Preservation of dependence concepts under bivariate weighted distributions

ABSTRACT

In this paper, we provide conditions under which some bivariate dependence structures are preserved under bivariate weighted distributions. Bivariate weighted distributions whose dependence structure is the same as the original distribution are characterized. Finally, we discuss some examples to show the usefulness of our results.     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

Note on the Problem of Class—Size Distributions

It is demonstrated that probabilities for the distribution of the sample correlation coefficient, such as those given by David [2] can be quickly and accurately calculated on modern desk calculators.     

A new measure in local influence

The concept of local influence, based on the curvature of the log-likelihood surface and introduced by Cook in 1986, has suffered by not having well-established criteria for deciding whether individual cases are important in this respect or not. Here a new method for measuring the closeness of an individual case to the direction of maximum curvature is presented and discussed.     

A measure of general functional dependence between two continuous variables

Existing measures in the literature that are specifically concerned with testing and measuring independence between two continuous variables are all based on examining the definition of independence, i.e., F_XY(x, y) = F_X(x)F_Y(y). A new measure is constructed uniquely in this paper that uses the absolute value of first difference on adjacent ranks of one variable with respect to the other. This measure captures the degree of functional dependence attributable to the amount of randomness and the complexity of the underlying bivariate dependence structure in a commensurate way that existing coefficients are incapable of. As a test statistic of independence, this measure is shown to have comparable or better power than existing statistics against a wide range of alternative hypotheses that consist of functional and multivalued relational dependence with additive noise.     

A class of weighted dependence measures for bivariate failure time data

This paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow-up region. The class consists of measures that are weighted averages of local dependence measures, and includes the cross-ratio-measure and finite region version of Kendall's τ; recently proposed by the authors. Two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators. Nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated. Properties of selected estimators are evaluated in a simulation study, and the method is illustrated through an analysis of Australian Twin Study data.     

A bivariate generalized linear model with an application to meteorological data analysis

A bivariate generalized linear model is developed as a mixture distribution with one component of the mixture being discrete with probability mass only at the origin. The use of the proposed model is illustrated by analyzing local area meteorological measurements with constant correlation structure that incorporates predictor variables. The Monte Carlo study is performed to evaluate the inferential efficiency of model parameters for two types of true models. These results suggest that the estimates of regression parameters are consistent and the efficiency of the inference increases for the proposed model for ρ≥0.50 especially in larger samples. As an illustration of a bivariate generalized linear model, we analyze a precipitation monitoring data of adjacent local stations for Tokyo and Yokohama.     

“Weak” stochastic orderings and dependence measures

Many notions of dependence rely upon orderings of random pairs. These orderings are generally partial orders, and thus there are many pairs of random vectors which are not comparable. By using a weakened version of stochastic dominance, many new orderings, as well as corresponding dependence measures, are created. The application to stock market data is explored.     

On a bivariate Kumaraswamy type exponential distribution

ABSTRACT

This paper considers a class of absolutely continuous bivariate exponential distributions whose univariate margins are the ordinary exponential distributions. We study different mathematical properties of the proposed model. The estimation of the parameters by maximum likelihood is discussed. Application is made to a real data example to illustrate the flexibility of theproposed distribution for data analysis.     

Modelling conditional distributions in bivariate survival

Conditional distributions for bivariate survival can be obtained via a model for the joint distribution, or, as has sometimes been suggested, by modelling the conditioned variable directly, with the conditioning variable included as a covariate. A quantitative comparison of estimated covariate effects and predictive distributions under the two approaches is given. The results are illustrated in a novel frailty application.     

Investigating dependence between frequency and severity via simple generalized linear models

Recently, a body of literature proposed new models relaxing a widely-used but controversial assumption of independence between claim frequency and severity in non-life insurance rate making. This paper critically reviews a generalized linear model approach, where a dependence between claim frequency and severity is introduced by treating frequency as a covariate in a regression model for severity. As an extension of this approach, we propose a dispersion model for severity. For this model, the information loss caused by using average severity rather than individual severity is examined in detail and the parameter estimators suffering from low efficiency are identified. We also provide analytical solutions for the aggregate sum to help rate making. We show that the simple functional form used in current research may not properly reflect the real underlying dependence structure. A real data analysis is given to explain our analytical findings.     

On the first time of ruin in two-dimensional discrete time risk model with dependent claim occurrences

This article is concerned with a two-dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present numerical results on this model by comparing the corresponding results of the bivariate compound binomial risk model.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.     

Association of progressively Type-II censored order statistics

The association of progressively Type-II censored order statistics from a sample of associated random variables _X1,…,_Xn$X_1,\dots ,X_n$ is established. Moreover, some bivariate dependence properties are discussed for independent but not necessarily identically distributed _X1,…,_Xn$X_1,\dots ,X_n$.     

A new measure of association

This paper presents a new measure of association. It is applicable to polytomies of either categorical or numerical type. It has the desirable property of being 0 if and only if the polytomies are independent. Its properties are studied and compared to those of existing measures. An interpretation of it is given. One situation where it is particularly useful is in measuring the ability to predict one polytomy given knowledge of the other. An example is given where the proposed measure is more relevant in describing the degree of association between two polytomies than are any of the existing measures. The corresponding sample quantity is presented and its asymptotic properties are studied. A discussion of its use in inference is given. The test for independence based on this measure is contrasted with the chi-square test.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Estimation and Comparison of Signed Symmetric Covariation Coefficient and Generalized Association Parameter for Alpha-stable Dependence Modeling

In this paper, we study the estimators of two measures of dependence: the signed symmetric covariation coefficient (scov) proposed by Garel and Kodia and the generalized association parameter (g.a.p.) put forward by Paulauskas. In the sub-Gaussian case, the scov and the g.a.p. coincide. The estimator of the scov proposed here is based on fractional lower-order moments. The estimator of the g.a.p. is based on estimation of a stable spectral measure. We investigate the relative performance of these estimators by comparing results from simulations.     

A new rank correlation measure

A new rank correlation measure β _n is proposed, so as to develop a nonparametric test of independence for two variables. β _n is shown to be the symmetrized version of a measure earlier proposed by Borroni and Zenga (Stat Methods Appl 16:289–308, 2007). More specifically, β _n is built so that it can take the opposite sign, without changing its absolute value, when the ranking of one variable is reversed. Further, the meaning of the population equivalent of β _n is discussed. It is pointed out that this latter association measure vanishes not only at independence but, more generally, at indifference, that is when the two variables do not show any “tendency” to positive or negative dependence. The null distribution of β _n needs an independent study: hence, the finite null variance and a table of critical values are determined. Moreover, the asymptotic null distribution of β _n is derived. Finally, the performance of the test based on β _n is evaluated by simulation. β _n is shown to be a good competitor of some classical tests for the same problem.     

Reliability analysis of systems with components having two dependent subcomponents

In this article, a system that consists of n independent components each having two dependent subcomponents (A_i, B_i), i = 1, …, n is considered. The system is assumed to compose of components that have two correlated subcomponents (A_i, B_i), and functions iff both systems of subcomponents A₁, A₂, …, A_n and B₁, B₂, …, B_n work under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented.