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     

Goodness-of-fit tests for type-I extreme-value and 2-parameterWeibull distributions

This study investigates the properties of the Kolmogorov-Smirnov (K-S), Cramer-von Mises (C-M) and Anderson-Darling (A-D) statistics for goodness-of-fit tests for type-I extreme-value and for 2-parameter Weibull distributions, when the population parameters are estimated from a complete sample by graphical plotting techniques (GPT). Three GPT-median ranks, mean ranks, symmetrical sample cumulative distribution (symmetrical ranks)-are combined with the least-squares method (LSM) on extreme-value and Weibull probability paper to estimate the population parameters. The critical values of the K-S, C-M, A-D statistics are calculated by Monte Carlo simulation, in which 10⁶ sets of samples for each sample size of 3(1)20, 25(5)50, and 60(10)100 are generated. The power of the K-S, C-M, A-D statistics are investigated for 3 graphical plotting techniques and for maximum likelihood estimators (MLE). A Monte Carlo simulation provided the power results using 10⁴ repetitions for each sample size of 5, 10, 25, 40. The power comparison showed that: Among 3 GPT, the symmetrical ranks give more powerful results than the median and mean ranks for the K-S, C-M, A-D statistics; Among 3 GPT and the MLE, the symmetrical ranks provide more powerful results than the MLE for the K-S and A-D statistics; for the C-M statistic, the MLE provide more powerful results than 3 GPT; Generally, the A-D statistic coupled with the symmetrical ranks and LSM is most powerful among the competitors in this study and is recommended for practical use     

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.     

A cautionary tale about Weibull analysis [reliability estimation]

This paper makes three points about possible perils of unguarded fitting of Weibull distributions to data: (1) bias is introduced by incomplete data, which may have counter-intuitive effects; (2) bias is introduced into percentile estimates by using regression on log-transformed variables to fit the Weibull parameters, particularly if the percentile to be predicted lies outside the range of the data; and (3) the amount of variation associated with such estimates can be very substantial. A partial solution to the incomplete data problem using simulation is presented, and the maximum likelihood approach to parameter estimation and its advantages relative to regression estimation are explained. The problem arose in predicting life expectancy of long-lived components subject to natural aging which cannot be investigated using accelerated testing and for which the collection of data provides an incomplete life record     

On estimating parameters in a discrete Weibull distribution

Two discrete Weibull distributions are discussed, and a simple method is presented to estimate the parameters for one of them. Simulation results are given to compare this method with the method of moments. The estimates obtained by the two methods appear to have almost similar properties. The discrete Weibull data arise in reliability problems when the observed variable is discrete. The modeling of such a random phenomenon has already been accomplished. Estimation of parameters in these models is considered. Since the usual methods of estimation are not easy to apply, a simple method is suggested to estimate the unknown parameters. The estimates obtained by this method are comparable to those obtained by the method of moments. The method can be applied in most inferential problems. Though the authors have restricted themselves to type I distribution, their method of proportions for the estimation of parameters can be easily applied to the type II distribution as well     

Order statistics in goodness-of-fit testing

Our new method uses order statistics to judge the fit of a distribution to data. A test-statistic based on quantiles of order-statistics compares favorably with the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) test statistics. The performance of this new goodness-of-fit test statistic is examined with simulation experiments. For certain hypothesis tests, the test statistic is more powerful than the K-S and A-D test statistics. The new test statistic is calculated using a computer algebra system because of the need to compute exact distributions of order statistics     

Evaluation of four probability distribution models for speckle in clinical cardiac ultrasound images

Segmenting cardiac ultrasound images requires a model for the statistics of speckle in the images. Although the statistics of speckle are well understood for the raw transducer signal, the statistics of speckle in the image are not. This paper evaluates simple empirical models for first-order statistics for the distribution of gray levels in speckle. The models are created by analyzing over 100 images obtained from commercial ultrasound machines in clinical settings. The data in the images suggests a unimodal scalable family of distributions as a plausible model. Four families of distributions (Gamma, Weibull, Normal, and Log-normal) are compared with the data using goodness-of-fit and misclassification tests. Attention is devoted to the analysis of artifacts in images and to the choice of goodness-of-fit and misclassification tests. The distribution of parameters of one of the models is investigated and priors for the distribution are suggested.     

