Misclassified Data from a Binomial Population

The method of maximum likelihood is employed to estimate the binomial parameter p whenever an erroneous observation or report results in recording c defective samples when there are actually c + 1. The proportion of such erroneous observations is also estimated, and asymptotic variances and covariances of the estimates are obtained. A numerical example is included.     

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.     

Degrees of Freedom for Chi-Square

In this paper we have obtained the maximum likelihood estimates of the fraction defective (p) when data from the curtailed sampling plans are subject to misclassification. We have assumed that the inspector classifies a defective as a nondefective when the kth defective appears at the kth inspection. The maximum likelihood estimate of the probability of misclassification is also obtained. The asymptotic variances and covariances of these estimates are derived. A numerical example is provided.     

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.     

Application du Théorème D'échantillonnage à la Restitution D'objets Partiellement Connus

On cherche à restituer le mieux possible un signal b(x)dont un appareil nous fournit un enregistrement e(x)=b(x)*?(x); la fonction ?(x) est supposée couper toutes les fréquences de b(x) supérieures à une fréquence de coupure μ₁ = a. Il s'agit de rechercher une représentation b _c(x) du signal, identique à celle qu'aurait fourni un appareil de réponse impulsionnelle A _c(x) plus fine que ?(x) (i.e. dont la fréquence de coupure c est supérieure à a). On montre que l'on doit utiliser toutes les informations, même vagues, que l'on possède a priori sur b(x) afin d'imposer ‘des contraintes’ à b _c(x). On recherche alors, parmi les développements qui satisfont à ces contraintes, le développement b _c ⁿ qui rend minimum l'écart quadratique On montre que, grâce aux contraintes introduites, on peut restituer correctement une fraction importante des fréquences coupées : le développement b _c ⁿ ne diffère en effet pratiquement pas de l'approximation en moyenne quadratique sur -∞, +∞ de la représentation cherchée du signal, dans le système des bases     

Thermal expansion of NZP-family alkali-metal (Na, K) zirconium phosphates

The thermal expansion of the A _x Zr_2.25-0.25x(PO₄)₃ phosphates with A = Na(x = 0.5,1.0,2.0,3.0,4.0, 5.0) and K(x = 1.0, 3.0, 5.0), crystallizing in structures of the NaZr₂(PO₄)₃ type (sp. gr.R3c or C2/c), was studied by high-temperature x-ray powder diffraction in the range 20–700‡C. The lattice parametersa and c and thea- andc-axis thermal expansion coefficients (α_a and α_c) were determined. The thermal expansion of the phosphates was found to be highly anisotropic (α_a < 0, α_c > 0). The strongest anisotropy was found in NaZr₂(PO₄)₃ (α_a = -4.89 x 10^-6 K^-1, α_c = 22.02 x 10^-6 K^-1), KZr₂(PO₄)₃ (α_a =-5.30 x 10^-6 K^-1, α_c = 5.41 x 10^-6 K^-1), and Na₅Zr(PO₄)₃ (α_a = -5.82 x 10^-6 K^-1, α_c = 20.73 x 10^-6 K^-1). K₅Zr(PO₄)₃ exhibited the smallest thermal expansion and weak anisotropy (α_a = -2.14 x 10^-6 K^-1, α_c = 2.65 x 10^-6 K^-1). The effects of M(l) and M(2) site occupancies on α_a, α_c,a, and c were assessed. The relative magnitudes of crystal-chemical and thermal expansion in the Na and K compounds were analyzed.     

Preparation of SiB4±x and SiB6 plates by chemical vapour deposition of SiCl4 + B2H6 system

SiB_4±x and SiB₆ plates were prepared by chemical vapour deposition (CVD) using SiCl₄, B₂H₆ and H₂ gases under the conditions of deposition temperatures (T _dep) from 1323–1773 K, total gas pressures (P _tot) from 4–40 kPa and B/Si source gas ratio (m _B/Si=2B₂H₆/SiCl₄) from 0.2–2.8. The effects of CVD conditions on the morphology, structure and composition of the deposits were examined. High-purity and high-density SiB_4±x and SiB₆ plates about 1 mm thick were obtained at the deposition rates of 71 and 47 nm s⁻¹, respectively. The lattice parameter, composition and density of CVD SiB_4±x plates were dependent on their non-stoichiometry. The lattice parameter,a, was 0.6325 nm, butc ranged from 1.262–1.271 nm.The B/Si atomic ratio ranged from 3.1–5.0, and the density ranged from 2.39–2.45×10³ kg m⁻³. The CVD SiB₆ plates showed constant values of lattice parameters (a=1.444 nm,b=1.828 nm,c=0.9915 nm), composition (B/Si=6.0) and density (2.42×10³ kg m⁻³), independent of CVD conditions.     

