On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.Research supported by an NSERC Canada operating grant at the University of Alberta.Part of this research was done while visiting the University of Alberta supported by the NSERC Canada grant of the first author.     

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are X_t=X?t | X>t and Y_t=Y?t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(X_t>Y_t) which enables one to obtain, at time t, the probability that the residual lifetime X_t is greater than the residual lifetime Y_t. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22 , 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.     

Sulla scelta tra variabili aleatorie

Date due variabili aleatorie stocasticamente indipendentiX eY può accadere cheX dominiY secondo il criterio della dominanza stocastica FSD anche seP(X>Y)0. In questo lavoro l'autore propone un nuovo criterio di dominanzaH fondato sulla teoria dell'utilità SSB e lo applica al caso di due variabili aleatorie dipendenti.X può dominareY secondo il criterioH solo seP(X>Y)0,5. Nel caso di variabili aleatorie indipendenti il criterioH risulta essere un affinamento del criterio FSD.

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Summary Given two independent random variablesX andY it can happen thatX dominatesY according to the usual stochastic dominance criterion FSD even thoughP(X>Y)0. In this paper, the author proposes a new criterionH involving the SSB utility theory (Fishburn 1982) and applies it to the case of dependent random variables. It happens thatX can dominateY according to the criterionH only ifP(X>Y)0,5.In the case of independent variables, the criterionH is finer than the usual one, i.e., ifX dominatesY according to FSD then the same is true with respect toH.

     

The association between two random elements: A complete characterization and odds ratio models

For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.Acknowledgements The author thanks both referees for their helpful comments which improved the first draft of the paper.     

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F:X→Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f:X→Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.     

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Beware of averages!

Summary Let X₁,.,., X_m, and Y₁, Y_n, be two independent samples from the same distribution and let X and Y be the means of these samples. What is the maximal value of P(X < Y)?     

Tendency towards normality of linear combinations of random variables

IfX andY are two random variables with the same means and variances, thenX is said to be nearer normal thanY if the absolute values of its cumulants are smaller than the corresponding cumulants ofY. Using this definition, it is shown that a linear combination of a finite number of independent identically distributed random variables is always nearer normal than its constituents, but that this is not necessarily true if not-identically distributed or not-independent variables are used. Some consequences of the results are reached for the testing of normality of time series and for the assumptions frequently made by social scientists about the distribution of their data.     

Sulla somma di due numeri aleatori egualmente distribuiti

In questo lavoro si studia il problema di ricerca della distribuzione di probabilità comune da assegnare a due numeri aleatori discreti che assumano i primin valori interi naturali in modo che la loro somma abbia moda di minima probabilità.Il problema è affrontato sia dal punto di vista teorico tramite gli strumenti della programmazione matematica, sia dal punto di vista numerico.

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LetX andY be two random numbers with the same distribution function; in this paper we consider the problem of finding a random numberX+Y having mode with minimal probability. In particular we have considered only the case ofX andY assuming the firstn integer values, so thatp (dimensionn) is the common distribution andq (dimension 2n–1) is the distribution ofX+Y; then the problem is to minimizem=maxq ₁.In the known literature it appears that theoretical results and numerical experience have brought to various conjectures not confermed. In this paper the problem is considered from the mathematical programming point of view. Several theoretical results are obtained even if the full solution of the problem is not reached. Anyway, such results, limiting the search range of a solution, suggested extended numerical testing, also for rather large values ofn, so that non trivial conclusions can be derived.

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Pervenuto il 22-1-82     

Some remarks on stochastic dominance

The present work proposes a definition of dominance (dominance in the strict sense), which is weaker than first order stochastic dominance, stating precisely that the r.v.Y dominatesX (XY) if Pr(YX)=1.Such a dominance in the strict sense is then compared with first and second order stochastic dominance and with dominance between descisions of the same decision problem summarised in a table of results, arriving at certain general remarks about decision problems and the choice between r.v.'s. Indications are also given about how it is possible to obtain simple and useful bounds for Pr(YX).

