Total Positivity Of Order Research Articles

Bivariate and Multivariate Weighted Kumaraswamy Distributions: Theory and Applications

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this paper, we derive the bivariate and multivariate weighted Kumaraswamy distributions via the construction method as discussed in B.C. Arnold, I. Ghosh, A. Alzaatreh, Commun. Stat. Theory Methods. 46 (2017), 8897–8912. Several structural properties of the bivariate weighted distributions including marginals, distributions of the minimum and maximum, reliability parameter, and total positivity of order two are discussed. We provide some multivariate extensions of the proposed bivariate weighted Kumaraswamy model. Two real-life data sets are used to show the applicability of the bivariate weighted Kumaraswamy distributions and is compared with other rival bivariate Kumaraswamy models.

A family of transformed copulas with a singular component

In this paper, we present a family of bivariate copulas obtained by transforming a given copula function by means of two increasing functions, named as transformed copula. One distinctive characteristic of the transformed copula is its singular component along the main diagonal. Conditions guaranteeing the transformed function to be a copula function are provided, and several classes of the transformed copulas are given. The singular component along the main diagonal of the transformed copula is verified, and the tail dependence coefficients of transformed copulas are obtained. Some properties of the transformed copula are discussed, such as the total positivity of order 2 and the concordance order. Finally, conditions for the proposed transformation being invariant are given, sufficient conditions guaranteeing that the iterative transformation is a copula function are provided, and the convergence of the iterative transformation is proved under some conditions.

Construction of bivariate and multivariate weighted distributions via conditioning

ABSTRACTWeighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this article, we propose an alternative method to construct a new family of bivariate and multivariate weighted distributions. For illustrative purposes, some examples of the proposed method are presented. Several structural properties of the bivariate weighted distributions including marginal distributions together with distributions of the minimum and maximum, evaluation of the reliability parameter, and verification of total positivity of order two are also presented. In addition, we provide some multivariate extensions of the proposed models. A real-life data set is used to show the applicability of these bivariate weighted distributions.

Ordinal aggregation results via Karlin's variation diminishing property

When is the weighted sum of quasi-concave functions quasi-concave? We answer this, extending an analogous preservation of the single-crossing property in QS: Quah and Strulovici (2012). Our approach develops a general preservation of n-crossing properties, applying the variation diminishing property in Karlin (1956). The QS premise is equivalent to Karlin's total positivity of order two, while our premise uses total positivity of order three: The weighted sum of quasi-concave functions is quasi-concave if each has an increasing portion more risk averse than any decreasing portion.

Ordinal Aggregation Results via Karlin's Variation Diminishing Property

When is the weighted sum of quasi-concave functions quasi-concave? We answer this, extending an analogous preservation of the single-crossing property in QS: Quah and Strulovici (2012). Our approach develops a general preservation of n-crossing properties, applying the variation diminishing property in Karlin (1956). The QS premise is equivalent to Karlin’s total positivity of order two, while our premise uses total positivity of order three: The weighted sum of quasi-concave functions is quasi-concave if each has an increasing portion more risk averse than any decreasing portion.

Exploring the total positivity of yields correlations

We test the plausibility of the total positivity assumption of interest rates changes recently introduced in order to justify the presence of shift, slope and curvature for yield curves. To this aim, we introduce and discuss a test of total positivity of order for covariance and correlation matrices. The explicit expressions of the test statistics are given for Gaussian samples and an extension to a distribution-free framework is made via a bootstrap method. After exploring with simulation the robustness of such tests, we show using real data how it is realistic to assume that correlation matrices of interest rates changes are totally positive of order two. Conclusions on total positivity of order three are more controversial.

Properties of a job search problem on a partially observable Markov chain in a dynamic economy

This paper observes a job search problem on a partially observable Markov chain, which can be considered as an extension of a job search in a dynamic economy in [1]. This problem is formulated as the state changes according to a partially observable Markov chain, i.e., the current state cannot be observed but there exists some information regarding what a present state is. All information about the unobservable state are summarized by the probability distributions on the state space, and we employ the Bayes' theorem as a learning procedure. The total positivity of order two, or simply TP 2, is a fundamental property to investigate sequential decision problems, and it also plays an important role in the Bayesian learning procedure for a partially observable Markov process. By using this property, we consider some relationships among prior and posterior information, and the optimal policy. We will also observe the probabilities to make a transition into each state after some additional transitions by empolying the optimal policy. In the stock market, suppose that the states correspond to the business situation of one company and if there is a state designating the default, then the problem is what time the stocks are sold off before bankrupt, and the probability to become bankrupt will be also observed.

