Estimation of the Parameters of the Gamma Distribution by Sample Quantiles

This paper deals with large sample estimation of the location parameter (α₁ and the scale parameter α₂ in the gamma distribution with known shape parameter. Best linear unbiased estimates based on k sample quantiles are used. For a given k, the optimum spacings of the sample quantiles can be replaced by simpler “nearly optimum” spacings at virtually no loss of asymptotic efficiency. The theory behind the nearly optimum spacings is briefly reviewed. The major part of the paper concerns estimation of α₂ when α₂ is known. Nearly optimum spacings together with the coefficients to be used in computing the estimates are presented in a number of tables for k = 1(1) 10, and various values of the shape parameter. The paper also contains brief discussions of estimation of α₁, when α₂ is known, and simultaneous estimation of α₁ and α₂.     

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.     

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.     

The Use of Incomplete Beta Functions for Prior Distributions in Binomial Sampling

Let θ be the proportion of defectives in a batch selected at random from a fixed population of batches. The paper deals with the determination of the prior distribution of θ, using prior knowledge obtained from batches examined in the past. Assuming a beta type prior distribution, the following three situations are investigated: a. Past records provide the numbers r ₁, r ₂, …, r _N of defectives found in N samples of size n from N previously inspected batches;

b. The expected fraction defective in a batch, E(θ), and the probability, P, of a batch exceeding twice the expected fraction defective are approximately known;

c. (c) The probabilities of θ exceeding a certain value θ₁ and of θ falling below another value θ₂ are approximately known.

Methods of estimating the parameters of the prior distribution are given, and charts are provided to facilitate their determination.

The effects of errors in the parameters on the posterior distribution are discussed in relation to sample size and number of defectives found in the sample. The effects of assuming a beta distribution when in fact some other distribution would have been appropriate are found to be negligible in most practical applications.     

Confidence Intervals from Censored Samples,II

In previous work [1] I have presented an interval estimation procedure applicable to singly censored samples from any univariate population depending only on a scale parameter and location parameter; the procedure applies to several censoring criteria. For any parametric function, the procedure gives an interval estimate based on two confidence statements each at the 1 – 2α level; these estimates were shown to be at least at the (1 – 2α)² level for any size sample and converge, for large samples, to population values. The present work concerns itself specifically with censored samples from N(μ, σ) and studies the possibility of reducing the conservatism of the procedure. It is shown that, asymptotically, the procedure is at least at the 1 – 2α level. For some special parametric functions, theoretical considerations and fairly extensive computations strongly suggest that the procedure is at least at the 1 1 – 2α level even for small samples. It is conjectured that this property carries over to other functions and other distributions involving only a location and a scale parameter.     

Comparisons of Designs and Estimation Procedures for Estimating Parameters in a Two-Stage Nested Process

For the usual two-stage nested random model, N observations are to be used to estimate some or all of these parameters: μ (the general mean), σ _a ² (the between classes variance component), σ _b ² (the within classes variance component) and ρ = σ _a ²/σ _b ² Using the traditional analysis of variance estimator, with fixed number of classes, (k), an allocation procedure to minimize the variance of the estimator is derived to estimate σ _a ² or ρ: p + 1 observations in each of r classes and p observations in each of k ? r classes, where N = pk + r, 0 ≤ r < k. The optimal number of classes (k) is conjectured to be the closest integer to

For large N, this implies that the optimal number of classes to estimate σ _a ² would be approximately k ₁ = N _ρ/(1 + ρ) with an average of 1 + ρ^?1 observations per class; the optimal number of classes to estimate ρ would be approximately Nρ/(1 + 2ρ) with an average of 2 + ρ^?1 observations per class. Hence if ρ > 1, p = 1 for σ _a ² and p = 2 for .

Investigations on the effect of using an incorrect value of ρ in k ₁ and k ₂ show that if 0.5 < ρ₁/ρ < 2.0, where ρ₁ is the value of ρ used, the loss in efficiency is generally less than 10%. Comparisons with the variances of hypothetical estimators in which k need not be an integer show that the integral requirement results in only a small loss of efficiency. Percent efficiencies of the estimates of the other parameters when a sampling plan is designed to minimize the variance of one estimator are presented for selected values of N and p.

