Pair Of Random Variables Research Articles

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.

Specifying bivariate distributions by polynomial regressions

We consider pairs of random variables X, Y with the property that regressions E(X n|Y), E(Y n|X) are polynomial. We show that the leading terms of these polynomials and the marginal distributions determine uniquely the bivariate distribution.

Bootstrap confidence intervals for ratios of expectations

We are concerned with computing a confidence interval for the ratioE[Y]/E[X, where (X,Y) is a pair of random variables. This ratio estimation problem arises in, for instance, regenerative simulation. As an alternative to confidence intervals based on asymptotic normality, we study and compare different variants of the bootstrap for one-sided and two-sided intervals. We point out situations where these techniques provide confidence intervals with coverage much closer to the nominal value than do the classical methods.

Correlation, Regression Lines, and Moments of Inertia

A student of engineering or physics discussing a random variable X with mean μ and variance σ2 might refer to μ as the center of gravity of the (probability) mass distribution of X, and to σ2 as the moment of inertia of X about μ. Is there a similar “physical” interpretation of ρ, Pearson's product-moment correlation coefficient for pairs of random variables? We answer this question affirmatively by showing that ρ is equal to the ratio of a difference and sum of two moments of inertia about certain lines in the plane. From this observation it is easy to derive familiar important properties of ρ. Similar results hold for the population version of the nonparametric correlation coefficient Spearman's rho. These ideas are readily accessible by students in an undergraduate mathematical statistics course.

Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Let (X,Y) be a pair of random variables with supp(X) \subseteq [0,1]?I and EY?2 < \infinity. Let m* be the best approximation of the regression function of (X,Y) by sums of functions of at most d variables (formula). Estimation of m* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at most d variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X,Y) the weak and strong L2-consistency of the estimate is shown. Furthermore, for every distribution of (X,Y) with supp(X) \subseteq [0,1]?I, Y bounded and m* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate n?{-\frac{2p}{2p+d}}. Let (X,Y) be a pair of random variables with supp(X) (formula) [0,1]1 and EY2,(formula). Let m* be the best approximation of the regression function of (X,Y0 by sums of functions of at most d variables (formula). Estimation of m* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at most d variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X,Y) the weak and strong L2-consistency of the estimate is shown. Furthermore, for very (formula) and every distribution of (X,Y) with supp(X) (formula) , Y bounded and m* p-smooth, the integrated squard error of the estimate achieves up to a logarithmic factor the (optimal) rate in (formula).

Rich and Saturated Adapted Spaces

In the paper [HK] the notions of an adapted distribution and of a saturated adapted probability space were introduced. The adapted distribution of a random variable on an adapted space (with values in a complete separable metric space) is the natural analogue of the distribution of a random variable on a probability space. An adapted space Ω is saturated if for any random variable x on Ω and pair of random variables x and ȳ on another adapted space Γ such that x and x have the same adapted distribution, there is a random variable y on Ω such that (x, y) and (x, ȳ) have the same adapted distribution. For stochastic differential equations and a wide variety of other existence problems, every existence theorem which holds on some adapted space holds on a saturated adapted space. The paper [FK1] introduced a new method for proving existence theorems in probability theory, based on the notion of a neocompact set of random variables. A set of random variables on an adapted space is said to be basic if it is either compact or is the set of all random variables which are measurable at time t and whose law belongs to a compact set C of measures, for some t and C. The family of neocompact sets is the closure of the family of basic sets under finite unions and Cartesian products, countable intersections, existential projections, and “universal

Tests of Independence and Zero Correlations Among P Random Variables

AbstractSuppose it is desired to determine whether there is an association between any pair of p random variables. A common approach is to test H0 : R = I, where R is the usual population correlation matrix. A closely related approach is to test H0 : Rpb = I, where Rpb is the matrix of percentage bend correlations. In so far as type I errors are a concern, at a minimum any test of H0 should have a type I error probability close to the nominal level when all pairs of random variables are independent. Currently, the Gupta‐Rathie method is relatively successful at controlling the probability of a type I error when testing H0: R = I, but as illustrated in this paper, it can fail when sampling from nonnormal distributions. The main goal in this paper is to describe a new test of H0: Rpb = I that continues to give reasonable control over the probability of a type I error in the situations where the Gupta‐Rathie method fails. Even under normality, the new method has advantages when the sample size is small relative to p. Moreover, when there is dependence, but all correlations are equal to zero, the new method continues to give good control over the probability of a type I error while the Gupta‐Rathie method does not. The paper also reports simulation results on a bootstrap confidence interval for the percentage bend correlation.

