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.     

A class of subset selection procedures for Weibull populations

Among distributions that represent component and system life, the Weibull family is presented. Goodness of a Weibull population is defined in terms of the shape parameter. A class of subset selection-procedures based on the U-statistic (derived via the concept of convex ordering) is proposed for two cases: (1) selecting a subset from k Weibull populations which contains the population with the largest shape parameter; and (2) selecting a subset containing all the populations whose shape parameters are at least that of the control population; unknown and known control populations are considered. The approximate implementation of the selection procedures with the help of the existing tables is discussed, and a numerical example is given. Statistical simulation is used to investigate the optimal members of the proposed class in case 1. The simulation shows that a common sample-size of 11 or more can be used to implement the asymptotic results     

Confidence Limits for Weibull Regression With Censored Data

The response variable in an experiment follows a 2-parameter Weibull distribution having a scale parameter that varies inversely with a power of a deterministic, externally controlled, variable generically termed a stress. The shape parameter is invariant with stress. A numerical scheme is given for solving a pair of nonlinear simultaneous equations for the maximum likelihood (ML) estimates of the common shape parameter and the stress-life exponent. Interval and median unbiased point estimates for the shape parameter, stress-life exponent and a specified percentile at any stress, are expressed in terms of percentage points of the sampling distributions of pivotal functions of the ML estimates. A numerical example is given.     

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.     

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     

A reliability model for stress vs strength problem

In this paper a general reliability model for the stress vs strength problem has been discussed. The model has been illustrated by considering the stress to follow normal distribution and the strength to follow the gamma distribution. A numerical estimate of the reliability under different values of the distribution parameters (for both the strength and stress distributions) has been presented.Exponential distribution, to which the gamma distribution reduces when its shape parameter α equals unity, has been discussed as a particular case.     

Signal level distributions and fading for a long over-water link at 5 GHz

Measurement results are presented for an 88-km multiple-diversity line-of-sight microwave link across the English Channel operating in the 4 to 5 GHz frequency range. Signal level and fade duration statistics derived from the measurements are compared with the applicable International Radio Consultative Committee (CCIR) models. Although signal level distributions are in fair agreement with CCIR predictions, the median fade duration tends to be shorter than values estimated from the CCIR model.     

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.     

Some Graphical Techniques for Estimating Weibull Confidence Intervals

Many methods for estimating the parameter and percentile statistical confidence intervals for the Weibull and Gumbel (extreme value) distributions have been described in the literature. Most of these methods depend on extensive computer programs, require reference to tables which do not cover all sample sizes of interest and/or are not widely available. This paper describes a semi-empirical technique which permits rapid estimation of the 2-sided 90% statistical confidence intervals for the Weibull or Gumbel distribution parameters, as well as for the 1, 5, 10 percentiles. The estimates can be obtained for type II censoring and sample sizes to 25. The statistical confidence intervals calculated using this method are not exact, but are very good approximations and are useful to engineers who do not have ready access to programs or lengthy tables, or who require quick estimates. If more accurate statistical confidence intervals are required, then the more complicated methods described in the references should be used.     

A new method for the prediction of fade duration statistics in satellite links above 10 GHz

A model for the statistical prediction of fade durations, applicable both to scintillation and rain-induced effects, is presented in this paper. It assumes that durations longer than about 1 minute are lognormally distributed whereas shorter durations follow a power-law distribution. The model is assessed using data contained in the CCIR data bank and from measurements performed with the SIRIO and Olympus satellites. The model is tested using the same data, and the results give multiplicative errors in the range from 0.6 to 1.8 (averages) and from 0.3 to 3.0 (r.m.s.) in the estimation of the fade durations exceeded for fixed probability.     

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.     

Analysis of Moderate and Intense Rainfall Rates Continuously Recorded Over Half a Century and Influence on Microwave Communications Planning and Rain-Rate Data Acquisition

This paper is concerned with the statistical investigation of a massive data bank of 49 years of rainfall rateRcontinuously recorded in Barcelona using a rain-rate gauge with ten seconds response time. With radio communications in mind, the paper addresses and reviews in detail: 1) reliable statistical model forR, 2) number of years required to obtain a database from which to derive a reliableR-distribution, and 3) the CCIR worst-month concept. The research has shown that the generalized Paretoaexp(-muR)/R^{b}gives nearly perfect fit for all ranges ofRfollowed closely by the gamma distribution, and the simpler square root (R^{1/2}) normal distribution gives excellent fit too. The log-normal distribution was unsatisfactory forR geq 60mm/h. The spread of the yearly distribution ofP(R)is cube root normally distributed([P(R)]^{1/3})and between 7 and 10 years are required before a reliable average distributionP(R)can be obtained. The study of theP(R)return time in years is also presented. High resolution ofP(R)is presented looking at the evolution of the annualP(R)in terms of the hourly and monthly contributing parts revealing statistical features such as the location in time of rain rates above 50 mm/h. Finally, the study shows that the calendar month contribution toP(R)remains at all times well below the synthetic CCIR worst month and recommendations are then given about its use.     

