On a generalization of polynomials in the ballot problem

This work gives two generalizations of polynomials in the ballot problem based on a dual relation for addition formulas by Watanabe (1984, 1985).One is given by solving an interpolation problem in the case of several variables. In the case of one variable, Niederhausen (1981) developed a theory of some rank order statistic based on this problem.On the other hand, Takács' urn model gives a multinomial generalization of the Gould polynomials. Since the Gould polynomials are the special case of basic polynomials, by obtaining a theory of multinomial basic polynomials with several variables, we give a generalization of polynomials in Takács' urn model.     

Recurrence relations of coefficients of the generalized hypergeometric function in multivariate analysis

James(1960) defined the zonal polynomials and used it to represent the joint distributions of latent roots of VVisfiart matrix. The zonal polviionnals played an important role to define the generalized hypergeometric function of symmetric matrix argument Since then, many density functions and moments based on Wishart matrix have been expressed in terms of the generalized hy¬pergeometric Function. The purpose of this paper is to get the recurrence relations for the coefficients of it. In Section 1 we derive a partial differen¬tial equations having the generalized hypergeometric function as the unique solution. Then we ubtain the recurrence relations until order 7 in Section 2.     

Statistical inference for reciprocal gamma diffusion process

We consider the problem of parameter estimation for an ergodic diffusion with reciprocal gamma invariant distribution. Spectral decomposition of the transition density of such a Markov process is presented in terms of a finite number of discrete eigenfunctions (Bessel polynomials) and eigenfunctions related to a continuous part of the spectrum of the negative infinitesimal generator of reciprocal gamma diffusion. Consistency and asymptotical normality of proposed estimators are presented. Based on the Stein equation for reciprocal gamma diffusion and Bessel polynomials, the hypothesis testing procedure is considered.     

CATANOVA for two-way cross classified categorical data

In this article we develop an extension of categorical analysis of variance for one response and two factors, based on a partitioning of a measure of predictability for three-way contingency tables, known as Gray and Williams's index. At the first instance moment the decomposition of this multiple measure of association in partial association measures is shown. Finally, for ordinal-scale variables, we propose an extension of this decomposition using a particular set of orthogonal polynomials.     

Tests of symmetry derived as components of pearson's phi-squared distance measure

Rank statistics which arise as estimates of the first and third components of a frequency decomposition of Pearson's phi-squared distance measure, introduced by Eubank, LaRiccia, and Rosenstein (1987), are examined for their usefulness as tests of symmetry about a known median against various asymmetric alternatives, Pitman asymptotic relative efficiencies are used to compare the efficacies of the new statistics with classical test procedures, and empirical powers of the new tests are compared via simulation for a variety of asymmetric distributions. Statistics which arise from components based on Legendre polynomials are of particular interest. The first component is the familiar Wilcoxon signed rank statistic. The third component, which is a new statistic for this problem, exhibits a high level of sensitivity to a variety of asymmetric alternatives both asymptotically and in small sample studies.     

Bayesian,classical and hybrid methods of inference when one parameter value is special

This paper considers the problem of making statistical inferences about a parameter when a narrow interval centred at a given value of the parameter is considered special, which is interpreted as meaning that there is a substantial degree of prior belief that the true value of the parameter lies in this interval. A clear justification of the practical importance of this problem is provided. The main difficulty with the standard Bayesian solution to this problem is discussed and, as a result, a pseudo-Bayesian solution is put forward based on determining lower limits for the posterior probability of the parameter lying in the special interval by means of a sensitivity analysis. Since it is not assumed that prior beliefs necessarily need to be expressed in terms of prior probabilities, nor that post-data probabilities must be Bayesian posterior probabilities, hybrid methods of inference are also proposed that are based on specific ways of measuring and interpreting the classical concept of significance. The various methods that are outlined are compared and contrasted at both a foundational level, and from a practical viewpoint by applying them to real data from meta-analyses that appeared in a well-known medical article.     

Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the case of non-response

In multivariate cases, usually the minimization of sampling variances is considered as an objective under a cost constraint. Since the variances are not unit free, it is more logical to consider the minimization of the squared coefficients of variation as an objective. In this paper, the problem of optimum compromise allocation in multivariate stratified sampling in the case of non-response as a multi-objective all-integer nonlinear programming problem is described. A solution procedure using four different approaches is considered, namely the value function, goal programming,∈-constraint and distance based, to obtain the compromise allocation for non-response. A numerical example is also presented to illustrate the computational details.     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.     

Age–period–cohort decompositions using principal components and partial least squares

Age–period–cohort decomposition requires an identification assumption because there is a linear relationship between age, survey period, and birth cohort (age+cohort=period). This paper proposes new decomposition methods based on factor models such as principal components model and partial least squares model. Although factor models have been applied to overcome the problem of many observed variables with possible co-linearity, they are applied to overcome the perfect co-linearity among age, period, and cohort dummy variables. Since any unobserved factor in the factor model is represented as a linear combination of the observed variables, the parameter estimates for age, period, and cohort effects are automatically obtained after the application of these factor models. Simulation results suggest that in almost all cases, the performance of the proposed method is better than that of a conventional econometric method. Empirical examples are also provided.     

