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.     

Bounds on Minimum Distance for Linear Codes over GF(5)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d ₅(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d ₅(n, k) k≤ 8, n≤ 100 is presented.     

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.     

The Distribution of a Variate Based on Independent Ranges from a Uniform Population

The statistic of interest in this article is W = R ₁/min_{1 ≤ j ≤ k} R _j where R _i , i = 1, …, k, denote k independent sample ranges from a standard uniform distribution each based on a common sample size n. Expressions are derived for the distribution function and moments of W, and a table of percentage points is included for selected valrles of k and n. The role of the statistic W in the construction of a subset selection procedure is indicated and operating characteristics of the proceduce are derived under a slippage configuration. A numerical example is provided.     

Book Reviews–10-Year Index 1959–1968

The dependence structure of a stationary time series {X_t } is usually discussed in terms of the observed variables X_t . Let F(x) be the common marginal c.d.f. of the X_t We propose a model involving only the transformed series {Z_t }, where Z_t = F(X_t ). The conditional distribution of Z_t given Z _t–1, Z _t–2, … is expressed in terms of Z_t , Z _t1, … and parameters β₁, β₂, …. The case where Z_t depends only on Z _t–1 is considered in detail. Estimators for the β's are found, and an example using a normal Markov process is given.     

Sequential Optimum Procedures for Unbiased Estimation of a Binomial Parameter

Let x ₁, x ₂, … x _n, … be a sequence of independent random variables with a common density function P(x = 1) = p, P(x = 0) = 1 – p, 0 < p < 1. This paper considers the non-randomized sequential procedures δ's for estimating p and the following three problems on choice of δ. (i) Choose δ to minimize E _v (Z _δ – P)² subject to E _p N _δ ≤ m where m ≥ 1 and Z _δ is an unbiased estimate of p; (ii) Choose δ to minimize E_p N _δ subject to E _p (Z _δ – p)² ≤ α where α is a real positive number; (iii) choose δ to minimize C E_p N _δ + E _p (Z _δ – p)² where C is the cost of an observation. In each case the minimization is to be done uniformly in p if possible; otherwise the supremum over p of the risk in question is to be minimized. A procedure is constructed for problem (i) when m is not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.     

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.     

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.     

Analysis and optimal control of discrete time‐varying systems Via Hahn polynomials

Abstract

Recently, the problem of analysis and optimal control of discrete time‐invariant systems has been extensively studied using finite series expansion of discrete orthogonal polynomials. This paper is to extend the applicable scope of discrete orthogonal polynomials to discrete time‐varying systems. The finite set of Hahn polynomials {q_ik)], i=0, 1, …, N} is chosen as the finite series expansion basis due to its general form and useful properties. First, for treating the product of two discrete‐time functions by Hahn series expansion, a new algorithm is derived to compute the Hahn series expansion coefficients of products qi(k)q_j (k), i, j=0, 1, …, N. These Hahn coefficients are then used to establish a product operational matrix for relating the Hahn coefficient vector of a product function to those of its component functions. This product operational matrix, along with the relations for connecting the Hahn coefficient vectors of a discrete function x(k) and its time‐shifted x(k+1), is finally applied to derive computational algorithms for solving the problems of analysis and optimal control of discrete time‐varying systems via finite Hahn series. Computed results are provided to illustrate the applicability of the proposed algorithms.     

Note on Separation of Variables

Necessary and sufficient conditions are given for the existence of a monotone transformation f(y) of a givenf unctiony (x ₁, …, x _k ) to separate the variables(transformation to additivity).     

An assessment of expressions for the apparent thermal conductivity of cellular materials

