of the symmetric alpha stable model is often supported by empirical evidence, where qualitative criteria are used to judge the fit, leading to subjective decisions. Objective decisions can only be made through quantitative statistical tests. Here, a goodness-of-fit hypothesis test for symmetric alpha-stable distributions is developed based on their unique stability property. Critical values for the test are found using both asymptotic theory and from bootstrap estimates. Experiments show that the stability test, using bootstrap estimates of the critical values, is better able to discriminate between symmetric alpha stable distributions and other heavy-tailed distributions than classical tests such as the Kolmogorov-Smirnov test.     

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.     

A simple goodness-of-fit test for the power-law process, based on the Duane plot

The PLP (power-law process) or the Duane model is a simple model that can be used for both reliability growth and reliability deterioration. GOF (goodness-of-fit) tests for the PLP have attracted much attention. However, the practical use of the PLP model is its graphical analysis or the Duane plot, which is a log-log plot of the cumulative number of failures versus time. This has been commonly used for model validation and parameter estimation. When a plot is made, and the coefficient of determination, R/sup 2/, of the regression line is computed, the model can be tested based on this value. This paper introduces a statistical test, based on this simple procedure. The distribution of R/sup 2/ under the PLP hypothesis is shown not to depend on the true model parameters. Hence, it is possible to build a statistical GOF test for the PLP. The critical values of the test depend only on the sample size. Simulations show that this test is reasonably powerful compared with the usual PLP GOF tests. It is sometimes more powerful, especially for deteriorating systems. Implementing this test needs only the computation of a coefficient of determination. It is much easier than, for example, computing an Anderson-Darling statistic. Further study is needed to compare more precisely this new test with the existing ones. But the R/sup 2/ test provides a very simple and useful objective approach for decision making with regard to model validation.     

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.     

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.     

The effect of conductivity values on ST segment shift in subendocardial ischaemia

The aim of this study was to investigate the effect of different conductivity values on epicardial surface potential distributions on a slab of cardiac tissue. The study was motivated by the large variation in published bidomain conductivity parameters available in the literature. Simulations presented are based on a previously published bidomain model and solution technique which includes fiber rotation. Three sets of conductivity parameters are considered and an alternative set of nondimensional parameters relating the tissue conductivities to blood conductivity is introduced. These nondimensional parameters are then used to study the relative effect of blood conductivity on the epicardial potential distributions. Each set of conductivity parameters gives rise to a distinct set of epicardial potential distributions, both in terms of morphology and magnitude. Unfortunately, the differences between the potential distributions cannot be explained by simple combinations of the conductivity values or the resulting dimensionless parameters.     

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.     

Approximate Tolerance Limits and Confidence Limits on Reliability for the Gamma Distribution

No exact method is known for determining tolerance limits or s-confidence limits for reliability for the gamma distribution when both parameters are unknown. Perhaps the simplest approximate method is to determine a tolerance limit assuming the shape parameter known and then replace the shape parameter with its ML estimate to obtain approximate limits. Simulated values of the true probability levels, achieved by this method, indicate that this method is not suitable, contrary to what has been anticipated. A second approach is to consider the corresponding tolerance limits assuming the distribution mean known and the shape parameter unknown, and then replace the distribution mean by the sample mean. This approach gives useful results for many practical cases. Simulated values of the true probability levels achieved are presented for some typical cases and limiting values are provided. This method appears satisfactory for all values of the shape parameter, for the common s-confidence levels, and moderate sample sizes.