应用不等权重支持向量机预测人民币汇率的变动

基于金融时间序列的近期数据对未来的影响会大于早期数据,对应用于金融时间序列预测的支持向量机方法进行改进,给出了不等权重支持向量机方法（USVM）及其多项式光滑化处理。将不等权重支持向量机方法应用于训练样本集的子集确定预测模型,实证分析表明USVM算法预测是有效的。     

Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs

It is common in regression discontinuity analysis to control for third, fourth, or higher-degree polynomials of the forcing variable. There appears to be a perception that such methods are theoretically justified, even though they can lead to evidently nonsensical results. We argue that controlling for global high-order polynomials in regression discontinuity analysis is a flawed approach with three major problems: it leads to noisy estimates, sensitivity to the degree of the polynomial, and poor coverage of confidence intervals. We recommend researchers instead use estimators based on local linear or quadratic polynomials or other smooth functions.     

Response surface methodology for non-equispaced experimental configurations applied to a discrete and stochastic simulator of a manufacturing company

This work is based on the construction and use of a SLAM II PC-based simulator for a medium-sized company manufacturing machine tools designed to work glass. Response surface methods and several experimental design techniques were applied to this company to optimize simulator performances. Since there is a feedback loop on the actual manufacturing system, the theory of orthogonal polynomials was applied to the case-study relative to a planned experimental configuration of non-equispaced tests performed on the real system, which were based on analysis characteristics. This discussion is the ideal extension of the basic case involving equispaced configurations. A summary is presented of the results obtained from the application of a 3X4 mixed level factorial experiment implemented with a discrete and stochastic industrial simulator.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

Linear generalizations of various matric-t distributions

The concept of a matric-t variate is extended to cases where the positive (definite) part of the variate, which is usually Wishart distributed independently of the normal part, is a linear sum of positive (definite) variates with positive coefficients. These distributions and their quadratic forms are of importance i.a, for the exact solution to the multi¬variate Behrens-Fisher problem. A few useful identities con¬cerning the invariant polynomials with matrix arguments are derived     

Quantile residual life regression based on semi-competing risks data

This paper investigates the quantile residual life regression based on semi-competing risk data. Because the terminal event time dependently censors the non-terminal event time, the inference on the non-terminal event time is not available without extra assumption. Therefore, we assume that the non-terminal event time and the terminal event time follow an Archimedean copula. Then, we apply the inverse probability weight technique to construct an estimating equation of quantile residual life regression coefficients. But, the estimating equation may not be continuous in coefficients. Thus, we apply the generalized solution approach to overcome this problem. Since the variance estimation of the proposed estimator is difficult to obtain, we use the bootstrap resampling method to estimate it. From simulations, it shows the performance of the proposed method is well. Finally, we analyze the Bone Marrow Transplant data for illustrations.     

Hypothesis testing for Fisher–Snedecor diffusion

We consider the problem of testing the hypothesis on marginal distribution of ergodic diffusion with Fisher–Snedecor invariant distribution, to be called Fisher–Snedecor diffusion. We propose a GMM approach to testing this statistical hypothesis where the moment condition is based on eigenfunctions of the diffusion infinitesimal generator—Fisher–Snedecor polynomials. Statistical test is observed in two different settings: (1) for known values of parameters of the process; (2) for consistent moment based estimators of parameters. Results are illustrated in a short simulation study.     

On the Skew-Normal-Cauchy Distribution

A fixed effects one-way layout model of analysis of variance is considered where the variances are taken to be possibly unequal. Conservative single-stage procedures based on Banerjee’s method for the solution of the Behrens-Fisher problem are proposed for the following multiple comparisons problems: 1) all pairwise comparisons with a control population mean, and 2) all pairwise comparisons and all linear contrasts among the means. Since these procedures are likely to be very conservative in practice, approximate procedures based on Welch’s method for the solution of the Behrens-Fisher problem are suggested as alternatives. Monte Carlo studies indicate that the latter are much less conservative and hence may be better in practice. Both these sets of procedures need only the tables of the Student’s t-distribution for their application and are very simple to use. Exact two-stage procedures are proposed for the following multiple comparisons problems: 1) all pairwise comparisons and all linear contrasts among the means, and 2) all linear combinations of the means.     

A unified treatment for the asymptotic normality of the coefficients of polynomials related to orthogonal ones

Suitable transformations on the orthogonal polynomials lead to polynomials with nonnegative coefficients. In this work, the asymptotic normality for the nonnegative coefficients of these polynomials is derived based on the nature of the weight function of the orthogonal polynomials. In particular, orthogonal polynomial cases from both classical and semi-classical systems are included as well as the singular behaved Pollaczek polynomial case.     

The generalized symmetric eigenproblem in multivariate statistical models

The solution of the generalized symmetric eigenproblem Ax = λBx is required in many multivariate statistical models, viz. canonical correlation, discriminant analysis, multivariate linear model, limited information maximum likelihoods. The problem can be solved by two efficient numerical algorithms: Cholesky decomposition or singular value decomposition. Practical considerations for implementation are also discussed.     

The ℓ1 oblique procrustes problem

In this paper, we reconsider the well-known oblique Procrustes problem where the usual least-squares objective function is replaced by a more robust discrepancy measure, based on the ₁ norm or smooth approximations of it.We propose two approaches to the solution of this problem. One approach is based on convex analysis and uses the structure of the problem to permit a solution to the ₁ norm problem. An alternative approach is to smooth the problem by working with smooth approximations to the ₁ norm, and this leads to a solution process based on the solution of ordinary differential equations on manifolds. The general weighted Procrustes problem (both orthogonal and oblique) can also be solved by the latter approach. Numerical examples to illustrate the algorithms which have been developed are reported and analyzed.