Unbiased Weibull capabilities indices using multiple linear regression

Although the recently proposed Weibull process capability indices (PCIs) actually measure the times that the standard deviation (σ_x) is within the tolerance specifications, because they not accurately estimate neither the log‐mean (μ_x) nor the σ_x values, then the actual PCIs are biased. This actually because μ_x and σ_x are both estimated without considering the effect that the sample size (n) has over their values. Hence, μ_x is subestimated and σ_x is overestimated. As a response to this issue, in this paper, μ_x and σ_x are estimated in function of n. In particular, the PCIs' efficiency is based on the following facts: (1) the derived n value is unique and it completely determines η, (2) the μ_x value completely determines the η value, and (3) the σ_x value completely determines the β value. Thus, now, since μ_x and σ_x are in function of n and they completely determine β and η, then the proposed PCIs are unbiased, and they completely represent the analyzed process also. Finally, a step by step numerical application is given.     

Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) ^bp–1 exp {–[(x – c)/a] ^p }/a ^bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results of applying this procedure to samples from Gamma, Weibull, and half-normal populations are tabulated, as are the asymptotic variances and covariances of the maximum-likelihood estimators.     

Summaries of New Books

Let X ₍₁₎, …, X _(k) be the first k ordered observations of a sample of size m from the distribution with p.d.f. f(x; β₁, θ) = (1/θ) exp [-(x – β₁)/θ] for x ≥ β₁ ≥ 0, θ > 0 and zero elsewhere (2 ≤ k ≤ m); let Y _(i), …, Y _(l) be the first l ordered observations of a sample of size n from the distribution with p.d.f. g(y; β₂, θ) = (1/θ) exp [—(y – β₂)/θ] for y ≥ β₂ ≥ 0, θ > 0 and zero elsewhere (2 ≤ l ≤ n). It is assumed that θ is unknown. A test based on (X ₍₁₎, …, X _(k), Y ₍₁₎, …, Y _(l)) is proposed for the null hypothesis β₁ = β₂ against the alternative β₁ ≠ β₂. The distribution of the test statistic under the null hypothesis is derived. The significance points for various values of k, I, m, n are tabulated for (α = .05 and .Ol.     

Conditional Maximum-Likelihood Estimators,from Singly Censored Samples,of the Scale Parameters of Type II Extreme-Value Distributions

Use of the functional relationships between the exponential and the Type II asymptotic distributions of largest and smallest values enables one to obtain conditional maximum-likelihood estimators, from singly censored samples, of the scale parameters (characteristic largest and characteristic smallest values) of the Type II asymptotic distributions of largest and smallest values, F ₁(y; v_n , K) = exp [–(y/v_n )^?K ] and F ₂(x; v ₁, K) = 1 – exp [– (x/v ₁)^?K ], by simple transformations of the corresponding estimator, , of the scale parameter of the exponential distribution, based on the first m order statistics of a sample of size n. Use is made of the fact that v _n , | K = and v₁, | K , where 2m /θ has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on the scale parameters, v_n , and v ₁, of the Type II asymptotic distributions of largest and smallest values. The probability densities of v _n , | K and v₁ K, each of which for given m is the same for any n > m, are obtained by simple transformations of that of . The expected values of v _n | K and v₁ | K are determined, and from them the unbiasing factors by which v₁, | K and v₁ | K must be multiplied to obtain unbiased estimators, v_n | K and v₁ | K. Expressions are found for the variances of these unbiased estimators and for the Cramér-Rao lower bounds. Values of the unbiasing factors, the variances of the unbiased estimators, and their efficiency relative to the Cramér-Rao lower bound, all of which are independent of n, are tabled for m = 1(1)20(4)100 and K = 0.5(0.5)4.0, 5.0. A numerical example and some remarks on applications are included.     