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Riassunto Nel presente lavoro si propone una definizione di dominanza (dominanza in senso stretto) più debole della dominanza stocastica del prim'ordine, precisamente dicendo che la v.a.Y dominaX (XY) se Pr(YX)=1.Si confronta poi tale dominanza in senso stretto con le dominanze stocastiche del primo e del secondo ordine e con la dominanza tra decisioni di uno stesso problema di decisione sintetizzato in una tabella dei risultati giungendo ad alcune precisazioni generali sui problemi di decisione e di scelta tra v.a. Si danno anche indicazioni su come sia possibile ottenere limitazioni per la Pr(YX).

     

Regression analysis and dependence

In this paper we study the relationship between regression analysis and a multivariate dependency measure. If the general regression model Y=f() holds for some function f, where 1i₁< i₂<···i_m k, and X₁,...,X_k is a set of possible explanatory random variables for Y. Then there exists a dependency relation between the random variable Y and the random vector (). Using the dependency statistic defined below, we can detect such dependency even if the function f is not linear. We present several examples with real and simulated data to illustrate this assertion. We also present a way to select the appropriate subset among the random variables X₁,X₂,...,X_k, which better explain Y.     

Estimation of P[Y < X] for generalized exponential distribution

This paper deals with the estimation of P[Y < X] when X and Y are two independent generalized exponential distributions with different shape parameters but having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is obtained. The asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are obtained. Different confidence intervals are proposed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a simulated data set has also been presented for illustrative purposes.Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council     

On the Ratio of Fréchet Random Variables

The distribution of the ratio X/Y is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.     

A new measure of association between random variables

We propose a new measure of association between two continuous random variables X and Y based on the covariance between X and the log-odds rate associated to Y. The proposed index of correlation lies in the range [\(-1\), 1]. We show that the extremes of the range, i.e., \(-1\) and 1, are attainable by the Fr\(\acute{\mathrm{e}}\)chet bivariate minimal and maximal distributions, respectively. It is also shown that if X and Y have bivariate normal distribution, the resulting measure of correlation equals the Pearson correlation coefficient \(\rho \). Some interpretations and relationships to other variability measures are presented. Among others, it is shown that for non-negative random variables the proposed association measure can be represented in terms of the mean residual and mean inactivity functions. Some illustrative examples are also provided.     

Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test

Summary For independents-variate samplesX ₁, ...,X _m i.i.d.f. (.),Y ₁, ...,Y _n i.i.d. g. (.), where the densitiesf (.),g (.) are assumed to be continuous on their respective sets of positivity, consider the numberT _m,n of pointsZ of the pooled sample (which are either of typeX or of typeY) such that the nearest neighbor ofZ is of the same type asZ. We show that, as , independently of (.). An omnibus test for the two sample problem f(.)g(.) orf(.)g(.)? may be obtained by rejecting the hypothesisf(.)g(.) for large values ofT _m,n.     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Modelling dependence using skew t copulas: Bayesian inference and applications

We construct a copula from the skew t distribution of Sahu et al. ( 2003 ). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete‐valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modelling of regional spot prices in the Australian electricity market. Here, we observe complex non‐Gaussian margins and nonlinear inter‐regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modelling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi‐site advertising campaigns. In both cases the skew t copula substantially outperforms symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modelling tool when coupled with Bayesian inference. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring process mean level using auxiliary information

In this study, a Shewhart‐type control chart is proposed for the improved monitoring of process mean level (targeting both moderate and large shifts which is the major concern of Shewhart‐type control charts) of a quality characteristic of interest Y. The proposed control chart, namely the M_r chart, is based on the regression estimator of mean using a single auxiliary variable X. Assuming bivariate normality of (Y, X), the design structure of M_r chart is developed for phase I quality control. The comparison of the proposed chart is made with some existing control charts used for the same purpose. Using power curves as a performance measure, better performance of the proposedM_r chart is observed for detecting the shifts in mean level of the characteristic of interest.