Poisson Parameter Estimation with Arbitrary Factors of Proportionality and Arbitrary Points of Left Truncation

Two unbiased estimators of the Poisson parameter are given for a truncation situation in whirh Cohen has suggested a near-efficient procedure. The situation is one in which each observation has its own distinctive left. truncation point, c_i , and its own distinctive Poisson parameter λt _i — it is the common λ which is to be estimated. The proposed estimators are obtained as adaptations of a pltblished simple estimator for the case all c_i = c, all t_i = 1. Illustrative data from Cohen are used for intercomparing the various estimators. including the maximum likelihood estimator.     

Electrical and structural characteristics of non-stoichiometric Cu-based delafossites

Non-stoichiometric Cu^xGaO² polycrystalline ceramics were prepared using solid-state synthesis methods and their structural and electrical properties were characterized as a function of x. While single phase delafossite could only be confirmed for the stoichiometric composition, X-ray diffraction results show that the a and c lattice parameters were stable from Cu^0.98GaO² to Cu^1.02GaO². Below x = 0.97, the structure underwent a 0.4% decrease in cell volume. For compositions rich in Cu, the a parameter remained constant while the c parameter shrank at a constant rate. Despite these changes in cell dimensions, the results of current-voltage and impedance spectroscopy measurements indicate that both the conductivity and the activation energy do not vary with x. The absence of a change in electrical properties as well as the formation of secondary phases in non-stoichiometric compositions suggests that CuGaO² does not allow significant deviations from the ideal CuGaO² stoichiometry. This implies that the p-type conductivity is fixed by a defect species that does not vary with the Cu stoichiometry.     

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.     

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.     

Mixed-Cation Thallium Phosphates Containing Di-, Tri-, and Pentavalent Cations

The mixed phosphates TlMn²⁺P₃O₉(a= 9.422 Å, b= 7.3199 Å, c= 12.227 Å sp. gr. Pnma), TlGaHP₃O₁₀(a= 12.01 Å, b= 8.471 Å, c= 9.098 Å, = 111.98° sp. gr. C2/c), and TlTa(PO₄)₂(a= 5.0308 Å, c= 8.497 Å sp. gr. P321) were obtained by precipitation from polyphosphoric-acid melts containing di-, tri-, and pentavalent cations in combination with Tl⁺. The crystal structures of TlMn²⁺P₃O₉and TlTa(PO₄)₂were determined by single-crystal x-ray diffraction.     

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.     

Cramer-Rao type integral inequalities for general loss functions

Cramer-Rao type integral inequalities are developed for loss functionsw(x) which are bounded below by functions of the typeg(x) =c|x| ^l ,l > 1. As applications, we obtain lower bounds of Hajek-LeCam type for locally asymptotic minimax error for such loss functions.     

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.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

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.     

Reduction of perovskite oxide LnCoO3(Ln = La-Eu) in a hydrogen atmosphere

The reduction of LnCoO₃(Ln = La-Eu) in a hydrogen atmosphere of 2×10⁶ Pa at 25 to 600° C was investigated by X-ray diffraction and thermogravimetric analysis, which were carried out to pursuein situ the reaction of reduction of LnCoO₃. The amount of lattice oxygen consumed at 600° C increases on going from LaCoO₃ to EuCoO₃. It was shown that the reduction process proceeded through the formation of a series of oxygen-deficient structures in keeping the perovskite structures; for example, the final X-ray powder diffraction pattern of NdCoO_3–x (x=1.1) could be indexed on a cubic cell (a = 0.39 nm) and an orthorhombic cell (a₀=0.522 nm,b ₀ = 0.559nm, c₀ = 0.795nm), the relationship with the cubic cell beinga ₀ 2^1/2 a _c,b ₀ 2^1/2 a _c,c 2a_c.     

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.     

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.