Diverse expressions for the thermal conductivity of cellular materials are reviewed. Most expressions address only the conductive contribution to heat transfer; some expressions also consider the radiative contribution. Convection is considered to be negligible for cell diameters less than 4 mm. The predicted results are compared with measured conductivities for materials ranging from fine-pore foams to coarse packaging materials. The dependencies of the predicted conductivities on the material parameters which are most open to intervention are presented graphically for the various models.Nomenclature a Absorption coefficient - C _itv(J mol^–1 K^–1) Specinc heat - E Emissivity - E _L Emissivity of hypothetical thin parallel layer - E _o Boundary surfaces emissivity - f Fraction of solid normal to heat flow - f _s Fraction of total solid in struts of cell - K(m^–1) Mean extinction coefficient - k(Wm^–1 K^–1) Effective thermal conductivity of foam - k _cd(Wm^–1 K^–1) Conductive contribution - k _cr(Wm^–1 K^–1) Convertive contribution - k _g(Wm^–1K^–1) Thermal conductivity of cell gas - k _r(Wm^–1 K^–1) Radiative contribution - k _s(Wm^–1 K^–1) Thermal conductivity of solid - L(m) Thickness of sample - L _g(m) Diameter of cell - L _s(m) Cell-wall thickness - n Number of cell layers - r Reflection coefficient - t Transmission coefficient - T(K) Absolute temperature - T _m(K) Mean temperature - T _N Fraction of energy passing through cell wall - T ₁(K) Temperature of hot plate - T ₂(K) Temperature of cold plate - V _g Volume fraction of gas - V _w Volume fraction of total solid in the windows - w Refractive index - (m) Effective molecular diameter - (Pa s) Gas viscosity - Structural angle with respect to rise direction - (Wm^–2 K^–4) Stefan constant     

The penetration depth of HgBa2CuO4+δ with 0.07 ≤ δ ≤ 0.35

The magnetic penetration depth λ(T) of three HgBa₂CuO_4+δ samples with 0.16 < δ ≤ 0.27 has been determined from the reversible magnetization. The obtained λ follows a BCS-like correlation of 1/λ² ∝ 1?(T/T_c)² over whole measured temperature range in an underdoped sample with T_c ～ 90 K, but deviates significantly from similar fits in an overdoped sample with the same T_c and an optimum doped sample, whose 1/λ² 's depends on T nearly linearly below T_c/2. This asymmetry between the underdoped and overdoped samples suggests that the T-dependence of 1/λ² is affected by doping in a complicated way.     

Some multiple comparisons using sample quasi ranges on censored data

We considerk(k)≥2 independent populations (treatments or systems) and an solutely continuous member of location-scale family of distributions, index by the location parameter μ _i (-∞ < μ _i < ∞) and scale parameter θ _i (θ _i > 0), is used to model the observations from the ith population,i=1,...k. The problem of simultaneous selection of two subsets, one containing population associated with the smallest ϕ-value and other containing population with the largest ϕ-value with probability at least a pre-specified value is considered when the data are censored. We also construct 100P ^*% simultaneous upper and two-sided confidence intervals for where θ_[1] ≤ ... ≤ θ_[k] denotes the ordered values of ϕs. The proposed procedures, based on sample quasi ranges, are useful when the experimenter has smaller samples or censored samples or there is suspicion of outliers in the samples. The results are applied to exponential populations model and, for thes casi: (i) the constants have been computed to apply the proposed multiple comparisons; (ii) two members of the proposed class have been compared with the existing procedure. A numerical example is also given.     

The three isentropic exponents of wet steam

Real gas isentropic changes may be described using the three well known ideal gas relations,pv ^k =const, p^(1−k)T^k=const andTv ^k-1=const, where exponent k has for each equation a different value k_{p, ν}, k_{p, T} and k_{T, ν} respectively. In this paper the three isentropic exponents theory for real gases is extended to the two phase region. As an application the numerical values of the three exponents are calculated for wet steam.     

General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

Based on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where $n×n$ real symmetric matrices $M$, $C$ and $K$ are constructed so that the quadratic pencil $Q(λ) = λ^{2}M+λC+K$ yields good approximations for the given $k$ eigenpairs. We discuss the case where $M$ is positive definite for $1≤ k≤n$, and a general solution to this problem for $n+1≤k≤2n$. The efficiency of our methods is illustrated by some numerical experiments.     

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.     

An assessment of expressions for the apparent thermal conductivity of cellular materials

Diverse expressions for the thermal conductivity of cellular materials are reviewed. Most expressions address only the conductive contribution to heat transfer; some expressions also consider the radiative contribution. Convection is considered to be negligible for cell diameters less than 4 mm. The predicted results are compared with measured conductivities for materials ranging from fine-pore foams to coarse packaging materials. The dependencies of the predicted conductivities on the material parameters which are most open to intervention are presented graphically for the various models.Nomenclature a Absorption coefficient - C _v (Jmol^–1 K^–1) Specific heat - E Emissivity - E _L Emissivity of hypothetical thin parallel layer - E ₀ Boundary surfaces emissivity - f Fraction of solid normal to heat flow - fics Fraction of total solid in struts of cell - K(m^–1) Mean extinction coefficient - k(W m^–1 K^–1) Effective thermal conductivity of foam - k _cd(W m^–1 K^–1) Conductive contribution - k _cr(W m^–1 K^–1) Convective contribution - k _g(W m^–1 K^–1) Thermal conductivity of cell gas - k _r(W m^–1 K^–1) Radiative contribution - k _s(W m^–1 K^–1) Thermal conductivity of solid - L(m) Thickness of sample - L _g(m) Diameter of cell - L _s(m) Cell-wall thickness - n Number of cell layers - r Reflection coefficient - t Transmission coefficient - T(K) Absolute temperature - T _m(K) Mean temperature - T _N Fraction of energy passing through cell wall - T ₁(K) Temperature of hot plate - T ₂(K) Temperature of cold plate - V _g Volume fraction of gas - V _w Volume fraction of total solid in the windows - w Refractive index - (m) Effective molecular diameter - (Pa s) Gas viscosity - Structural angle with respect to rise direction - (W m^–2 K^–4) Stefan constant     

Sample size required to estimate a parameter in the power function distribution

The purpose of this paper is to establish a two step sampling procedure for estimating the parameter θ of the power function distribution to within givend units of its true value with a given probability 1—α;(0<α<1). The density of the power function distribution is a function of two parameters, the second of whichk is assumed known. Given a preliminary sample sizem, tables and formulas are presented by which one may establish the sizen of the second sample such thatP(|y _n —θ|<d)>1—α is true, wherey _n is the largest observation in the second sample. The method used is deriving the results of this paper is similar to that given by Graybill and Connell (1964) and since the power function density reduces to the uniform density whenk=0, their results can be derived from the formulas given here. Also a table of comparisons between the expected second sample size in this paper and two other solutions is given.

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Sumario El propósito de este escrito es el de establecer un procedimiento de muestreo de dos pasos, para estimar el parámetro de la distribución de funciones de potencia cerca ded unidades de su valor verdadero con una probabilidad 1—α;(0<0<α<1). la densidad de la distribución de funciones de potencia es una función de dos parámetros, el segundo de los cualesk se supone es conocido. Dado un tama?o de muestra preliminarm, se presentan tabulaciones y fórmulas con las cuales se puede establecer el tama?on de la segunda muestra, de tal manera queP(|y _n —θ|<d)>1—α sea cierto, dondey _n es la observación más grande en la segunda muestra. El método usado en la derivación de los resultados de este escrito es similar al dado por Graybill y Connell (1964, Journal of the American Statistical Association) y ya que la densidad de funciones de potencia se reduce a la densidad uniforme cuandok=0, sus resultados pueden ser derivados a partir de las fórmulas dadas aquí. Además, se presenta una tabulación de comparaciones entre el segundo tama?o de muestra dado en este escrito y otras dos soluciones. Lafdp de la distribución de funciones de potencia es de la formaf(u)=(k+1)θ ^−(k+1) u ^k ,0<u<θ, θ>0 y cero en cualquiera otra parte.

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Research supported under ONR contract N00014-68-A-0515.     

Fourier transform of the Morse-V DD potential

The Fourier transform (k) of the Morse-V _DD potential is derived analytically. An expansion of (k) valid for smallk is given and comments are made about the occurrence of both even and odd powers ofk in the expansion.     

On the location of roots of non-reciprocal integer polynomials

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M(P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R ₀ < |z| < R ₁, where R ₀ = R ₀(d, k, M(P)) and R ₁ = R ₁(d, k, M(P)) are given explicitly. As an application, for non-reciprocal conjugate algebraic numbers α, α′ of degree d ≥ 2 and of Mahler’s measure M(α), we prove the inequality ${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc |z| ≤ α ⁻¹ + 0.1999α ⁻² contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then |α α′ − 1| > (19α ²)⁻¹.