Estimation Of Shape Parameter Research Articles

Accuracy of Univariate, Bivariate, and a `Modified Double Monte Carlo' Technique for Finding Lower Confidence Limits of System Reliability

The purpose of this paper is to compare the accuracy of two asymptotic (univariate and bivariate) techniques and a modified double Monte Carlo technique to obtain system reliability s-confidence limits from component test data modeled by the 2-parameter Weibull distribution. The bivariate technique assumes that the maximum likelihood estimators of the component shape and scale parameter have their asymptotic bivariate normal distribution. The univariate technique assumes that the component reliability estimates have their asymptotic s-normal distribution. A variation of the univariate technique using the beta distribution as the distribution of the component reliability estimates is also examined. The analysis reveals that: 1. The bivariate technique is generally superior to the other asymptotic techniques for the four system-configurations examined. 2. The accuracy of each technique increases as the component test-data sample-size increases, as anticipated. 3. For low system-reliability all four techniques are close in their performance. 4. The s-confidence intervals are in general s-biased (upward), that is the actual s-confidence levels are lower than desired. 5. When sample size reaches 30, the bivariate and modified double Monte Carlo techniques are comparable for three of four systems studied. 6. As sample-size for component test-data increases, the accuracy of the double Monte Carlo method increases more rapidly than other methods and it can be used in a much wider variety of situations.

On maximum likelihood estimators of shape and scale parameters and their application in constructing confidence contours.

The seminal result of a “Student”, which initiated exact statistical inference, stated that for the normal distribution the statistic where [Xbar] is the sample mean and s the sample standard deviation, has a t-distribution which depends upon the sample size but not upon the population mean μ or the standard deviation σ, see “Student” the Probable Error of the Mean Biometrika, 1908. This result can be generalized to other situat,ions by using maximum likelihood estimates, namely the statistic has a distribution independent of the location and scale parameters μ and σ, respectively, where denote the correspondi corresponding MLE's based on complete or type II censored samples, see Antle and Bain, “ A Property of Maximum Likelihood Estimators of Location and Scale Parameters” SIAM Review, 1969. In this report the MLE's of shape and scale parameters,a and fi respectively, for the survival distribution [Fbar]of a r.v. X defined by with R a prescribed reliability function, are shown to have the property that h...

The Effect of Data Analysis Techniques on the Interpretation of Wide-Angle Longwave Radiation Measurements

Three different data analysis techniques—shape factor, parameter estimation, and deconvolution—have been applied to the same set of satellite radiation measurements to determine their effect on the estimated radiation field. The measurements are from a wide-angle, horizon-to-horizon, nadir-pointing sensor. The shape factor technique reduces each measurement to a radiant exitance at the top of the atmosphere by simple division by a constant. The parameter estimation technique processes all measurements together as a batch and defines the radiant exitance as a least-squares fit to the data. The deconvolution technique takes advantage of the fact that spherical harmonies are the eigenfunctions of the measurement operator. All three techniques are derived, and their assumptions, advantages and disadvantages are discussed. Their results are compared globally, zonally, regionally and on a spatial spectrum basis. All three techniques give comparable results for global parameters. However, results on a regional scale were quite different. The standard deviations of the regional differences in radiant exitance varied from 7.4 to 13.5 W m−2. Of the three techniques, the parameter estimation technique produced the best regional results and is the choice of the author.

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.

Control of hyphal orientation in colonies of Mucor hiemalis

This is a study of factors controlling the development of hyphae of Mucor hiemalis on a plane uniform agar surface, whereby a single germ tube elongating linearly gives rise to a colony enlarging symmetrically on an approximately uniform front through 360° from its centre. Most branches arise closely behind an existing elongating hyphal tip, and only a small number are formed on older parts of hyphae. The observed distances between successive points of of origin of hyphae fit a γ distribution with estimated shape parameter 1.5 and an estimated scale parameter of 0.011μm. The observed angles at which branches diverged from their parents' axes ranged within a normal distribution (mean 56° σ 17°). The first formed hyphae extend at about 20 μm per h for about 1 h after germination. Subsequently some stop growing permanently or temporarily; variation in the rates of elongation at the remaining tips, and at those branches formed after this first hour fits a half normal curve (mean 85 μm h−1). Growing hyphal tips repel each other, the maximum distances between tips at which this interaction has been seen range from about 10 μm to about 20 μm. No appreciable reorientation has been seen when living tips come close to dead ones, or if they approach agar sites from which another growing tip has been removed. The effects of moving growing tips close to older parts of the mycelium are less regular. Half of 68 observed cases stopped elongating, approximately one quarter (16) grew away from the older hyphae before making contact, approximately one quarter (18) crossed the older hyphae with no visible change in direction. Analysis of patterns produced by computer simulation indicates that the circular shape is the expected result of any system having the observed variation in branching and rates of elongation. No other controlling factor is needed to explain the patterns seen. In particular there is no evidence that orientating interactions between hyphal tips have any biologically important effect on colony shape. The amount to which the observed variation could be increased or reduced without affecting the growth patterns in any biologically important way is being studied.

Estimation of Weibull Shape Parameter for Samples of Size 2

Maximum likelihood (ML) estimation of the Weibull shape parameter in samples of size 2 is considered in the context of life testing. For single samples the shape parameter is estimable even when one of the two items in the sample is censored at life L. The ML shape-parameter estimate is the same as if the suspended test had failed at life L1.88. For k uncensored samples of size 2, with possibly different scale parameters, the ML estimate of a common shape-parameter is a function of the ratios of the second to first failure times within each of the k samples. Applying the ML method directly to the joint distribution of these k order-statistic ratios results in a shape-parameter estimate that is a different function of these same ratios. The s-normal distribution behaves similarly.

Analysis of single classification experiments based on censored samples from the two-parameter Weibull distribution

Methodology and supporting tables are given for the detailed analysis of a one way layout or single classification experiment in which the data comprise censored samples from two parameter Weibull populations. The complete analysis includes a test for the equality of shape parameters and, when the shape parameters can be assumed homogeneous, 1. (i) an “omnibus” test for the equality of scale parameters 2. (ii) a fixed range multiple comparison test for the equality of scale parameters 3. (iii) interval estimation of the common shape parameter and 4. (iv) interval estimation of an arbitrary percentile of one of the populations. Two omnibus tests are considered, viz., the likelihood ratio test and a test analogous to the F ratio test of the analysis of variance. This latter test utilizes the ratio of the two maximum likelihood estimates of the shape parameter obtained with and without the condition that the scale parameters are distinct. Numerical values of key percentage points of the relevant distributions required for carrying out the analysis, obtained by Monte Carlo sampling, cover the range from k = 2 samples of size n = 5 censored at order statistic number r = 3 to the case k = r = n = 10. The power of the shape parameter ratio test is shown to depend on a single scale invariant symmetric function of the scale parameters. The power of the likelihood ratio test is governed by both this same function and by one additional scale invariant symmetric function. The acceptance probability is tabled for choices of four values of each of these two functions for eight n, r, k combinations in the range from k = 2, n = 5, r = 3 to k = r = n = 10. The shape parameter ratio test is found to be more powerful than the likelihood ratio test against the alternative hypothesis that k − 1 of k scale parameters are homogeneous. The likelihood ratio test is superior against more diffuse alternatives. The distribution of the likelihood ratio test statistic is compared to its asymptotic chi square distribution. A factor less than unity is found for each n, r, k whereby the likelihood ratio test statistic may be corrected into good accord with its asymptotic distribution. Using critical values from the asymptotic distribution with the uncorrected likelihood ratio test statistic is shown to result in a substantially higher than nominal error rate when the sample size is small and/or the amount of censoring is high. A numerical example is given of the analysis of a one way layout and of three alternative methods for the selection of sample size in single classification experiments.

The Weibull Distribution: A New Method of Summarizing Survivorship Data

Survivorship data can be effectively summarized using the shape and scale parameters of the Weibull frequency distribution. The shape parameter controls the rate of change of the age—specific mortality rate and, therefore, the general form of the survivorship curve. Estimates of shape and scale parameters and their confidence intervals can be easily calculated and used to compare survivorship curves of different populations.

