Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Kernel estimation of the regression function with random sampling times

Summary LetZ = (X, Y)= {Z(t)} _t∈ℛ be a stationary continuous-time process taking values in ℛ. By means of the corresponding discrete-time process {X(t _i ),Y(t _i )} _i=1 ⁿ , sampled at random instants {t _i }, a nonparametric kernel estimator of the regression function,m(x)=E(Y|X=x), is studied. Under mixing dependence conditions, the Mean Integrated Square Error (MISE) is derived and asymptotic normality is proven. In addition, we compare our results with those obtained under a classical approach for evenly-spaced observations. Finally, similar issues arising in the recursive case are considered. This work was awarded the Ramiro Melendreras price, offered to young researchers by the Spanish Statistical Society.     

3-D stress intensity factors for arrays of inner radial lunular or crescentic cracks in thin and thick spherical pressure vessels

Some spherical pressure vessels are manufactured from a series of double curved petals welded along their meridional lines. Such vessels are susceptible to multiple radial cracking along the welds. For fatigue life assessment and fracture endurance of such vessels, one needs to evaluate the stress intensity factors (SIF) distribution along the fronts of these cracks. In a recent paper by the authors, mode I SIF distributions for a wide range of lunular and crescentic internal, surface, radial cracks were evaluated for a typical spherical pressure vessel of an outer to inner radii ratio of η = R_o/R_i = 1.1. The present analysis is aimed to determine the influence of the spherical vessel geometry in terms of its outer to inner radii ratio η = R_o/R_i on the prevailing SIFs. Mode I SIF distributions for a wide range of lunular and crescentic crack array configurations are evaluated. The 3-D analysis is performed by means of the FE method, employing singular elements along the crack front, for five geometries representing thin, moderately thick, and thick spherical pressure vessels with outer to inner radius ratios of η = R_o/R_i = 1.01, 1.05, 1.1, 1.7, and 2.0. SIFs are evaluated for arrays containing n = 1–20 cracks; for a wide range of crack depth to wall thickness ratio, a/t, from 0.025 to 0.95; and for various ellipticities of the crack, i.e., the ratio of crack depth to semi crack length, a/c, from 0.2 to 1.5. The obtained results clearly indicate that the SIFs are affected considerably by the geometry of the spherical pressure vessel-η, and by the following parameters: the number of cracks in the array-n, the depth of the cracks-a/t, and their ellipticity-a/c.     

Testing the hypothesis of a general linear model using nonparametric regression estimation

Summary Given the modelY _i =m(χ _i )+ɛi,whereE(ɛ _i) =0,X _i ≠Ci=1, ...,n, andC is ap-dimensional compact set, we have designed a new method for testing the hypothesis that the regression function follows a general linear model,m(·) ∈ {m _θ(·) =A ^t (·)θ}_{θ∈Θ⊂ℛq} , withA a function fromℜ ^p toℜ ^q. The statistic, denoted ΔASE, used fortesting the given hypothesis is defined to be the difference between the average squared errors (ASE) associated with the non-parametric estimator ofm and the minimum distance parametric estimator ofm. The asymptotic normality of both ΔASE and the minimum distance estimators is proved under general conditions. Alternative bootstrap versions of ΔASE are also considered.     

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.     

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.     

COMPARISON BETWEEN TWO ASSESSMENT METHODS FOR DEFECTS IN THE CREEP RANGE

In recent years considerable effort has been made to understand the behaviour of creep defects at elevated temperature. A large number of experimental studies have been devoted to creep crack growth behaviour. In Europe, both in U.K. (CEGB) and in France (EMP), attention is focussed not only on crack growth but also on creep crack initiation behaviour in the assessment of defects at high temperature. This paper describes and applies both methods, first to laboratory test results and then to a cracked component. The comparison with test data is made with three materials, a ferritic 1 Cr-1Mo-0.25 V steel and two austenitic stainless steels, while the application to a cracked pressure vessel deals with a ½ Cr-Mo-V steel. For these materials which are creep ductile, the relevant load-geometer parameter for stationary creep cracks is the C* parameter, which relies on the concept of reference stress and reference length. The expressions used for these parameters in both procedures are compared. Then the methods used for calculating the time for incubation prior to crack extension, t₁, the time for subsequent growth, t_g, and the time to failure, t_F=t_i+t_g are compared. This comparison is made not only for laboratory test data but also for a cracked pressure vessel. It is shown that, in spite of different approaches, especially in the assessment of t_g, both methods provide comparable t₁ - C* and t_F=C* diagrams. The reasons for this situation are briefly discussed.     

