Estimation of Weibull parameters with linear regression method

Abstract

Based on minimum mean square error, a modified probability estimator is proposed by a Monte Carlo simulation for estimating the Weibull parameters with the linear regression method. It is shown that compared with the commonly used estimators, the modified probability estimator gives a more accurate estimation of the Weibull modulus and the same estimation precision of the scale parameter. Furthermore, it is more conservative than the commonly used estimator recommended by previous authors and hence results in a higher safety in reliability predictions.     

On estimating Weibull modulus by the linear regression method

Statistical models were developed to estimate the bias of the shape parameter of a 2-parameter Weibull distribution where the shape parameter was estimated using a linear regression. These models were formulated for 27 sample sizes from 5 to 100 and for 35 probability estimators, , by varying “a” and “b”. In each simulation, 20,000 trials were used. From these models, a class of unbiased estimators were developed for each sample size. The standard deviation and coefficient of variation of these estimators were compared to the bias of the estimators. The standard deviation increased while the coefficient of variation decreased with increasing bias of the shape parameter. Also, the Anderson–Darling statistics was used to determine that the normal, log-normal, 3-parameter Weibull, and 3-parameter log-Weibull distributions did not provide good fit to the estimator of the shape parameter.     

On estimating percentiles of the Weibull distribution by the linear regression method

Probability estimators developed previously by the authors have been used to obtain unbiased estimates of the Weibull parameters by the linear regression method. Using these unbiased estimators, percentiles of the Weibull distribution have been estimated. Since these percentiles are determined from the estimated parameters, they also have distributions and subsequently are determined for five sample sizes. Analysis has shown that the distributions of these estimated percentiles are neither normal, lognormal, three-parameter Weibull nor three-parameter log-Weibull. A new methodology to estimate the percentile with a specified level of confidence has been introduced. The step-by-step use of the methodology is demonstrated by examples in this paper.     

Estimating the parameters of the weighted l p norm by linear regression

Empirical distance functions are used for such purposes as predicting travel distances in a transportation network, estimating the lengths of conductors used in piping and wiring systems, verifying distance data, and formulating location models. It has been previously found that when the reference axes are correctly rotated, die weightedl _p to norm is a robust distance predictor. However, a specialized computer program is required to fit this model to a geographical region. In this paper we show how the parameters of the model may be accurately calculated with a standard linear regression program, thus eliminating die need to develop or buy specialized computer software. The method has been verified empirically for several geographical regions.     

Optimization of parameters of the linear TOF-SIMS spectrometer by DOE method

Time of flight SIMS spectrometers are widely used in many industrial and scientific applications. Mass spectra of secondary ions ejected from a bombarded surface supply valuable data on its composition and give insights into physical and chemical processes at the surface. The data quality depends on several parameters, which control the spectrometer functioning. This contribution presents our work on an application of a so-called Design of Experiments (DOE) procedure to optimize working parameters of a home-assembled linear TOF-SIMS spectrometer. The efficiency of optimization procedures was tested for mass spectra of positive ions sputtered from Al surface under keV Ar ion bombardment. In particular, we show that use of the method leads to a large increase both of the mass resolution and of the signal/noise ratio (S/N), i.e. improves two quantities, which play a crucial role in mass spectrometry measurements. In addition, it is shown that DOE' method allows to reduce significantly the number of test measurements, and therefore to shorten the adjusting procedure to a minimum.     

Evaluation of the Weibull distribution parameters

Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 12, pp. 92–95, December, 1988.     

Optimal method of linear regression in laser remote sensing

Remote lidar sensing in the photon-counting mode is now the commonly accepted method for studying atmospheric processes in the lower and free atmosphere. However, when processing signals obtained from lidar measurements, investigators necessarily face the problem of achieving accuracy in reconstructing the atmospheric parameters despite the presence of inhomogeneous noise in the measured signals. We propose an optimal method of linear regression (OMLR) of signals. The accuracy of the the method for the reconstructed signal is estimated. An example of application of the OMLR to the reconstruction of the temperature profile from the data obtained with a Raman lidar at the Siberian Lidar Station of the Institute of Atmospheric Optics (Tomsk, Russia) is given. The proposed method is distinguished by simplicity of interpretation of the criteria used, based on careful adherence to statistical principles. This method is shown to be an efficient auxiliary tool for the processing of measured data.     

Evaluation of the statistical parameters of a Weibull distribution

A simple iterative procedure for determination of the statistical parameters of a Weibull distribution is proposed. All experimental results on specimens of different size are considered together as a statistically representative population. The procedure can be used for a population in which each specimen has a unique size. The statistical reliability of the iterative procedure is illustrated by comparison with a minimization analysis and confirmation with existing methods. Experimental confirmation of the analysis is developed using six types of glass and carbon fibres at four gauge lengths each. It is shown that Weibull parameters, obtained separately for populations of fixed length, vary with the fibre length. This revised version was published online in November 2006 with corrections to the Cover Date.     

On comparing the shape parameters of two Weibull distributions

Probability estimators developed previously by the authors have been used to compare unbiased estimates of the shape parameters (Weibull modulus) from two distributions by the linear regression method for thirteen sample sizes ranging from 10 to 100. Percentiles of the ratio of two estimated Weibull moduli for various confidence levels have been developed. The use of these percentiles for hypothesis testing, i.e., comparison of the two Weibull moduli is demonstrated by using datasets from the literature.     

