Moments of the product and ratio of two correlated chi-square variables

The exact probability density function of a bivariate chi-square distribution with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution.      

Uniform Tail Asymptotics for the Sum of Two Correlated Classes with Stochastic Returns and Dependent Heavy Tails

Consider a continuous-time risk model with two correlated classes of insurance business and risky investments whose price processes are geometric Lévy processes. By assuming that the correlation comes from a common shock, and the claim sizes are heavy-tailed and pairwise quasi-asymptotically independent, we investigate the tail behavior of the sum of the stochastic present values of the two correlated classes, and a uniform asymptotic formula is obtained.     

Tail behavior for the sum of two correlated classes of discounted aggregate claims in a time-dependent risk model

Consider a continuous-time risk model with two correlated classes of insurance business and a constant force of interest. Suppose that the correlation comes from a common shock, and that the claim sizes and inter-arrival times correspondingly form a sequence of random pairs, with each pair obeying a dependence structure. By assuming that the claim sizes are heavy tailed, a uniform tail asymptotic formula for the sum of the two correlated classes of discounted aggregate claims is obtained.     

Some sampling effects of pairwise correlated observations on likelihood ratio tests for the difference between two means

We study the effects of the inclusion of pairs of correlated observations in a sample on likelihood ratio tests for the difference in two means. In particular, we assess how the inclusion of correlated data pairs (e.g., such as data inadvertently collected from sib-pairs) affects the sample size requirements necessary for the implementation of a Likelihood Ratio (LR) test for the difference between two means. Our results suggest that correlated data pairs beneficially or adversely effect sample size requirements for an LR test to a degree functionally related to the mixture parameters dictating their relative frequencies in the larger sample on which the test will be performed, the strength of the correlation between the observations, and the size of imbalances in the sample with respect to the number of observations in each group. The relevance and implications of the results for genetic and epidemiologic research are discussed.     

A note on the correlation betweenS 2 and the least squares estimator in the linear regression model

In the following we consider the correlation betweenS ² and the least squares estimator in the linear regression model. We are interested in situations where these two statistics are uncorrelated though the errors are correlated. Conditions are developed without normality assumption, only assuming finite fourth moments of the error distributions. Support by Deutsche Forschungsgemeinschaft Grant No. Tr 253/1-2 is gratefully acknowledged.     

Approximated distributions of the weighted sum of correlated chi-squared random variables

We study the asymptotic behavior of the weighted sum of correlated chi-squared random variables. Both chi-squared and normal distributions are proved to approximate the exact distribution. These two approximations are established by matching the first two cumulants. Simulation comparison is made to study the performance of two approximations numerically. We find that the chi-squared approximation performs better than the normal one in the study.     

Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80?years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein’s method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.     

Performance Evaluation Based on the Robust Mahalanobis Distance and Multilevel Modeling Using Two New Strategies

In this article, we propose a general framework for performance evaluation of organizations and individuals over time using routinely collected performance variables or indicators. Such variables or indicators are often correlated over time, with missing observations, and often come from heavy-tailed distributions shaped by outliers. Two new double robust and model-free strategies are used for evaluation (ranking) of sampling units. Strategy 1 can handle missing data using residual maximum likelihood (RML) at stage two, while strategy two handles missing data at stage one. Strategy 2 has the advantage that overcomes the problem of multicollinearity. Strategy one requires independent indicators for the construction of the distances, where strategy two does not. Two different domain examples are used to illustrate the application of the two strategies. Example one considers performance monitoring of gynecologists and example two considers the performance of industrial firms.     

An Analysis of Two-Dimensional Agricultural Field Trials in the Presence of Outliers and Fertility Jumps

This paper proposes a model for outliers and fertility jumps in agricultural field trials when the fertility is correlated in two directions. Uncertainty concerning the occurrence of these anomalies can be substantial, especially when outliers occur near a fertility jump. The proposed estimation technique can take this uncertainty into account rather than condition on a particular set of anomalies. Diagnostics for the detection of these anomalies are also discussed. The procedures are illustrated with a wheat trial.     

Testing the difference between spectral densities of two independent periodically correlated (cyclostationary) time series models

In this work, the asymptotic distribution for the discrete Fourier transform of periodically correlated (PC) processes is applied to test the equality of two PC time series. Then the performance of the proposed method is investigated through the Monte Carlo simulations.     

An improved alternative estimation procedure for current population mean in presence of positively and negatively correlated auxiliary variables in two-occasion rotation patterns

Abstract

The present study confirms the influential role of a positively and a negatively correlated auxiliary variables in enhancing the precision of estimates of current population mean in two occasion rotation (successive) sampling. Exponential-type estimators of current population mean have been proposed for three different situations: (i) the information on a positively correlated auxiliary variable is readily available on both occasions (ii) the information on a negatively correlated auxiliary variable is readily available on both occasions and (iii) the information on both positively and negatively correlated auxiliary variables are readily available on both the occasions. The characteristics of the proposed estimators have been explored and their efficacious performances are compared with the natural and recent contemporary estimators. Optimum replacement strategies of the proposed estimation procedures have been formulated. Simulation and empirical studies are carried out to justify the proposition of the proposed estimators and appropriate recommendations have been put forward to the survey practitioners.     