Robust parameter-estimation using the bootstrap method for the2-parameter Weibull distribution

This paper proposes bootstrap robust estimation methods for the Weibull parameters; it applies bootstrap estimators of order statistics to the parametric estimation procedure. Estimates of the Weibull parameters are equivalent to the estimates using the extreme value distribution. Therefore, the bootstrap estimators of order statistics for the parameters of the extreme value distribution are examined. Accuracy and robustness for outliers are examined by Monte Carlo experiments which indicate adequate efficiency of the proposed reliability estimators for data with some outliers     

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.     

Inference Based on Type-II Hybrid Censored Data From a Weibull Distribution

A hybrid censoring scheme is a mixture of type-I and type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are type-II hybrid censored. The maximum likelihood estimators, and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals. Bayes estimates, and the corresponding highest posterior density credible intervals of the unknown parameters, are obtained using suitable priors on the unknown parameters, and by using Markov chain Monte Carlo techniques. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyse one data set for illustrative purposes.     

基于统计相关差异的多基地雷达拖引欺骗干扰识别

针对长基线多基地雷达系统在目标跟踪阶段的拖引欺骗干扰识别问题,考虑到真实目标回波在不同节点雷达中的幅度相互独立,而拖引欺骗干扰假设来自同一干扰机,其在不同节点雷达中的幅度完全相关,从而跟踪波门内只有目标回波与同时存在目标回波与拖引欺骗干扰这两种不同情形下的信号幅度存在统计相关差异。该文提出利用这一差异实现多基地雷达系统的拖引欺骗干扰识别。通过在分析统计相关差异的基础上,对不同节点雷达接收到的回波信号幅度序列进行相关性度量及参数估计,构建检验统计量,在给定的虚警概率下实现了对欺骗干扰的识别。仿真实验结果表明,该方法对欺骗干扰具有较好的识别效果,相较于基于拟合优度的AD检测算法,识别概率平均提高18.63%。     

A general formula for the failure-rate function when distributioninformation is partially specified

This paper presents a new formula for the failure-rate function (FRF), derived from a recently introduced 4-parameter family of distributions. The new formula can be expressed in terms of its Cdf, is characterized by algebraic simplicity, and can replace more complex hazard functions by using routine distribution fitting. When the actual Cdf is unknown and partial distribution-information is available (or can be extracted from sample data), new fitting procedures that use only first-degree or first- and second-degree moments are used to approximate the unknown FRF. This new approach is demonstrated for some commonly used Cdfs and shown to yield highly accurate values for the FRF. Relative to current practice, the new FRF has four major advantages: it does not require specification of an exact distribution thus avoiding errors incurred by the use of a wrong model; since estimates of only low-degree (at most first- or second-degree) moments are required to determine the parameters of the FRF, the associated mean-square-deviations are relatively small; the new FRF can be easily adapted for use with censored data; and simple maximum likelihood estimates can be developed     

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     

Complete Sample Estimation Techniques for Reparameterized Weibull Distributions

The Weibull distribution, frequently employed to assign probabilities to the lifetimes of components and systems operating under stress, is habitually characterized by a pair of positive parameters, termed the scale and shape parameters. Two fundamental reparameterizations of the Weibull probability density function are proposed. The first reparameterization replaces the shape parameter by its inverse, the resulting positive parameter thereafter termed the shaping parameter. This permits a more facile exposition of the properties of parameter estimates, derived in the event that a complete random sample from the Weibull distribution is available. The characteristics of these parameter estimation techniques are then reviewed and compared, and their variances and distributional properties are delineated whenever possible. A second reparameterization extends the parameter space so as to include nonpositive values of the shape parameter. This extension augments the utility and applicability of the Weibull distribution without requiring radical alteration of the standard parameter estimation procedures applicable to the original parameter space.     