Estimation of Weibull parameters by omission of some data in a sample

Methods of estimating Weibull parameters by omission of some data in a sample are considered for various functions for survival probability in linear Weibull plots. Computer simulation is used for the consideration. A use of the survival probability function, (P(i) = 1 – (i – 0.5)N, in whichP(i) is survival probability,i is thei-th order of failure strength andN is the total number of data in the sample, and the omission of 2 to 4% of the data, which are of the smallest value, the largest one and their neighbours, are recommended to get the best value of the Weibull parameters.     

The sensitivity of estimates of regression to the mean

Estimations of the effectiveness of remedial treatments in road safety analysis are frequently bedevilled by the problem of regression to the mean (RTM). The number of accidents x observed at a site in the “before” period is a “noisy” quantity: x is Poisson distributed about an (unknown) true mean m for that site, so that x = m + e. Sites selected for treatment tend to have a positive random error component e, which will on average be zero in the “after” period, even if no treatment is applied.Methods for estimating RTM usually require some assumption about the underlying (prior) between-site distribution of the true means f₀(m): for example, in the empirical Bayes method, a gamma distribution is assumed. The paper considers the impact of different assumptions for this distribution and, indeed, whether any distributional form needs to be assumed. Using Markov Chain Monte Carlo methods, a variety of distributional forms are assumed for f₀(m) and applied to each of a number of real data sets, including that from a major study on the effectiveness of speed cameras. It is shown that, in some cases, the size of the estimated RTM effect can be quite sensitive to the choice of distribution.     

一元线性回归方法的应用

按常规方法检测大型钢筋水泥预制件内置钢筋的张紧拉力,其数据很粗略,不能完全满足设计和工艺要求。为科学反映真实内在规律性,达到数据准确可靠且操作快捷方便,提出对待一组常规的测量数据应用一元线性回归数据处理方法,通过最小二乘法原理来处理两个变量之间的关系,最后得出数学模型,借助计算器即可圆满解决。该方法在工程领域有着一定的应用前景。     

An investigation of the loading rate dependence of the Weibull stress parameters

This paper examines the dependence of the Weibull stress parameters on loading rate for a 22NiMoCr37 pressure vessel steel. Extensive fracture tests, including both quasi-static and dynamic tests, are conducted using deep- and shallow-cracked SE(B) specimens. The fracture specimens are carefully prepared to ensure the crack fronts are placed at the location where the material is homogeneous. Three dynamic loading rates (in terms of the stress intensity factor rate, in the low-to-moderate range are considered. The load-line velocities for the dynamic tests are chosen so that the resulted values for the deep- and shallow-cracked specimens are the same. Independent calibrations performed at each loading rate (quasi-static and the three dynamic loading rates) using deep- and shallow-cracked fracture toughness data show that the Weibull modulus, m, is invariant of loading rate. The calibrated m-value is 7.1 for this material. Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The demonstrated loading rate invariant of m, when combined with the master curve for dynamic loading, can provide a practical approach which simplifies the process to estimate σ_u as a function of loading rate.     

Calibration of Weibull stress parameters using fracture toughness data

The Weibull stress model for cleavage fracture of ferritic steels requires calibration of two micromechanics parameters . Notched tensile bars, often used for such calibrations at lower-shelf temperatures, do not fracture in the transition region without extensive plasticity and prior ductile tearing. However, deep-notch bend and compact tension specimens tested in the transition region can provide toughness values under essentially small-scale yielding (SSY) conditions to support Weibull stress calibrations. We show analytically, and demonstrate numerically, that a nonuniqueness arises in the calibrated values, i.e., many pairs of provide equally good correlation of critical Weibull stress values with the distribution of measured (SSY) fracture toughness values. This work proposes a new calibration scheme to find which uses toughness values measured under both low and high constraint conditions at the crack front. The new procedure reveals a strong sensitivity to m and provides the necessary micromechanical values to conduct defect assessments of flawed structural components operating at or near the calibration temperature in the transition region. Results of a parameter study illustrate the expected values of m for a typical range of material flow properties and toughness levels. A specific calibration is carried out for a mild structural steel (ASTM A36). This revised version was published online in August 2006 with corrections to the Cover Date.     

Temperature dependence of Weibull stress parameters: Studies using the Euro-material

This work demonstrates the temperature invariance of the Weibull stress modulus, m, for a 22Ni-MoCr37 pressure vessel steel through calibrations at two extreme temperatures of the ductile-to-brittle transition. This temperature invariance reflects the characterization of microcrack size distribution in the material described by the modulus. The calibrations performed here also demonstrate the clear dependence of the Weibull scale parameter, σ_u, on temperature. The increase of σ_u with temperature reflects the increase in microscale toughness of ferritic steels. The calibration procedure employs a three-parameter Weibull stress model which includes the effects of a minimum (threshold) toughness, K_min. The calibrations suggest that K_min increases gradually with temperature. Finally, an engineering procedure is presented to enable practical applications of the Weibull stress model for defect assessments. This procedure combines the demonstrated temperature invariance of m, a recently developed method for predicting the variation of σ_u with temperature using the Master Curve, and calibration of the Weibull stress parameters at one temperature. The (calibrated) temperature invariant m and the estimated σ_u as a function of temperature are used to predict the cumulative probability of fracture for several large datasets without direct calibration.