Asymptotic properties in partial linear models under dependence

Consider the regression modely _i =ζ _i ^T β+m(t _i )+ε _i fori=1,...,n. Here (ζ _i ^T ,t _i ) ^T ∈ℝ ^p ×[0,1] are design points, β is an unknownp×1 vector of parameters,m is an unknown smooth function from [0,1] to ℝ andε _i are the unobserved errors. We will assume that these errors are not independent. Under suitable assumptions, we obtain expansions for the bias and the variance of a Generalized Least Squares (GLS) type regression estimator, and for an estimator of the nonparametric functionm(·). Furthermore, we prove the asymptotic normality of the first estimator. The obtained results are a generalization of those contained in Speckman (1988), who studied a similar model with i.i.d. error variables. Research supported by the Xunta de Galicia (Spain) and the DGES (Spain) under research projects XUGA 10503A98 and PB98-0182-c02-01, respectively.     

Algorithms for Computing Sparse Shifts for Multivariate Polynomials

In this paper, we investigate the problem of finding t-sparse shifts for multivariate polynomials. Given a polynomial f∈ℱ[x ₁, x ₂, …, x _n ] of degree d, and a positive integer t, we consider the problem of representing f(x) as a ?-linear combination of the power products of u _i where u _i = x _i −b _i for some b _i ∈?, an extension of ℱ, for i = 1, …, n, i.e., f = ∑ _j F _j u ^αj , in which at most t of the F _j are non-zero. We provide sufficient conditions for uniqueness of sparse shifts for multivariate polynomials, prove tight bounds on the degree of the polynomial being interpolated in terms of the sparsity bound t and a bound on the size of the coefficients of the polynomial in the standard representation, and describe two new efficient algorithms for computing sparse shifts for a multivariate polynomial. Received: January 30, 1996; revised version: January 15, 2000     

Influence of impurities on the etching of NaCl crystals

The effect of CdCl₂, CuCl₂·2H₂O, MnCl₂·4H₂O and FeCl₃·6H₂O impurities and undersaturation on the rates of macroscopic dissolution,v _p, lateral etching away from a dislocation line,v _t, and normal etching along the dislocation line,v _n, and on the surface micromorphology of the {100} face of NaCl single crystals in water, methanol and 96% ethanol is investigated. The dependence of etch rates on impurity concentration,c _i, showed that the addition of a salt to the solvent always leads to a decrease inv _p, which attains a minimum value after a particular value ofc _i. The concentration dependence ofv _t andv _n is relatively complex, but often both decrease or increase simultaneously. A change in etch-pit morphology is caused by increasing the concentrations of all additives in ethanol and methanol. The dependence of etch rates on the undersaturation of methanol and methanol containing 10^–3 M CdCl₂ showed that dislocation etch pits are formed only for undersaturations greater than 0.02 and 0.06, respectively. These results as well as the roughening of etched surfaces at low impurity concentrations, the formation of terraced etch pits and the difference between etch pits at aged and fresh dislocations are discussed.     

Consistencia de un estimador no parametrico, recursivo, de la regresion bajo condiciones generales

Resumen Se define un estimador no paramétrico, recursivo, de la función de regresiónr(x)=E(Y/X=x), que se calcula a partir de un conjunto den observaciones {(X ₁,Y _i ):i=1,…n} del vector aleatorio (X, Y). Bajo la hipótesis de que los datos son idénticamente distribuidos pero no necesariamente independientes, lo que permite utilizar el estimador definido para estimar la función de autorregresión de una serie de tiempo, se obtienen resultados sobre la consistencia puntual débil (en probabilidad) y fuerte (completa) del estimador definido. Finalmente, se presentan ejemplos de utilización del estimador con datos simulados. Summary With a initial sample of sizen: {(X _i ,Y _i ):i=1,…n} of a radom vector (X,Y) a general recursive nonparametric estimator of a regression function,r(x)=E(Y/X=x) is obtained. Assume the data are identically distribuited but non necessarily independent. So, the defined estimator may be used to estimate the autoregression function of a time series. Weak and strong pointwise consistency are obtained for such estimator. Finally some examples of simulation are presented.

     

On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a ₁, a ₂,) where a ₁ is a residue class (mod 2) and a ₂ is a residue class (mod 2t), the addition being defined by (a ₁, a ₂) + (b ₁, b ₂) = (c ₁, c ₂) where a ₁ + b ₁ = C ₁ (mod 2) and a ₂ + b ₂ = c ₂ (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.     