Modelling attenuation by rain in tropical regions

The data in the CCIR data banks were employed for a statistical study of the relative performance of several rain attenuation prediction procedures in temperate and tropical regions. The results show that the models worked well, in general, when used for prediction at latitudes more than 30° from the equator, but, in the equatorial region, significant prediction errors were observed for all the models. Three sources of error were discovered. The most important is the use of too few rain climate zones to span the wide range of rain conditions present in the equatorial region. The second is an inadequate procedure for taking the naturally occurring vertical variations of specific attenuation into account. Finally, for the CCIR attenuation prediction model, the use of a universal shape for the cumulative distribution of path attenuation must be called into question.     

Texture retrieval using mixtures of generalized Gaussian distribution and Cauchy–Schwarz divergence in wavelet domain

This paper presents a novel similarity measure in a texture retrieval framework based on statistical modeling in wavelet domain. In this context, we use the recently proposed finite mixture of generalized Gaussian distribution (MoGG) thanks to its ability to model accurately a wide range of wavelet sub-bands histograms. This model has already been relied on the approximation of Kullback–Leibler divergence (KLD) which hinders significantly the retrieval process. To overcome this drawback, we introduce the Cauchy–Schwarz divergence (CSD) between two MoGG distributions as a similarity measure. Hence, an analytic closed-form expression of this measure is developed in the case of fixed shape parameter. Otherwise, when the shape parameter is variable, two approximations are derived using the well-known stochastic integration with Monte-Carlo simulations and numerical integration with Simpson׳s rule. Experiments conducted on a well known dataset show good performance of the CSD in terms of retrieval rates and the computational time improvement compared to the KLD.     

A statistical design approach for gigabit-rate fiber-optictransmission systems

A statistical design approach is proposed for the design of fiber-optic transmission systems (FOTSs). It yields increased regenerator spacings up to about 50% over those provided by the traditional worst-case design. The method involves assuming most of the system parameters to be random variables with known probability distributions. The probability distribution of the regenerator spacing is then numerically found by a Monte Carlo simulation from which the regenerator spacing with a given success probability can be determined. A thorough statistical treatment of attenuation, dispersion, and chirp limits is presented, and the effects of varying the probability density functions of underlying optical system parameters are examined. The numerical results shown demonstrate the repeatability of the proposed statistical design approach     

Corner Function Analysis of Microstrip Transmission Lines

A new method of analysis of microstrip transmission lines using corner functions (eigenfunctions) is presented. Assuming the TEM mode propagation for the shielded microstrip line structure, solutions are set up in a series of corner functions, which isolates the singularity at the edge of the strip. The boundary conditions are satisfied by using a method of successive integration of boundary errors. Numerical results are obtained for the charge distribution from which the microstrip line characteristic impedance is determined. Excellent agreement with published theoretical results and experimental data is obtained. The numerical results clearly bring out the power of the method.     

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.     

Gaussian class multivariate Weibull distributions: theory and applications in fading channels

Ascertaining on the suitability of the Weibull distribution to model fading channels, a theoretical framework for a class of multivariate Weibull distributions, originated from Gaussian random processes, is introduced and analyzed. Novel analytical expressions for the joint probability density function (pdf), moment-generating function (mgf), and cumulative distribution function (cdf) are derived for the bivariate distribution of this class with not necessarily identical fading parameters and average powers. Two specific distributions with arbitrary number of correlated variates are considered and studied: with exponential and with constant correlation where their pdfs are introduced. Both cases assume equal average fading powers, but not necessarily identical fading parameters. For the multivariate Weibull distribution with exponential correlation, useful corresponding formulas, as for the bivariate case, are derived. The presented theoretical results are applied to analyze the performance of several diversity receivers employed with selection, equal-gain, and maximal-ratio combining (MRC) techniques operating over correlated Weibull fading channels. For these diversity receivers, several useful performance criteria such as the moments of the output signal-to-noise ratio (SNR) (including average output SNR and amount of fading) and outage probability are analytically derived. Moreover, the average symbol error probability for several coherent and noncoherent modulation schemes is studied using the mgf approach. The proposed mathematical analysis is complemented by various evaluation results, showing the effects of the fading severity as well as the fading correlation on the diversity receivers performance.     

Dispersion Characteristics of Open Microstrip Lines

A method based on spectral-domain analysis is derived to compute effectively and automatically the effective relative permittivity of an open microstrip line. Simple but accurate closed-form expressions are used as the basis functions of the longitudinal and transverse current distributions. The numerical results are shown in tables and figures for various cases and compared WMI other available results. The results presented here are seen as having a high degree of accuracy and may be used as reference standards,