Comparison of torsion and compression constitutive analyses for elevated temperature deformation of Al–Li–Cu–Mn alloy

Abstract

Experimental alloy 2090 (Al – 2.1Li – 2.6Cu – 0.4Mn) was deformed in torsion and compression at temperatures T=300, 400, and 500°C and strain rates ε?=10^–2 to 10 s^–1. The torsion and compression results were analysed using equation A(sinh ασ)ⁿ=Z=ε? exp(Q_HW/RT), where Z is the Zener – Hollomon parameter. The variation of stress exponent n, Arrhenius slope s, and activation energy Q_HW were calculated with variation of the stress multiplier a from 0.01 to 0.08 MPa^–1. The change of α caused the stress exponent to decrease at a declining rate and Arrhenius slope s to increase linearly, thus causing the activation energy Q_HW to become stable above α=0.04 MPa^–1. The two experimental test techniques gave very similar flow stresses and constitutive constants. At α=0.052 MPa^–1 activation energy values showed reasonable consistency with other age hardenable aluminum alloys. The compression tests were also analysed using the power law expression ε? exp(Q′_HW/RT)=Z=A″σ^n′.     

Series Estimation of a Probability Density Function

A class of nonparametric estimators of f(x) based on a set of n observations has been proved by Parzen [l] to be consistent and asymptotically normal subject to certain conditions. Although quite useful for a wide variety of practical problems, these estimators have two serious disadvantages when n is large: 1. All n observations must be stored in rapid-access storage.

2. Evaluation of f(x) for a particular value of x requires a long computation involving each of the observations.

The Parzen estimator, which has n terms, can be replaced by a series approximation which has a number of terms determined by the accuracy required in the estimate rather than by the number of observations in the sample. The summation over all of the observations is performed only to establish the value of the coefficients in the series.

Although no member of the class of estimators has been proved “best” for estimating an unknown density from a finite sample, a power series expansion for a particular member of the class was singled out because of computational simplicity. A comparison is made between the proposed estimator and the Gram–Charlier Series of Type A. A multidimensional counterpart of the proposed estimator has also been derived.     

Conditional Weibull Control Charts Using Multiple Linear Regression

Because in Weibull analysis, the key variable to be monitored is the lower reliability index (R(t)), and because the R(t) index is completely determined by both the lower scale parameter (η) and the lower shape parameter (β), then based on the direct relationships between η and β with the log‐mean (μ_x) and the log‐standard deviation (σ_x) of the analyzed lifetime data, a pair of control charts to monitor a Weibull process is proposed. Moreover, because the fact that in Weibull analysis, right censored data is common, and because it gives uncertainty to the estimated Weibull parameters, then in the proposed charts, μ_x and σ_x are estimated of the conditional expected times of the related Weibull family. After that both, μ_x and σ_x are used to monitor the Weibull process. In particular, μ_x was set as the lower control limit to monitor η, and σ_x was set as the upper control limit to monitor β. Numerical applications are used to show how the charts work. Copyright © 2016 John Wiley & Sons, Ltd.     

Some Comments on Spectral Analysis of Time Series

The Average Run Length of a Cusum chart for controlling a normal mean is calculated by solving the systems of linear equations which approximate the integral equations for the required quantities. The accuracy of approximation by this method is numerically evaluated and the results are compared with those obtained by other approximate methods. The construction and use of a new nomogram based on the contours of Average Run Lengths L_a . and L_r drawn in the h√n/σ—|μ – k|√n/σ plane is discussed. Numerical examples are given to illustrate the flexibility and convenience provided by this nomogram in the design of Cusum charts.     

The Moments of Log-Weibull Order Statistics

Let X _1n ≤ X _2n ≤,…, ≤ X_nn be the order statistics of a random sample of size n. For any integrable function g(x) define E(i, n) = E(g(X _in )) and M(n) = E(1, n) = E(g(X _1n )). A number of formulae expressing E(i, n) in terms of M(j), j ≤ n, are developed. For example,

These results are applied to obtain the means and variances of the order statistics of a log-Weibull distribution (F(z) = 1 – exp (? exp x)). Tables of these means and variances are given for 1 ≤ i ≤ n, n = 1 (1) 50 (5) 100. The computations were made using a set of 100 decimal place logarithms of integers. Examples of the use of these tables in obtaining weighted least squares estimates from censored samples from a Weibull distribution are also given.     

