Simulation of weibull and gamma autoregressive stationary process

This paper presents two simple non-Gaussian first-order autoregressive markovian processes which are easy to simulate via a computer. The autoregressive Gamma process {X_n:} is constructed according to the stochastic difference equation X_n:=V_n:X_n?1+?_n:, where {?_n:} is an i.i.d. Exponential sequence and {V_n:} is i.i.d. with Power-function distribution defined on the interval [0,1). The autoregressive Weibull process {X_n:} is constructed from the probabilistic model X_n:= k.min (X_n?1:, Y_n:) where {Y_n:} is an i.i.d. Weibull sequence and k > 1.     

On the maxima of heterogeneous gamma variables

In this article, we establish some new results on stochastic comparisons of the maxima of two heterogenous gamma variables with different shape and scale parameters. Let X₁ and X₂ [X*₁ and X*₂] be two independent gamma variables with X_i?[X*_i] having shape parameter r_i?[r*_i] and scale parameter λ_i?[λ*_i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X_{2: 2} and X*_{2: 2} when λ₁ = λ*₁ ? λ₂ = λ*₂ and r₁ ? r*₁ ? r₂ = r*₂. We also prove that, if r_i, r*_i ∈ (0, 1], (r₁, r₂) majorizes (r*₁, r₂*), and (λ₁, λ₂) is p-larger than (λ*₁, λ₂*), then X_{2: 2} is larger than X*_{2: 2} in the sense of the hazard rate order [dispersive order]. Some numerical examples are provided to illustrate the main results. The new results established here strengthen and generalize some of the results known in the literature.     

On the normality of transformed beta and unit- gamma random variables

In this paper we consider properties of the logarithmic and Tukey's lambda-type transformations of random variables that follow beta or unit-gamma distributions. Beta distributions often arise as models for random proportions, and unit-gamma distributions, although not well- known, may serve the same purpose. The latter possess many properties similar to those of beta distributions. Some transformations of random variables that follow a beta distribution are considered by Johnson (1949) and Johnson and Kotz (1970,1973). These are used to obtain a -new"random variable that potentially approximately follows a normal distribution, so that practical analyses become possible. We study normality -related properties of the above transformations. This is done for the first time for unit-gamma distributions. Under the logarithmic transformation the beta and unit-gamma distributions become, respectively, the logarithmic F and generalized logistic distributions. The distributions of the transformed beta and unit-gamma distributions after application of Tukey's lambda-type transformations cannot be derived easily; however, we obtain the first four moments and expressions for the skewness and kudos is of the transformed variables. Values of skewness and kurtosis for a variety of different parameter values are calculated, and in consequence, the near (or not near) normality of the transformed variables is evaluated. Comments on the use of the various transformations are provided..     

On accurate and precise generation of generalized Poisson variates

This article is concerned with comparison of a few random variate generation techniques for the generalized Poisson distribution. An evaluation is conducted on the degree of proximity between the estimates for its two distributional parameters and first four moments, and the specified or computed true population values via commonly accepted accuracy and precision measures in a simulated setting.     

PoisNor: An R package for generation of multivariate data with Poisson and normal marginals

In this article, the operational details of the R package PoisNor that is designed for simulating multivariate data with count and continuous variables with a prespecified correlation matrix are described, and examples of some important functions are given. The data-generation mechanism is a combination of the “NORmal To Anything” principle and a recently established connection between Poisson and normal correlations. The package provides a unique and useful tool that has been lacking for generating multivariate mixed data with Poisson and normal components.     

On the convolution of normal and t random variables

The need for the convolution of normal and Student's t random variables arises in many areas. Since the 1930s, various authors have attempted to derive closed-form expressions for the probability density function (pdf) of the convolution, but with little success. Here, general closed-form expressions are derived for the pdf.     

Distributions of ratios of random variables from the power-quadratic exponential family and applications

We derive the closed form exact expression of the density of the ratio X ₁/X ₂, where X ₁ and X ₂ are independent and belong to the very versatile family of power-quadratic exponential distributions. The case of X ₂ as a gamma variable is studied in detail, and new applications found in different domains show the varied uses of the results presented.     

Concurrent generation of binary and nonnormal continuous data through fifth-order power polynomials

Data collection process in most observational and experimental studies yield different types of variables, leading to the use of joint models that are capable of handling multiple data types. Evaluation of various statistical techniques that have been developed for mixed data in simulated environments requires concurrent generation of multiple variables. In this article, I present an important augmentation to a unified framework proposed in our previously published work for simultaneously generating binary and nonnormal continuous data given the marginal characteristics and correlation structure, via fifth-order power polynomials that are known to extend the area covered in the skewness-elongation plane and to provide a better approximation to the probability density function of the continuous variables. I evaluate how well the improved methodology performs in comparison to the original one, in a simulated setting with illustrations of algorithmic steps. Although the relative gains for the associational quantities are not substantial, the augmented version appears to better capture the marginal quantities that are pertinent to the higher-order moments, as indicated by very close resemblance between the specified and empirically computed quantities on average.     

On a confidence interval about a parameter estimated by a ratio of normal random variables

A confidence interval is geometrically constructed about a parameter estimated by the ratio of bivariate normal random variables. The resulting confidence interval is equivalent to that of Fieller's theorem. The geometric construction shown that such intervals are conservative. Bioassay examples are used to demonstrate the technique.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.     

On the finite sample performance of exogeneity tests of revankar, revankar and hartley and wu-hausman

In this paper we compare the performance of the exogeneity tests of Revankar, Revankar and Hartley and Wu-Hausman for the cases of two and three included endogenous variables. The distribution and power functions are evaluated using the conditional distributions given in Kariya and Hodoshima. Our results indicate that the Revankar's test is the most powerful for large values of the concentration parameter and the Revankar and Hartley test is the most powerful for small values of the concentration parameter.     

On complete and complete moment convergence for weighted sums of ANA random variables and applications

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated (ANA, for short) random variables are studied. Several sufficient conditions of the complete convergence and complete moment convergence for weighted sums of ANA random variables are presented. As an application, the complete consistency for the weighted estimator in a nonparametric regression model based on ANA random errors is established by using the complete convergence that we established. We also give a simulation to verify the validity of the theoretical result.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

On the relationship between Spearman's rho and Kendall's tau for pairs of continuous random variables

It has long been known that for many joint distributions exhibiting weak dependence, the sample value of Spearman's rho is about 50% larger than the sample value of Kendall's tau. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho to tau approaches 3/2$3/2$ as the joint distribution approaches that of two independent random variables. We also find sufficient conditions for determining the direction of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely continuous.     

On the distribution of the maximum of bivariate normal random variables with general means and variances

A great amount of effort has been devoted to achieving exact expressions for moments of order statistics of independent normal random variables, as well as the dependent case with the same correlation coefficients, means and variances. It does not seem as if there are handy formulae for the order statistics of even the simple bivariate normal random variables when the means and variances are allowed to be different. In this paper we give an explicit formula for the Lanl ace-Stielties Transform of the maximum of bivariate normal random variables by which we obtain formulae for the first two moments in the standard way.     

The probability distribution and the expected value of a stopping variable associated with one-sided cusum procedures for non-negative integer valued random variables

The structure of a stopping variable N based on one-sided CUSUM procedures is analyzed. Stopping occurs when a Markovian sequence of maxima of partial sums {M } crosses a certain boundary. On the basis of a recursive relationship between the M_n+1 and M_n a recursive equation is derived for the determination of the defective distributions K_n(x) = P{M ≤ x, N ≤n} . This recursive equation yields a recursive algorithm for the determination of P {N > n} . The paper studies the case when the basic random variables are non-negative integers-valued. In these cases the values of P{N > n} and E{N} can be determined by solving proper systems of linear equations.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

On the correct regression function (in L2) and its applications when the dimension of the covariate vector is random

We derive the optimal regression function (i.e., the best approximation in the L₂ sense) when the vector of covariates has a random dimension. Furthermore, we consider applications of these results to problems in statistical regression and classification with missing covariates. It will be seen, perhaps surprisingly, that the correct regression function for the case with missing covariates can sometimes perform better than the usual regression function corresponding to the case with no missing covariates. This is because even if some of the covariates are missing, an indicator random variable δ$\delta$, which is always observable, and is equal to 1 if there are no missing values (and 0 otherwise), may have far more information and predictive power about the response variable Y than the missing covariates do. We also propose kernel-based procedures for estimating the correct regression function nonparametrically. As an alternative estimation procedure, we also consider the least-squares method.