A Note on the Distribution of Linear Functions of Two Ordered Correlated Normal Random Variables

We consider the problem of finding the distribution of linear functions of two ordered correlated normal random variables. We derive some distributional properties for these linear statistics and briefly discuss the use of them in location estimation. The connection of the subject with the skew normal distribution is also noted.     

Expectations of the maximum of sums and the sum of maxima for correlated normal variables

This paper compares expectations of the maximum of sums and the sum of maxima for correlated normal variables. General formulas are given for the expected differences and relative expected differences. Numerical values are computed for the special case of a correlated structure with common variance and covariance to illustrate the general magnitude of the differences which may occur. It is numerically shown that an approximation to the relative differences based on the extreme value distribution is very good. General correlated models with location-scale parameters are also considered. This result should be useful in application. Examples are discussed.     

Sampling strategy for optimal classification into one of two correlated normal populations

A unit ω is to be classified into one of two correlated homoskedastic normal populations by linear discriminant function known as W classification statistic [T.W. Anderson, An asymptotic expansion of the distribution of studentized classification statistic, Ann. Statist. 1 (1973), pp. 964–972; T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd edn, Wiley, New York, 1984; G.J. Mclachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley and Sons, New York, 1992]. The two populations studied here are two different states of the same population, like two different states of a disease where the population is the population of diseased patient. When a sample unit is observed in both the states (populations), the observations made on it (which form a pair) become correlated. A training sample is unbalanced when not all sample units are observed in both the states. Paired and also unbalanced samples are natural in studies related to correlated populations. S. Bandyopadhyay and S. Bandyopadhyay [Choosing better training sample for classifying an individual into one of two correlated normal populations, Calcutta Statist. Assoc. Bull. 54(215–216) (2003), pp. 167–180] studied the effect of unbalanced training sample structure on the performance of W statistics in the univariate correlated normal set-up for finding optimal sampling strategy for a better classification rate. In this study, the results are extended to the multivariate case with discussion on application in real scenario.     

Leverage,influence and residuals in regression models when observations are correlated

This paper considers the concepts of leverage and influence in the linear regression model with correlated errors when the error covariance structure is completely specified. Generalizations of the usual measures are given. Extensions of residuals also naturally arise. The theory is illustrated using two examples     

A GEE approach to estimating accuracy and its confidence intervals for correlated data

In this paper, we provide a method for constructing confidence interval for accuracy in correlated observations, where one sample of patients is being rated by two or more diagnostic tests. Confidence intervals for other measures of diagnostic tests, such as sensitivity, specificity, positive predictive value, and negative predictive value, have already been developed for clustered or correlated observations using the generalized estimating equations (GEE) method. Here, we use the GEE and delta‐method to construct confidence intervals for accuracy, the proportion of patients who are correctly classified. Simulation results verify that the estimated confidence intervals exhibit consistent/appropriate coverage rates.     

Moments for a ratio of correlated gamma variates

This paper considers the finite integral moments for the ratio, R = X/Y, where X and Y re correlated gamma distributed variables. An analytical and numerical comparison is given for two classes of underlying bivariate gamma distributions. It is shown that the two bivariate gamma structures provide indentical experessions for the mth unadjussted moment, E(R^m), if and only if either of the following conditions hold : 1) X and Y are uncorrelated of 2) m=1. A numerical evaluation is performed to determine the extent that the two methods differ whenever the variables are correlated     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

Process capability index for bivariate exponentially distributed quality characteristics and its sampling properties

In this paper, a process capability index for two correlated quality characteristics jointly following bivariate exponential distribution has been proposed. The expectation and sampling variance of the estimated index have been derived. Choice of the natural process interval corresponding to a specified coverage probability has been discussed.     

Interval estimation of the joint survival function for successive duration times under left truncation and right censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, we propose two inverse-probability-weighted (IPW) estimators for the estimation of the joint survival function of two successive duration times. One of them is similar to the estimator proposed by Chang and Tzeng [Nonparametric estimation of sojourn time distributions for truncated serial event data – a weight adjusted approach, Lifetime Data Anal. 12 (2006), pp. 53–67]. The other is the extension of the nonparametric estimator proposed by Wang and Wells [Nonparametric estimation of successive duration times under dependent censoring, Biometrika 85 (1998), pp. 561–572]. The weak convergence of both estimators are established. Furthermore, the delete-one jackknife and simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to compare the two IPW approaches.