Investigation of Rules for Dealing With Outliers in Small Samples from the Normal Distribution: I: Estimation of the Mean

The performance of three rules for dealing with outliers in small samples of size n from the normal distribution N(μ, σ²) are investigated when the primary objective of sampling is to obtain an accurate estimate of μ. It is assumed that at most one observation in the sample may be biased, arising from either N(μ + aσ, σ²) or N(r, (1 + b)σ²). Performance of each rule is measured in terms of “Protection”, the fractional decrease in the Mean Square Error (MSE) obtained by using the rule when a biased observation actually is present in the sample. Although numerical results have been obtained for n ≤ 10 when σ² is known, computational difficulties have prevented evaluation of protections when σ² is unknown except when n = 3.     

Modified trapezoidal quadratures and accurate potential evaluation for numerical solutions of S.I.E's on the unit circle

Plane elasticity problems that can be reduced to singular integral equations over the unit circle are considered, the method of using a dislocation density function ω(σ) is adopted. First, the trapezoidal approximation is modified for the evaluation of the contour integral ∮ _y [ω(σ)/(σ – z)]dσ at the complex field point z; extra correction terms for known singularities of ω(z) can easily be determined from the error expression. Based on the quadratures obtained, expressions for the accurate evaluation of the potentials are derived using their analyticity properties. The numerical techniques proposed are used for the solution of the problem of an infinite plate weakened by a circular hole and subjected to a dislocation.     

Topology-based denoising of chaos

In this article, we propose a denoising algorithm to denoise a time series y _i = x _i + e _i , where {x _i } is a time series obtained from a time-T map of a uniformly hyperbolic or Anosov flow, and {e _i } a uniformly bounded sequence of independent and identically distributed (i.i.d.) random variables. Making use of observations up to time n, we create an estimate of x _i for i < n. We show under typical limiting behaviours of the orbit and the recurrence properties of x _i , the estimation error converges to zero as n tends to infinity with probability 1.     

The Construction of Good Linear Unbiased Estimates from the Best Linear Estimates for a Smaller Sample Size

A general procedure is found for constructing good unbiased linear estimators of the location and scale parameters of a distribution for use with an uncensored sample of size n.

It is presupposed that the coefficients of the best linear estimates are available for an uncensored sample of size m < n for the distribution under investigation. The coefficients of the proposed estimators are obtained as linear combinations of these with the aid of tabled values of the hypergeometric probability function.     

Generating a Variable from the Tail of the Normal Distribution

The performance of three rules for dealing with outliers in small samples of size n from the normal distribution N(μ, σ²) is investigated when the primary objective of sampling is to obtain an accurate estimate of σ². It is assumed that at most one observation in the sample may be biased, arising from either N(μ + aσ, σ²) or N(μ, (1 + b) σ²). Performance of each rule is measured in terms of “Protection”, the fractional decrease in MSE obtained by using the rule when a biased observation actually is present in the sample. Numerical results have been obtained for n 5 ≤ 11 when μ is known and n = 3 when μ is unknown.     

X-ray photoelectron spectroscopy of lead fluorosilicate glasses

X-ray photoelectron spectra were measured of F 1s, O 1s, Pb 4f, and Si 2p core levels for lead fluorosilicate glasses of analysed compositionsxPbF₂·(69−x)PbO·(27–29)SiO₂ (x<18 mol%). The observed binding energies were discussed in terms of atomic charges and repulsion with taking the binding energy data of relevant compounds in the literature as the reference. The 688 eV component of an F 1s doublet was attributed to fluorines of [O_4−α SiF_α] units and the 684 eV component to free fluoride ions under ionic interaction with lead ions. The fraction of the fluoride ions increased withx up to 51 %. The Si-F bonds were confirmed in the glasses with 3<x<18 mol%.     