MEASUREMENT OF STRETCH ZONE HEIGHT AND ITS RELATIONSHIP TO CRACK TIP OPENING DISPLACEMENT AND INITIATION J-VALUE IN AN AISI 316 STAINLESS STEEL

An accurate method using SEM for the measurement of stretch zone height (SZH) on fracture surfaces has been established. This method does not make any assumption regarding crack blunting angle θ, and for 316 stainless steel in the present investigation θ was in the range of 50 to 67°, contrary to the common assumption of 45°. The semi-empirical equations, SZH(T) = 2.5 SZW (stretch zone width) and SZW = 89(J/E), reported in the literature were found to give very conservative predictions of initiation toughness J_i for an AISI 316 stainless steel. Lowerbound values of the coefficient m in the equation, J = mσ_yδ, relating J to CTOD, δ, is found to be 1.5–1.7 for the 316 SS used in this investigation; the upperbound value for m is found to be 2.6. The plastic CTOD values (δ_max) corresponding to the maximum load-point on the instrumented impact test traces are not sensitive to the aging conditions and as they incorporate significant crack growth effects they cannot be used for predicting J_i, by a procedure similar to that discussed above. The ratio t₁/t_T, where t_i is the time to crack initiation and t_i is the time to total fracture, in precracked Charpy tests of unaged and aged 316 SS used in this investigation (a/W=0.55 to 0.8) corresponds to 0.15 to 0.17.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

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.     

Effects of Mill Speed on Kinetic Breakage Parameters of Four Different Particulate Pumices

In this study, the effect of mill speed was investigated on different particulate pumice samples at batch grinding conditions based on a kinetic model. For this purpose, S _i and B _i,j equations were determined from the size distributions at different grinding times, and the model parameters (S _i , a _T , α, γ, and φ _j ) were compared for three different mill speed (fraction of mill critical speed; 65%, 75%, and 85%).

The effect of fraction of mill critical speed (φ_c) on the grinding for model parameter a _T found to be different for four different pumices. Additionally, in this study it was found that optimum grinding occurs at φ_c = 85% for all pumice samples, in contrast to the 75% of critical speed of the ball mill.     

Scaling of the Resistance and Magnetization for High Temperature Superconductors under Pressure

The scaling relation of single parameter scaling hypothesis is applied to the study of the scaling behavior of high temperature superconductors under pressure. The data of resistance and magnetization under various pressures are scaled onto a universal curve according to this scaling relation. The scaling parameters are pressure dependent while temperature independent. It is found that the controlling parameter B _i equals the relative critical temperature t _cP , which indicates that the superconducting energy gap at the zero temperature 2Δ _s0 is the controlling parameter in this scaling.      

Stability of supercurrents in cylindrical films of tin

Using pulsed currents of 80-nsec duration the stability of supercurrents in cylindrical films with current return through the center has been investigated. It is found that the voltage along the film rises with timet approximately asV = a[1 – erf(t/2(t – c)) ^1/2] where the time constantt is an exponentially decreasing function ofi(1 + 420/d)/(1 –(T/T _c)⁴). Herei is the current through the sample,d the thickness of the film in angstroms, andT/T _c the reduced temperature. Comparison with older data by Mydosh and Meissner and by Hagedorn indicates that it is indeed the currenti, rather than the current densityJ, which determinest. It is suggested that the instability is of the nature of those investigated by Kramer and by Fink and Presson for currents in a superconducting surface sheath.Supported in part by the National Science Foundation and in part by the Cryogenics Center through a Grant from the DOD Themis Program.     

Adaptive Distribution-free Regression Methods and their Applications

We consider the model Y = βx + Z, where the random variable Z has a continuoustype distribution that can be badly skewed, contaminated, or censored. To test the hypothesis H ₀ : β = β₀, we use the distribution-free statistic K(β₀) = Σc(Q _i )a(R _i ), where c(·) and a(·) are increasing score functions and Q _i and R _i are the respective ranks of x _i and y _i – β₀ x _i . The score functions c(·) and a(·) can be adapted or chosen after observing the data without destroying the distribution-free nature of the test. A Monte Carlo study is presented which illustrates the excellent performance of an adaptive test when a wide range of distributions is considered for the residuals. Interval and point estimates of sβ can be found by employing the “inverse” of the testing procedure. These results are used to find estimates of the percentile lines. Two examples are given which involve lifetimes of electric motor insulation and grade point averages of beginning university students, respectively.