Alternative estimators are considered for μ and σ _a ² and are found to be useful when the sampling is decidedly unbalanced and ρ is quite large. A computing procedure is developed for restricted maximum likelihood estimation of the variance components.     

Unbiased estimates of the Weibull parameters by the linear regression method

For sample sizes from 5 to 100, the bias of the scale parameter was investigated for probability estimators, P = (i − a)/(n + b), which yield unbiased estimates of the shape parameter. A class of unbiased estimators for both the shape and scale parameters was developed for each sample size. In addition, the percentage points of the distribution of unbiased estimate of the shape parameter were determined for all sample sizes. The distribution of the scale parameter was found to be normal by using the Anderson-Darling goodness-of-fit test. How the results can be used to establish confidence intervals on both the shape and scale parameters are demonstrated in the paper.     

The Distribution of the Maximum Sum of Rank

In this paper a c-sample slippage analogue of the Wilcoxon [11] test is considered. Given a sample of size n for each of c populations, the test rejects the hypothesis that the c populations are identical when max_1≤i≤c σ _k r _ik > λ, where r _i1, …, r _in are the ranks of the observations from the i-th population in the combined sample of size cn. The small and large sample distributions of the test statistic are derived. Tables of the exact distribution are given for c = 2(1)5, n = 2(1)5. Tables of critical values are given for c = 2(1)6, n = 2(1)8 for values of α = 0.001, 0.005, 0.01, 0.025, 0.05, 0.10, and 0.20.     

Handbook of Nonparametric Statistics

A sample is thought to be the result of mixing a random sample of size N ₁ from a p-variate normal population with mean and covariance matrix Σ with an independent random sample size N ₂ from a p-variate normal population with mean and covariance matrix Σ. It is wished to identify the population from which each observation has come and to estimate the parameters N ₁, N ₂, , and Σ. The paper discusses the application of the method of moments and the maximum likelihood method to the solution of this problem. The methods are illustrated by a numerical example.     

Genesis of Bimodal Distributions

Conditions for which the density function of a mixture of two normal distributions is bimodal are investigated. For fixed values of the variances σ₁ ² and σ₂ ² of the normal distributions if the difference between the means is sufficiently small, the distribution of the mixture will be unimodal, independent of the proportions p and 1 – p, 0 < p < 1. If the difference exceeds a critical value which depends on σ₁ ² and σ₂ ² the bimodality property then depends on p. Values of p sufficiently close to zero and one always exist for which the distribution is unimodal.     

Testing the Mean and Standard Deviation of a Normal Distribution Using Quantiles

Assuming a normally distributed population and a large sample size, three tests of hypotheses are presented using sample quantiles. First, we test whether μ = μ₁ and σ = σ₁ or μ = μ₂ and σ = σ₂, using one optimum quantile. Then test statistics and acceptance regions are given for testing the mean and standard deviation independently. The efficiency of each test is given relative to the best test using all the sample values.     

Experimental estimation of the probability distribution of fatigue crack growth lives

This paper deals with a method to estimate numerically the reliability of fatigue sensitive structures with respect to fatigue crack growth. A method is proposed to experimentally determine the probability distribution functions of material parameters of the Paris law, da/dN = C(ΔK/K₀)^m, using stress intensity factor controlled tests. The auto-correlation function of the resistance to fatigue crack growth, 1/C, is also estimated from the experimental data. The results of a high tensile strength steel show that the distribution of the parameter, m, is approximately normal and that of 1/C is a 3-parameter Weibull. The merit of the proposed method is that only a small number of tests are required to determine these functions. The probability distribution of the fatigue crack length after a given number of load cycles or the number of load cycles for a crack to reach a given length can be estimated by simulations of non-Gaussian random processes having these functions.     