Transferability of the two‐parameter fracture criterion for 2219 aluminium alloy cracked configurations

Two‐dimensional elastic–plastic finite‐element fracture simulations with the critical crack‐tip‐opening‐angle fracture criterion were used to evaluate the two‐parameter fracture criterion (TPFC). Three different crack configurations under tension and bending loads made of thin‐sheet 2219‐T87 aluminium alloy were analysed. A very wide range of widths (w = 76 to 2440 mm) and initial crack‐length‐to‐width ratios (c_i/w = 0.05 to 0.95) were considered. A relation from the original TPFC was shown to fit the simulated fracture behaviour fairly well for the three different specimen types for net‐section stresses less than the yield stress (σ_y) of the material. Comparisons were also made on measured and simulated fracture tests on middle‐crack‐tension specimens. A relation between the elastic stress‐intensity factor, K_Ie, and net‐section stress, S_n, at failure was found to be linear for S_n < σ_y. The results demonstrated the transferability of the TPFC for different crack configurations for S_n < σ_y, but further study is needed for S_n > σ_y.     

Constructing Tables of Minimum Aberration Pn–m Designs

Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 ^n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 ^n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 ^n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.     

Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples

Iterative procedures are given for joint maximum-likelihood estimation, from complete and censored samples, of the three parameters of Gamma and of Weibull populations. For each of these populations, the likelihood function is written down, and the three maximum-likelihood equations are obtained. In each case, simultaneous solution of these three equations would yield joint maximum-likelihood estimators for the three parameters. The iterative procedures proposed to solve the equations are applicable to the most general case, in which all three parameters are unknown, and also to special cases in which any one or any two of the parameters are known. Numerical examples are worked out in which the parameters are estimated from the first m failure times in simulated life tests of n items (m ≤ n), using data drawn from Gamma and Weibull populations, each with two different values of the shape parameter.     

Further Remarks on Maximum Likelihood Estimators for the Gamma Distribution

The maximum likelihood estimators , â, for the parameters ρ, a of the gamma density f(x) = k(x/a)^ρ–1 exp(?x/a) are solutions of the equations In – ψ() = ln(A/G), â = A, where Ψ is the logarithmic derivative of the gamma function, A and G being the sample (of size n) arithmetic and geometric means, respectively. The moments of and â are developed in descending powers of ρ. A comparison of the assessments of the moments by the present approach and a method involving series in descending powers of n is made.

Approximate expressions are also given for the first four moments of ? = l/â, which is the maximum likelihood estimator of c = l/a.     

A Note on Estimation from a Type I Extreme-Value Distribution

If the random variable T has the ta-o-parameter Weibull distribution with cumulative distribution function F(t; θ, K) = 1 – exp[–(t/θ) ^k ], where θ is the scale parameter and K is the shape parameter, then the random vatiable X = In T has the Type I extreme-value distribution of smallest values with cumulative distribution function F(x; u, b) = 1 – exp {–exp [(x – u)/b}, where u = In θ is the location parameter (mode) and b = 1/K is the scale parameter. It is therefore possible to obtain the maximum-likelihood estimator û _mn | b of u, based on the first m order statistics of a sample of size n, when b is known, by a simple transformation of the corresponding estimator of θ when K is known. Use is made of the fact that û _mn | b = In _mn | K, where 2m( _mn | K) ^k /θ ^k has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on u. The probability density function of û _mn | b which for given m is the same for any n ≥ m, is obtained by a simple transformation of that of _mn | K. Integration yields expressions, involving digamma and trigamma functions, for the bias E = E[(û _mn |b) – u] and the variance V = V(û _mn | b). By subtracting the bias E](û _mn |b) – u] from û _mn |b, one obtains an unbiased estimator û|b which has the same variance as the maximum-likelihood estimator. Values of E/b(6DP) and of V/b ²(6DP) are tabulated for m = 1(1)100. The use of the table is discussed and illustrated by a numerical example.