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.     

Testing Goodness-of-Fit for the Singly Truncated Normal Distribution Using the Kolmogorov-Smirnov Statistic

This paper proposes the singly truncated normal distribution as a model for estimating radiance measurements from satellite-borne infrared sensors. These measurements are made in order to estimate sea-surface temperatures which can be related to radiances. Maximum-likelihood estimation is used to provide estimates for the unknown parameters. In particular, a procedure is described for estimating clear radiances in the presence of clouds and the Kolmogorov-Smirnov statistic is used to test goodness-of-fit of the measurements to the singly truncated normal distribution. Tables of quantile values of the Kolmogorov-Smirnov statistic for several values of the truncation point are generated from Monte Carlo experiments. Finally a numerical example using satellite data is presented to illustrate the application of the procedures.     

Generalized Linear Mixed Models for Reliability Analysis of Multi-Copy Repairable Systems

The power law process (PLP) is usually applied to failure data from a single repairable system. When a system has a number of copies for analysis, the usual approach is to assume homogeneity among all system copies, and then to pool data from these copies. In the real world, however, it may be more reasonable to assume heterogeneity among the system copies. Therefore, this paper proposes a new generalized linear mixed model (GLMM), called PLP-GLMM, to analyse failure data from multi-copy repairable systems. In the PLP-GLMM, the underlying model for each system copy is assumed to be a PLP at Stage 1, and parameters vary among copies at Stage 2. The PLP-GLMM can make inferences about both the population, and each system copy when accounting for copy-to-copy variance. A modified Anderson-Darling test is adapted to the goodness-of-fit test of the PLP-GLMM. A numerical application is given to show the effectiveness of the model     

Minimum Distance Estimation of the Three Parameters of the Gamma Distribution

Procedures have been investigated to establish robust, adaptive estimating techniques for the 3-parameter Gamma distribution, The procedures incorporate minimum distance statistics for determining the location parameter for a range of sample sizes and shape parameters. Seven new estimators were developed of which six incorporate minimum distance estimation for determining the location parameter or guaranteed life with the remaining parameters estimated by maximum likelihood. All the estimators were compared with maximum likelihood estimators (MLEs) using 1000 Monte Carlo repetitions. The criteria of comparison was the ratio of the mean square errors of the parameter estimates. All the new estimators give better results than the MLE. The minimum distance estimation of the location parameter using the Anderson Darling goodness of fit statistic provided the overall best estimates of the parameters. As the sample size increased the relative position of MLEs improved but were still very inefficient with respect to best of the new estimators at sample size 20.     

On the Weibull shape factor of intrinsic breakdown of dielectric films and its accurate experimental determination. Part I: theory, methodology, experimental techniques

Critically examined several important aspects concerning the experimental determination of Weibull shape factors (slopes). Statistical characteristics of breakdown distribution such as area scaling property and the extreme-value distribution are reviewed. We discuss the experimental measurement methodology of time-to-breakdown (T/sub BD/) or charge-to-charge (Q/sub BD/) distributions with the emphasis on the accuracy. The influence of sample numbers on the estimation of Weibull distribution parameters such as characteristic T/sub BD/ and Weibull slopes are investigated in the context of confidence limits. Some examples of the measurement fallacy on Weibull slopes are given. Three different experimental techniques to measure Weibull slopes are described and compared in terms of their advantages and disadvantages. Finally, we will give a comparison of these three methods. Having established these fundamental aspects of the Weibull slope measurements, we will present our extensive experimental data on thickness, voltage, temperature, and polarity dependence of Weibull slopes in part II.     

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.