Symbolic computing the exact distributions of L-statistics from a uniform distribution

The exact probability density function of linear combinations of k=k(n) order statistics selected from the whole order statistics (L-statistic) based on a random sample of size n from the uniform distribution on [0, 1] was derived by Matsunawa (1985, Ann. Inst. Statist. Math., 37, 1–16). As the main expression for the density function given by Matsunawa is not complete for the general situation, we first provide the corrections for this formula. Second, we propose a simple scheme involving symbolic computing for evaluating the corrected version of the density function. The cumulative distribution function and the r-th mean of his L-statistic are also derived.     

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

Quantitative Approximation to the Ordered Dirichlet Distribution under Varying Basic Probability Spaces

An approximate expansion of a sequence of ordered Dirichlet densities is given under the set-up with varying dimensions of the relating basic probability spaces. The problem is handled as the approximation to the joint distribution of an increasing number of selected order statistics based on the random sample drawn from the uniform distribution U(0, 1). Some inverse factorial series to the expansion of logarithmic function enable us to give quantitative error evaluations to our problem. With the help of them the relating modified K-L information number, which is defined on an approximate main domain and different from the usual ones, is accurately evaluated. Further, the proof of the approximate joint normality of the selected order statistics is more systematically presented than those given in existing works. Concerning the approximate normality the modified affinity and the half variation distance are also evaluated.     

Estimation of a structural linear regression model with a known reliability ratio

In this paper, we consider the estimation of the slope parameter of a simple structural linear regression model when the reliability ratio (Fuller (1987),Measurement Error Models, Wiley, New York) is considered to be known. By making use of an orthogonal transformation of the unknown parameters, the maximum likelihood estimator of and its asymptotic distribution are derived. Likelihood ratio statistics based on the profile and on the conditional profile likelihoods are proposed. An exact marginal posterior distribution of , which is shown to be at-distribution is obtained. Results of a small Monte Carlo study are also reported.The first author acknowledges partial finantial suport from CNPq-BRASIL.     

The exact distribution of indefinite quadratic forms in noncentral normal vectors

The exact density of the difference of two linear combinations of independent noncentral chi-square variables is obtained in terms of Whittaker's function and expressed in closed forms. Two distinct representations are required in order to cover all the possible cases. The corresponding expressions for the exact distribution function are also given.     

Stability of Order Statistics under Dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.     

Analytic,Computational, and Approximate Forms for Ratios of Noncentral and Central Gaussian Quadratic Forms

Many useful statistics equal the ratio of a possibly noncentral chi-square to a quadratic form in Gaussian variables with all positive weights. Expressing the density and distribution function as positively weighted sums of corresponding F functions has many advantages. The mixture forms have analytic value when embedded within a more complex problem. The mixture forms also have computational value. The expansions work well with quadratic forms having few components and small degrees of freedom. A more general algorithm from earlier literature can take longer or fail to converge in the same setting. Many approximations have been suggested for the problem. A positively weighted noncentral quadratic form can always have two moments matched to a noncentral chi-square. For a single quadratic form, the noncentral form performs neither uniformly more or less accurately than older approximations. The approach also gives a noncentral F approximation for any ratio of a positively weighted noncentral form to a positively weighted central quadratic form. The method provides better accuracy for noncentral ratios than approximations based on a single chi-square. The accuracy suffices for many practical applications, such as power analysis, even with few degrees of freedom. Naturally the approximation proves much faster and simpler to compute than any exact method. Embedding the approximation in analytic expressions provides simple forms which correctly guarantee only positive values have nonzero probabilities, and also automatically reduce to partially or fully exact results when either quadratic form has only one term.     

The empirical distribution function and strong laws for functions of order statistics of uniform spacings

Let N points be arbitrarily chosen on the circle with unit circumference, and order them clockwise. The uniform mth order spacings are then defined as the clockwise distances between any pair of points having m − 1 other points in between. A Glivenko-Cantelli theorem and nonlinear almost sure bounds for the empirical distribution function based on these uniform spacings are derived. The parameter m is allowed to increase with N to infinity. Applications to linear combinations of functions of mth order spacings are given.     

Approximation by Partial Sums of Fourier Series and the Best Approximation of Certain Function Classes

Continuing investigations of S. A. Telyakovskii, S. B. Stechkin, A. I. Stepanets and R. M. Trigub, the author studies two questions about approximation of the class of periodic functions with a bounded -derivative by polynomials in the spaces C and L.First, the asymptotics of approximation by partial sums of Fourier series has been found without limitations for the decreasing order of . Second, the exact decreasing order of the best approximation of the class has been found. New examples are also given.     

Best Approximation with Walsh Atoms

We consider the approximation in L ² R of a given function using finite linear combinations of Walsh atoms, which are Walsh functions localized to dyadic intervals, also called Haar—Walsh wavelet packets. It is shown that up to a constant factor, a linear combination of K atoms can be represented to relative error ɛ by a linear combination of orthogonal atoms. In finite dimension N, best approximation with K orthogonal atoms can be realized with an algorithm of order . A faster algorithm of order solves the problem with indirect control over K. Therefore the above result connects algorithmic and theoretical best approximation. Date received: July 6, 1995. Date revised: January 8, 1996.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

The variance function of the Erlang process

Summary A relatively simple exact expression of closed form is obtained for the varianceσ ²(t) of the asynchronous counting distribution for a counting period of lengtht,t>0, in an Erlang process. Useful bounds are placed upon the error of the linear approximation toσ ²(t). Implications of these results are examined. In particular, a new exact expression and related bounds are obtained for the mean function of the synchronous counts (also known as the renewal function of the process). All bounds given are sharp in asymptotic order of magnitude as the length of the counting period is allowed to increase. Prepared under Contract No. FH-11-6890, Bureau of Public Roads, Federal Highway Administration, U.S. Department of Transportation. The opinions, findings and conclusions expressed herein are those of the author and not necessarily those of the Bureau of Public Roads.     

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R^a, as multinomial goodness-of-fit statistics. Each R^a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R^a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R^a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R^a. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.This research was partially supported by the ISM Cooperative Research Program (94-ISM-CRP-5).     

Exact Stratified Linear Rank Tests for Ordered Categorical and Binary Data

Abstract

We present an efficient algorithm for generating exact permutational distributions for linear rank statistics defined on stratified 2 × c contingency tables. The algorithm can compute exact p values and confidence intervals for a rich class of nonparametric problems. These include exact p values for stratified two-population Wilcoxon, Logrank, and Van der Waerden tests, exact p values for stratified tests of trend across several binomial populations, exact p values for stratified permutation tests with arbitrary scores, and exact confidence intervals for odds ratios embedded in stratified 2 × c tables. The algorithm uses network-based recursions to generate stratum-specific distributions and then combines them into an overall permutation distribution by convolution. Where only the tail area of a permutation distribution is desired, additional efficiency gains are achieved by backward induction and branch-and-bound processing of the network. The algorithm is especially efficient for highly imbalanced categorical data, a situation where the asymptotic theory is unreliable. The backward induction component of the algorithm can also be used to evaluate the conditional maximum likelihood, and its higher order derivatives, for the logistic regression model with grouped data. We illustrate the techniques with an analysis of two data sets: The leukemia data on survivors of the Hiroshima atomic bomb and data from an animal toxicology experiment provided by the U.S. Food and Drug Administration.     

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.     

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofx ^k by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann ² and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p _k,n , which can be identified with a certain probability. The numberp _k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.