Testing First- and Second-Order Stochastic Dominance

Tests for stochastic dominance are important tools for the comparison of distributions between pairs of random variables. In finance, dominance criteria can potentially provide an unambiguous ranking of the desirability of different assets while placing only general restrictions on the preferences of investors. In welfare economics, dominance concepts allow for the ranking of income distributions using generally accepted welfare criteria. Existing dominance test procedures suffer from a number of weaknesses. One weakness concerns restrictions on the class of distribution functions that may be used as a basis for testing. While early literature in this area typically ignored sampling errors, this issue has been addressed in more recent literature, but only at the cost of restricting the class of parametric distributions discussed; see e.g. Stein and Pfaffenberger (1986). Since dominance criteria are attractive primarily because they allow for a ranking of returns on risky assets or income distributions, while placing only weak restrictions on preferences, it is important that dominance tests retain a degree of generality, and hence remain nonparametric in nature. Another weakness of existing dominance tests is that these often specify the null hypothesis improperly; see e.g. Bishop et al. (1989). For example, most test procedures make use of the null hypothesis that two distribution or quantile functions are identical. The hypothesis of dominance may be viewed as an hypothesis of inequality in a particular direction between two distribution or quantile functions. If such an hypothesis is rejected, then dominance cannot be sustained, a result that may or may not be caused by the equality of the two distribution or quantile functions. On the other hand, if the null hypothesis of equality is rejected, then the two distributions cannot be said to be equal, but the cause may or may not be that one dominates the other. Thus the null hypothesis of equality is not very helpful in providing information about dominance; see Levy (1992, p. 574).

On an empirical bayesian estimation of the location parameter

The estimation of the value of a certain unobserved parameter is a classical problem of mathematical statistics. The situation when this parameter is random leads to the problem of the empirical Bayes estimation. We consider (X, {theta}), a pair of random variables on the product spaces ({infinity}, {infinity}, b = + {infinity} * [a, b]; {beta} * {beta} 1; {mu} * {mu}), where {beta}, {beta} 1 are Borel {sigma}-algebras on R and on [a, b], respectively in particular, a = -{infinity}, b = +{infinity}, and {mu} is the Lebesgue measure. Let p(x) and g({theta}) be the unknown distribution densities of the r.v. X and {theta}, let p(x{vert_bar}{theta}) be the known distribution density of the r.v.X for a fixed {theta}, and assume that there exists a sequence of independent, identically distributed pairs (X{sub 1}, {theta}{sub 1}), ..., (X{sub N},{theta}{sub N}), are observed, while {theta}{sub 1}...,{theta}{sub N} are unobserved. Knowing the (N + 1)st observation X{sub N+1} = y, we have to estimate the corresponding random parameter {theta}{sub N+1} = t.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

The asymptotic distribution of the rènyi maximal correlation

Renyi (1959) defined the maximal correlation ρ between a pair of random variables (U, W) as where the supremum is taken over all functions of U and W with finite second moments. In this paper we derive the asymptotic distribution of the estimate of the Rbnyi correlation coefficient based on a sample of independent observations under the assumption that (U, W) are independent and assume only a finite number of values.

Nonparametric sequential estimation of zeros and extrema of regression functions

Let (X,Y), (X_{l}, Y_{l}), (X_{2}, Y_{2}), \cdots be independent identically distributed pairs of random variables, and let m(x)=E(Y|X = x) be the regression curve of Y on X . The estimation of zeros and extrema of the regression curve via stochastic approximation methods is considered. Consistency results of some sequential procedures are presented and termination rules are defined providing fixed width confidence intervals for the parameters to be estimated.