Testing for positive association in contingency tables with fixed margins

An exact conditional approach is developed to test for certain forms of positive association between two ordinal variables (e.g. positive quadrant dependence, total positivity of order 2). The approach is based on the use of a test statistic measuring the goodness-of-fit of the model formulated according to the type of positive association of interest. The nuisance parameters, corresponding to the marginal distributions of the two variables, are eliminated by conditioning the inference on the observed margins. This, in turn, allows to remove the uncertainty on the conclusion of the test, which typically arises in the unconditional context where the null distribution of the test statistic depends on such parameters. Since the multivariate generalized hypergeometric distribution, which results from conditioning, is normally intractable, Markov chain Monte Carlo methods are used to obtain maximum likelihood estimates of the parameters of the constrained model. The Pearson's chi-squared statistics is used as a test statistic; a p-value for this statistic is computed through simulation, when the data are sparse, or exploiting the asymptotic theory based on the chi-bar squared distribution. The extension of the present approach to deal with bivariate contingency tables, stratified according to one or more explanatory discrete variables, is also outlined. Finally, three applications based on real data are presented.

Grade correspondence analysis applied to contingency tables and questionnaire data

An alternative approach methodology for Correspondence Analysis is presented. This approach, called Grade Correspondence Analysis (GCA), utilizes Spearman's rho to detect underlying associations and trends. Two examples are presented using: (1) a contingency table (Heuer's suicide data) with cause of death, gender, and age; and (2) a survey questionnaire (data matrix) concerning employment, personal economics, computer skills, and disability level of handicapped computer specialists in Poland. GCA uses a search strategy (multi-starts / starts) to detect trends (not forced to be orthogonal) among rows and columns. (A similar strategy permits the determination of significance levels.) Results are discussed using measures of the representativness of the trends, as well as measures of their regularity. Visualization of trends (as well as outlier trend detection) is via the concept of overrepresentation maps. Survey data may be measured on any non-negative scale. Meaningful disjoint aggregation (or division) of sub-populations and variables are possible. This paper is written for the practitioner and includes a concepts example in an appendix. There is also, however, an appendix with GCA theory relating to: grade distributions; local maxima of Spearman's rho and their representativness, regularity and regions of attraction; total positivity of order 2 (TP2); similarity measures; suitable random references for the determination of significance levels; and the application of GCA to non-negative data matrices.

A generalization of multivariate total positivity of order two with an application to a Bayesian learning procedure

This paper investigates to generalize the MTP2 (multivariate total positivity of order two), and deals with the Bayesian learning procedure for sequential decision problems on a partially observable Markov process as an application of this property. This property has an order preserving property for the expectation of a non-decreasing function. As for a partially observable Markov process, all informations are summarized by probability measures on the state space. Among informations, an order is induced by means of the generalized MTP2. We employ the Bayes’ theorem as the learning procedure, and investigate the sequential decision problems under two assumptions. There also exist some relationships between prior and posterior information. These properties are essential for observing the sequential decision problems on this process.

Preservation of certain dependent structures under bivariate homogeneous Poisson shock models

A bivariate Poisson shock model resulting from two devices receiving shocks from two independent sources is shown to preserve certain bivariate dependent structures such as total positivity of order 2 (TP 2), stochastic increasing (SI), right tail increasing (RTI) etc. However, when two devices are subjected to the same source of shocks it is observed through a counter example that some of these preservation results do not hold any more. In such cases sufficient conditions are given under which the bivariate random vector denoting the life lengths of two devices is shown to have the above-mentioned bivariate dependent structures.

Functional Characterizations of Some Positively Dependent Bivariate Random Vectors

In this paper, we consider three different notions of positive dependence for a bivariate random vector: (i) total positivity of order 2, (ii) stochastic increasingness, and (iii) positive quadrant dependence. By defining three classes of arrangement-increasing functions, we show that these three different notions can be unified by functional inequalities. Using these functional inequalities, we derive the relations between the positively dependent notions (i) and (iii) and their corresponding counterparts in stochastic majorization orderings. Moreover, these classes of functions also lead to equivalent characterizations of two random vectors with independent components ordered in the sense of likelihood ratio ordering and stochastic ordering.