A Note on Moments of Gamma Order Statistics

Previously quadrature approximations were developed to determine the moments of a distribution of the response of a multivariable function when each of the variables is a random variable from a normal distribution. The error was shown to be of the order of the sixth power of the standard deviations of the random variables, but a more useful bound is desired in applied work. Only limited success has been achieved in this direction. It is shown that the best approximating distribution is a Beta distribution of the first kind with β₂ equal to 3 and mean, variance, and β₁ obtained by quadrature, where β₁ and β₂ are standard measures of skewness and kurtosis, respectively. A parametric study of the function X = C_o (a _o, ± y ₁ ± y ₂ ± … ±y_n ) ^m + b _o where the y_i all have the same standard deviation, σ, is conducted both analytically and by quadrature. The mean and variance obtained by quadrature are essentially exact in the range of interest. It is shown that for a large range of σ the above distribution is both a good approximation and a much better approximation than either a normal approximation with the same mean and variance or a linear approximation. The example also shows that the β₂ obtained by quadrature is a poor indicator of the precision of the quadrature approximation.     

THE EFFECT OF PRESSURE OVERLOAD ON FATIGUE CRACK GROWTH IN PRESSURE VESSELS

Abstract— The pressure-overload method for proof-testing pressure-vessels was investigated for hydrofilters made of an Al-alloy. After a vessel had been overloaded using a hydrostatic fluid, the fatigue crack growth rate of the material was studied at three levels of hydrodynamic pressure. The period of fatigue crack growth N_p and the uniaxial equivalent stress σ_e were calculated on the basis of fractographic analyses for various combinations of the hydrostatic overload and the hydrodynamic pressure. Increasing the overload level increased N_p, and decreased σ_e. For a pressure overload of 3 times the normal pressure the initiation of a fatigue crack did not occur after 2 × 10⁶ cycles of loading at an hydrodynamic pressure of 20.6 MPa. Striation spacing could not be correlated against the value of the hydrodynamic pressure after an overload of 2.5 times the normal pressure. A unified relationship between the durability N_d and the ratio N_p/N_d (=N^d_p) at different hydrodynamic fluid pressures was demonstrated.     

The Theory of Branching Processes

Given a population of N items D of which are defective. Let the population be divided at random into k lots of equal size s (i.e. N = ks). This paper derives the asymptotic distribution (k → ∞) of the number of lots containing z or more defectives (x = 0, 1, 2, .…, s). Now suppose it is reasonable to define a lot as defective if it contains x or more defectives, and one is interested in the proportion of lots in a grand lot which are defective. Then on the basis of a random sample of items one can obtain confidence limits on various functions of the distribution of the number of defective lots. This allows high power against an excessive number of defective lots in an average sense (as in variables sampling) even for rather small samples. If, in fact, allocation of items to lots is not random but occurs with local positive contagion it appears that the random sample will tend to overstate the proportion of defective lots.     

Measurement of the internal local stress distribution of composite materials by means of laser imaging methods

A new method, based on photoelastic and holographic interferometric techniques, is proposed for observing the distribution of internal local stress in composite materials. Isochromatic and isopachic fringe patterns can be observed simultaneously to obtain the principal stresses σ₁ and σ₂ quantitatively. The proposed method is applied to observe the stress/strain characteristics of epoxy materials and glass fibre-reinforced epoxy composites. The thermal residual stress distribution in the thermosetting composite is also studied.     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.     

A Bayesian approach to the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices

Summary Two independent random samples of sizesN ₁ andN ₂ from multivariate normal populationsN _p (θ₁,∑₁) andN _p (θ₂,∑₂) are considered. Under the null hypothesisH ₀: θ₁=θ₂, a single θ is generated from aN _p(μ, Σ) prior distribution, while underH ₁: θ₁≠θ₂ two means are generated from the exchangeable priorN _p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.     

Some Remarks on the Bayesian Solution of the Single Sample Inspection Scheme

Batches of items, assumed independent of one another, are to be sentenced using an acceptance sampling plan based upon a random sample of size n and an acceptance number c_n Granting that the prior distribution of the probability of a defective item is a mixed binomial distribution, and using a simple cost criteria, a rapid technique is proposed for computing n and c_n . The method appears robust over wide changes in the parameters of the model.     

Sample Size Determination for Tolerance Limits

The problem of determining sample size to take for a tolerance limit L(X), where L(X) is a function of a random sample X ₁, …, X_n from a distribution with density f(x : θ), and

is investigated. A criteriorl of “goodness” of tolerance limits is developed and a method given, using this criterion, for solving the sample size problem. Examples are given using the uniform, exponential, and normal distributions as underlying models.