Evaluation of minimum and maximum of a correlated pair of random variables via the bivariate laguerre transform

In a recent paper by Sumita and Kijima (1985), the bivariate Laguerre transform has been developed for mechanizing various bivariate continuum operations such as multiple bivariate convolutions, marginal convolutions, tail integration, partial differentiation and multiplication by bivariate polynomials. The transform has been shown to be quite useful and powerful in studying bivariate distributions and bivariate stochastic processes. In this paper, we study the minimum and maximum of a correlated pair of random variables via the bivariate Laguerre transform, thereby further enhancing the applicability of the bivariate Laguerre transform. A numerical procedure is developed for calculating the Laguerre coefficients of the probability density functions for the minimum and maximum in terms of the original bivariate Laguerre coefficients corresponding to the correlated pair of random variables. This enables one to evaluate the distributions and moments of the minimum and maximum. The joint distribution of the ...

Umvu estimation for covariances and first product moments of transformed variables

Suppose that a pair of random variables has a bivariate normal distribution with mean vector and covariance matrix denoted by N2:(μ, ∑), and the functions f and g are of recursive type. Then the UMVU estimators for Cov[f(X), g(Y)] and E[f(X)g(Y)] are given from independent samples having N2:(μ, ∑) for all i = 1,…n (n ≧ 3).

Estimable Versions of Griffiths' Measure of Association

SummaryMany techniques have been proposed for measuring the degree of association between a pair of random variables X and Y. However, the measures often suffer from two drawbacks: they are difficult to calculate when the joint distribution of (X, Y) is known, and difficult to estimate when the joint distribution is unknown. In the present paper we present two modifications of a measure proposed by Griffiths (1972), and show that they exhibit neither of these deficiencies.

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x 0, x1, …. Based onn independent observations ofX, x (n), we are to estimate the true (unknown) parameter vectorα=(α 1, α2, ...,αm) of the probability function ofX, Pα(X=∫ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector\(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex (n)=(x1, x2,…,x n) is a random sample ofX andf n(xi)=[number ofx i inx (n)]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of\(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-\nulldelimiterspace} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.

On some characterizations in probability by using minima and maxima of random variables

Let X, Y = ( Y 1 , . . . , Ym), Z = ( Z 1 . . . . . Z , ) be three independent random vectors of total dimension 1 + m + n with nonvanishing characteristic functions. Denote T I = X + Y j , j = l , . . . , m ; U k = X Z k , k = l . . . . ,n . Then ( T , U ) = (7"1 . . . . . T~, U1 . . . . . U,) is an (m + n)-dimensional random vector. It is known that the distribution of (T, U) determines the distributions of X, Y, Z up to shifts (see [1], Corollary 1). In this paper two theorems on similar characterizations are given, the sums and differences are replaced by maxima and minima of the corresponding pairs of random variables. Throughout this paper we shall denote for q , c2 real

Probability Functions which are Proportional to Characteristic Functions and the Infinite Divisibility of the von Mises Distribution

Reciprocal pairs of continuous random variables on the line are considered, such that the density function of each is, to within a norming factor, the characteristic function of the other. The analogous reciprocal relationship between a discrete distribution on the line and a continuous distribution on the circle is also considered. A conjecture is made regarding infinite divisibility properties of such pairs of random variables. It is shown that the von Mises distribution is infinitely divisible for sufficiently small values of the concentration parameter.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞&lt; b &lt; ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas

where the infimum is taken over all pairs of random variables X and Y defined on (f2, P) and distributed according to f and g respectively; here g is the Gaussian distribution with mean 0 and variance a 2 =~2 (f)e I-f] is sometimes denoted by e IX] when X is a random variable with distribution f. It should be noticed that the value of e [ f ] does not depend upon a choice of the probability space (f2, P). The purpose of this paper is to present some basic properties of e (especially, the inequality (2.2)) together with an application to the central limit theorem and then to show that the functional e is monotone decreasing along Boltzmann solutions of Kac's one-dimensional model of a Maxwellian gas. Some of our results can be generalized to the case of R 3; for example, the functional e similarly defined in R 3 decreases along solutions of Boltzmann's problem for the 3-dimensional Maxwellian gas, but this will be discussed in another occasion.