Maximum Likelihood Estimation of the Parameters of the Beta Distribution from Smallest Order Statistics

Numerical methods, useful with high-speed computers, are described for obtaining the maximum likelihood estimat.es of the two (shape) parameters of a beta distribution using the smallest M order statistics, 0 < u ₁ ≤ … ≤ ≤ … ≤ u _M , in a random sample of size K(≥M). The maximum likelihood estimates are functions only of the ratio, n = M/K, the Mth ordered observation, u _M , and the two statistics, G ₁ = [II ^M _i=1 u _i ]^1/M , and G ₁ = [II ^M _i=1 (1 – u _i )]^1/M . For the case of the complete sample (i.e., R = 1), however, the estimates are functions only of G ₁ and G ₂, and hence, for this case, explicit tables of the estimates are provided. When R < 1, the methods described depend crucially for their usefulness on the availability of a high-speed computer. Some esamples are given of the use of the procedures described for fitting beta distributions to sets of data. In one example, the fit is studied by using beta probability plots.     

A new relaxation method for obtaining the lowest eigenvalue and eigenvector of a matrix equation

The matrix eigenvalue problem Hu_i = λ_i u_i is considered. It is shown that when a new approximate vector v⁽ⁿ⁺¹⁾ to u₁ (the eigenvector of the lowest eigenvalue) is computed from the present one v⁽ⁿ⁾ by the relation v(n+1) = (1? αH + βH²) v⁽ⁿ⁾ or v⁽ⁿ⁺¹⁾ = (1? αH + βH² – γH³) v⁽ⁿ⁾, the convergence rate is at least double that of the gradient method which corresponds to set β = γ = 0. Moreover, by choosing parameters α, β, or γ properly, one can get about three to five times faster convergence rate than that of the latter method, for H having very small γ₂–γ₁ and very large λ_N (the largest eigenvalue), further modifications are suggested. The relation with the Richardson method is also discussed.     

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.     

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.     

Upper and lower bounds of the solution for an elliptic plate problem using a genetic algorithm

Summary This paper presents a new method of treating engineering problems, in which Mathematical Programming is combined with the Method of Weighted Residual (MWR), and is, therefore, referred to as Mathematical Programming MWR (MP-MWR). If a solutionZ(x) exists in the defining domainV of a problem, it is limited bound by any two functionsZ _u(x) andZ _l(x). These functions satisfy the definite condition that ifRZ _u(x)>0>RZ₁(x), thenZ _uZZ_l inV, whereR is the residual operator. By using the optimization method of genetic algorithms (GAs), it is also possible to obtain the values of minimumZ _u(x) and maximumZ _l(x) which satisfy the above inequality. One advantage of MP-MWR is that the demands on computer memory are less than those required when applying the finite element method. In this paper a boundary value problem is studied as an example. The efficiency, accuracy, and simplicity of the MP-MWR approach are fully illustrated, indicating that the proposed method may be readily extended to solve a wide range of physical engineering problems.     

Self Modeling Nonlinear Regression

The paper is concerned with parametric models for populations of curves; i.e. models of the form y_i (Z) = f(θ _i ; x) + error, i = I, 2, …, n. The shape invariant model f(θ _i ; x) = θ_0i + θ_1i g([x – θ_2i /θ_3i ) is introduced. If the function g(x) is known, then the θ _i may be estimated by nonlinear regression. If g(x) is unknown, then the authors propose an iterative technique for simultaneous determination of the best g(x) and θ _i . Generalizations of the shape invariant model to curve resolution are also discussed. Several applications of the method are also presented.     

Global existence and blow-up analysis for a nonlinear diffusion equation with inner absorption and boundary flux

This article deals with a nonlinear diffusion equation with inner absorption u _t ?=?(log^σ(1?+?u)u _x ) _x ???λ(1?+?u)log ^p (1?+?u), in ?₊?×?(0,?+∞), subject to a logarithmic boundary flux ? log^σ(1?+?u)u _x (0,?t)?=?(1?+?u)log ^q (1?+?u)(0,?t), t?∈?(0,?+∞). We establish the critical global existence curve and give the asymptotic behaviour close to the blow-up time.     

Convergence,accuracy and stability of finite element approximations of a class of non-linear hyperbolic equations

Numerical stability criteria and rates of convergence are derived for finite element approximations of the non- linear wave equation u_tt?F(u_x) = f(x, t), where F(u_x) possesses properties generally encountered in non-linear elasticity. Piecewise linear finite element approximations in x and central difference approximations in t are studied.     

A Bayes Sampling Allocation Scheme for Stratified Finite Populations With Hyperbinomial Prior Distributions

When sampling is carried out independently for the K strata of a finite stratified dichotomous population (defectives vs. standard items), and the number Z_i of defectives per stratum sample is observed, the corresponding probability function for X = (X_i , …, x_K ) is the product of hypergeometric functions which depend on the sample sizes n_i , the stratum sizes N_i , and the number of defectives M_i in the stratum (i = 1, …, K). It is assumed that prior information is available about the M_i 's which can be expressed, by suitable choice of the parameters a_i and b_i , as the product of independent hyperbinomial functions.

In each stratum the cost per observation is a known constant. Using squared error loss function, the prior Bayes risk is found for the linear function of interest,

and the optimum allocation of sample sizes is found, the one for which the prior Bayes risk is minimum when the total sampling budget is fixed.     

Distributions in Statistics: Continuous Univariate Distributions-1 and 2

The density of a symmetric statistic T = g(X ₁, X ₂, …, X_n ), for a random sample from a mixed population with density f(x) = pf ₁(x) + pf ₂), is a binomial mixture of the densities of the statist.ics T_k = g(X_k1 , X_k2 , X_kn ), k = 0, 1, … n. where X_ki 's are independent with density f ₁(x) if i ≤ k and density f ₂(x) if i > k. It is shown how to find the distributions of some important symmetric statistics like sample mean, sample variance, and order statistics by using T_k 's. The results are applied to normal and exponential mixtures.     

Secruential Rank Tests I. Monie Carlo Studies of the Two-Sample Procedure

Monte Carlo studies of a sequential, two-sample, rank-sum test developed earlier by Wilcoxon, Rhodes and Bradley are reported. Ranking is accomplished within groups of observations in the sequential procedures and the theory is based on a model suggested by Lehmann. Design parameters are α and β, probabilities of Type I or Type II errors, m and n, the numbers of X- and Y-observations in a group of observations, and k ₁, the power in the Lehmann model wherein the alternative hypothesis states that the cdf of the Y-population is G(u) = F ^{k ₁} (U), F(u) being the cdf of the X-population. Studies are made for designs with (α = β = .05, m = n = 2(1)5, k _k = 1.5, 2.33, 4 and 9. Data appropriate to the Lehmann model are generated and studies are also made when G(u – μ _y ) = F(u).

Average sample numbers and powers of tests are estimated. The effects of truncation are examined. It is concluded that the Monte Carlo results substantiate the use of the usual Wald theory of sequential analysis and that the use of the sequential methods for data from normal populations differing only in locations is satisfactory for most practical applications.     

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     

Second-Order Derivatives of the Gibbs Energy for Liquid Mixtures of Alcohol + Heptane at Pressures up to 100 MPa

Second-order thermodynamic derivative properties, such as isobaric thermal molar expansions, isothermal and adiabatic molar compressibilities, and isochoric molar heat capacities of (ethanol, decan-1-ol, 2-methyl-2-butanol) +  heptane mixtures at pressures up to 100 MPa and in the temperature range from 293.15 K to 318.15 K were derived from experimental speed-of-sound u(T, p), density ρ(T, p = 0.1 MPa), and isobaric heat-capacity C _p (T, p = 0.1 MPa) data using appropriate thermodynamic relations. Excess values for the given properties were calculated according to the criterion of thermodynamic ideality of a mixture (Douhéret et al., Chem. Phys. Chem. 2, 148 (2001)), i.e., assuming that the chemical potential of component i in the ideal liquid mixture is equal to the chemical potential of component i in the mixture of perfect gases. The deviations from ideality for the mixtures under test have been explained in terms of the self-association of alcohols in solution which produces a strong departure from random mixing, the change in the non-specific interactions during mixing, and the packing effects.     

High-pressure polymorph of LaNbO4

A new polymorph of LaNbO₄ with a BaMnF_4-type structure(a = 0.3938(2) nm,b = 1.442(1) nm,c = 0.5682(3) nm,Z = 4) and octahedral oxygen coordination of Nb was prepared at high pressures and temperatures (8 GPa, 1570 K). At normal pressure, the reverse transformation, from high-pressure LaNbO₄ to theM-fergusonite form, occurs at temperatures above 870 K and is close to a second-order phase transition. The spontaneous polarization in high-pressure LaNbO₄, evaluated from the second-order nonlinear optical response, is about 2 ώC/cm².     

Mechanical wave interaction at a diffuse interface

An investigation is made of the reflection coefficientR of an ultrasonic wave impinging on a boundary at which there is a gradual transition of material properties from one material to another. The study is confined to the one-dimensional case of compression waves incident at 90° on the boundary. Both analytical and numerical solutions forR are presented for various interfacial profiles, and are seen in general to be strongly dependent on wave frequency. Measurements are made on diffusion bonded samples of nickel to copper, and the results compared with theory. Significant scatter is found in the experimental results due primarily to the migration of oxygen into the diffusion zone. A second set of measurements on diffusion bonded samples of nickel to carbon steel show general agreement with theory.Nomenclature A _i ,B _i ,C _i ,i=1, 2 coefficients in the expression forX (x) - c _i concentration of speciesi - c _L compression wave velocity - D _i diffusion coefficient for speciesi - diffusion coefficient for a binary mixture - E(x) Young's modulus - stiffness for the case of one-dimensional strain, =(1–v)E(x)/(1+v)(1–2v) - one-dimensional stiffness within bluk materiali - J _i ,i=1, 2 flux of speciesi - k(x) wave number at spatial co-ordinatex, equal to 2/ - k _i ,i=1, 2 wave number within bulk materiali - K(x) pseudo-wavenumber, defined by Eq. (8). - k _B Boltzmann's constant - L thickness of interfacial zone - f wave frequency - m, N material profile parameters, used to specifyK(x) - Q activation energy for diffusion - R reflection coefficient - t time - T transmission coefficient - u(x, t) displacement of molecules due to wave action,u(x, t)=X(x) (t) - X(x) spatial portion of functionu(x, t) - x spatial variable - Y(x, t) (x, t)/ - Z _i ,i=1, 2 mechanical impedance of bulk materiali - strain - wavelength of sound - Poisson's ratio - material density - (x, t) stress - (t) temporal portion of functionu(x, t) - _i (x),i=1, 2 functional form ofX(x) within interfacial region - angular frequency of wave     

Self Modeling Curve Resolution

This paper presents a method for determining the shapes of two overlapping functions f ₁(x) and f ₂(x) from an observed set of additive mixtures, [α _i f ₁(x) + β _i f ₂(x); i = 1, …, n), of the two functions. This type of problem arises in the fields of spectrophotometry, chromatography, kinetic model building, and many others. The methods described by this paper are based on the use of principal component techniques, and produce two bands of functions, each of which contains one of the unknown, underlying functions. Under certain mild restrictions on the f_j (x), each band reduces to a single curve, and the f_i (x) are completely determined by the analysis.     

A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption

We give a new and explicit estimate for the asymptotic behavior of the solutions of the problem u_t − u_xx + u₊^P = 0, x > 0, t> 0, with conditions u(0, t) = 1, t > 0 and u(x, 0) = U₀(x) ≥ 0, x > 0, for a class of functions U₀ and parameter 0 < p < 1. We use an approximate solution given by the heat balance integral method with the innovation property which fixes appropriately the asymptotic limit of the corresponding approximate free boundary.     

Smooth interpolation of large sets of scattered data

Methods for solving the following data-fitting problems are discussed: given the data (x_i, y_i, f_i),i = 1,…,N construct a smooth bivariate function S with the property that S(x_i, y_i) = f_ii = 1,…,N. Because the desire to fit this type of data is encountered frequently in many areas of scientific applications, an investigation of the available methods for solving this problem was undertaken. Several aspects, such as computational efficiency, fitting characteristics and ease of implementation, were analysed and compared. Within the context of a general-purpose method for large sets of data, two of these methods emerged as being generally superior to the others. It is the purpose of this paper to describe these two methods and present examples illustrating their use and application.     

Dendritic solidification microstructure affecting mechanical and corrosion properties of a Zn4Al alloy

In the present article some important trends have been shown regarding the relationship between solidification variables, microstructure, mechanical and corrosion properties of Zn-4 wt%Al alloy castings. The aim of the present work is to investigate the influence of heat transfer solidification variables on the microstructure of Zn-4 wt%Al castings and to develop correlations with mechanical and corrosion properties. Experimental results include transient metal/mould heat transfer coefficient (h_i), secondary dendrite arm spacings (λ₂), corrosion potential (E_Corr), corrosion rate (i_Corr), ultimate tensile strength (σ_u) and yield strength (σ_y) as a function of solidification conditions imposed by the metal/mould system. It was found that a structural dendritic refinement provides both higher corrosion resistance and better mechanical properties for a hypoeutectic Zn4Al alloy.     

The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations

We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of (x,t,τ). When the group-preserving scheme is used to integrate these equations under a semi-discretization of u(x,t,τ) at the spatial-temporal grid points, we can achieve rather accurate solutions.