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.

Yield Probabilities for Cotton and Grain Sorghum Grown Under Dryland Conditions on the Texas High Plains1

AbstractYield probabilities were calculated for cotton (Gossypium hirsutum L.) and grain sorghum [Sorghum bicolor (L.) Moench] grown continuously and in rotations under dryland conditions on the Texas High Plains. The probabilities were calculated using the gamma function technique on data from three rotations (or cropping systems) of a 36‐year dryland study conducted at Lubbock, Texas during the period 1914‐1949. Differences in yields between rotations were also tested for statistical significance.The mean lint yield of the continuous‐cotton cropping system was 19 kg/ha greater than that of the cotton in the cotton and grain sorghum rotation, but this difference was not statistically significant. Also, the shape parameter estimates (“g” values) for these two rotations were the same. Therefore, calculated yield probabilities for them were essentially equal. Yields of dryland cotton should exceed 150, 250, 400, and 600 kg/ha approximately 30, 50, 25, and 10% of the time, respectively.The mean yield of grain sorghum grown in rotation with cotton was significantly higher than that from the continuous‐grain sorghum cropping system. Also, the “g” value obtained from the rotation was slightly higher than that obtained from the continuous‐grain sorghum cropping system. Therefore, the grain sorghum yield for a given probability level will be higher in rotations with cotton than from a continuous cropping system. A dryland, continuous‐grain sorghum cropping system should produce yields exceeding 750, 1,200, 2,000, and 2,750 kg/ha approximately 70, 50, 25, and 10% of the time, respectively. Slightly higher, yields can be expected when grain sorghum is grown m alternate years with cotton. Also, l0 to 20% higher dryland yields can be expected if hybrids are substituted for open‐pollinated varieties.

Evaluating Weibull Endurance Data by the Method of Maximum Likelihood

The method of maximum likelihood has been found to offer many advantages over existing methods for analyzing Weibull distributed rolling bearing fatigue data. The purpose of this paper is to provide the rolling bearing engineer with what he needs to use the method of maximum likelihood for, 1) Unbiased estimation of Weibull L10 life and shape parameter. 2) Setting confidence limits and testing hypotheses about Weibull L10 life and shape parameter. 3) Unbiased estimation of the ratio of two population L10's and shape parameters. 4) Testing the significance of differences between two estimates of L10 life or shape parameter. Presented as an American Society of Lubrication Engineers paper at the ASLE/ASME Lubrication Conference held in Houston, Texas, October 14–16, 1969

A non-linear regression method for calculating surface tension and contact angle from the shape of a sessile drop

A non-linear regression calculation procedure has been developed which permits calculation of surface tension and contact angle for any sessile drop shape with a contact angle greater than zero. Initial estimates of the drop shape and size parameters (β, b) are needed to start the calculation. Many points on the drop surface are measured, and a minimum sum of squares fit between the measured and calculated coordinates is obtained. No special attention must be given to any one point as is necessary when using the tables of Bashforth and Adams. Accuracy is improved by a factor of two compared to the classical calculation procedure. A computer program has been written to perform all necessary operations. Surface tension and contact angle data for the lead-bismuth eutectic on 316 stainless steel are reported.

Miscellanea: Separate maximum likelihood estimation of scale or shape parameters of the gamma distribution using order statistics

Miscellanea: Separate maximum likelihood estimation of scale or shape parameters of the gamma distribution using order statistics Get access M. B. WILK, M. B. WILK Bell Telephone Laboratories IncorporatedMurray Hill, New Jersey Search for other works by this author on: Oxford Academic Google Scholar R. GNANADESIKAN, R. GNANADESIKAN Bell Telephone Laboratories IncorporatedMurray Hill, New Jersey Search for other works by this author on: Oxford Academic Google Scholar MARILYN J. HUYETT MARILYN J. HUYETT Bell Telephone Laboratories IncorporatedMurray Hill, New Jersey Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 50, Issue 1-2, June 1963, Pages 217–221, https://doi.org/10.1093/biomet/50.1-2.217 Published: 01 June 1963