Nonparametric Tests for Positive Quadrant Dependence

We consider distributional free inference to test for positive quadrant dependence, i.e. for the probability that two variables are simultaneously small (or) large being at least as great as it would be were they dependent. Tests for its generalisation in higher dimensions, namely positive orthant dependences, are also analysed. We propose two types of testing procedures. The first procedure is based on the specification of the dependence concepts in terms of distribution functions, while the second procedure exploits the copula representation. For each specification a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US and Danish insurance claim data. Practical implications for the design of reinsurance treaties are also discussed.

[1]  LevyHaim.  Stochastic Dominance and Expected Utility , 1992 .

[2]  Harshinder Singh,et al.  Testing for Second-Order Stochastic Dominance of Two Distributions , 1994, Econometric Theory.

[3]  Takemi Yanagimoto,et al.  Partial orderings of permutations and monotonicity of a rank correlation statistic , 1969 .

[4]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[5]  Armand M. Makowski,et al.  Multidimensional Stochastic Ordering and Associated Random Variables , 1989, Oper. Res..

[6]  H. Levy.  Stochastic dominance and expected utility: survey and analysis , 1992 .

[7]  Hans U. Gerber.  Life Insurance Mathematics , 1990 .

[8]  工藤 昭夫,et al.  A Multivariate Analogue of the One-Sided Testについての一注意 (多次元統計解析の数理的研究) , 1979 .

[9]  C. Newman.  Asymptotic independence and limit theorems for positively and negatively dependent random variables , 1984 .

[10]  Roger Kaufmann,et al.  Introduction to Dynamic Financial Analysis , 2001 .

[11]  Russell Davidson,et al.  Distribution-Free Statistical Inference with Lorenz Curves and Income Shares , 1983 .

[12]  Franz C. Palm,et al.  Wald Criteria for Jointly Testing Equality and Inequality , 1986 .

[13]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[14]  Frank A. Wolak,et al.  Local and Global Testing of Linear and Nonlinear Inequality Constraints in Nonlinear Econometric Models , 1989, Econometric Theory.

[15]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[16]  D. Andrews.  Estimation When a Parameter is on a Boundary , 1999 .

[17]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[18]  B. Arnold.  Majorization and the Lorenz Order: A Brief Introduction , 1987 .

[19]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[20]  Michel Denuit,et al.  Does Positive Dependence between Individual Risks Increase Stop-Loss Premiums? , 2001 .

[21]  O. Scaillet.  Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall , 2004 .

[22]  Marco Scarsini,et al.  Stochastic Orders and Applications: A Classified Bibliography , 1993 .

[23]  R. Davidson,et al.  Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality , 1998 .

[24]  P. Embrechts,et al.  Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls , 2002 .

[25]  Jean Pinquet,et al.  Designing Optimal Bonus-Malus Systems from Different Types of Claims , 1998, ASTIN Bulletin.

[26]  D. Pollard.  New Ways to Prove Central Limit Theorems , 1985, Econometric Theory.

[27]  Y. L. Tong.  The multivariate normal distribution , 1989 .

[28]  P. Embrechts,et al.  Correlation and Dependency in Risk Management , 2002 .

[29]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[30]  Marc Goovaerts,et al.  Dependency of Risks and Stop-Loss Order , 1996, ASTIN Bulletin.

[31]  Peter C. Fishburn,et al.  Continua of stochastic dominance relations for bounded probability distributions , 1976 .

[32]  D. J. Bartholomew,et al.  A TEST OF HOMOGENEITY FOR ORDERED ALTERNATIVES. II , 1959 .

[33]  Emiliano A. Valdez,et al.  1 UNDERSTANDING RELATIONSHIPS USING COPULAS * , 1998 .

[34]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[35]  Stuart A. Klugman,et al.  Fitting bivariate loss distributions with copulas , 1999 .

[36]  A. Kudô,et al.  A multivariate analogue of the one-sided test , 1963 .

[37]  Stephen G. Donald,et al.  Consistent Tests for Stochastic Dominance , 2003 .

[38]  F. Wolak.  Testing inequality constraints in linear econometric models , 1989 .

[39]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[40]  Donald W. K. Andrews,et al.  Empirical Process Methods in Econometrics , 1993 .

[41]  R. Nelsen.  An Introduction to Copulas , 1998 .

[42]  Tomasz Rolski,et al.  Order relations in the set of probability distribution functions and their applications in queueing theory , 1976 .

[43]  A. McNeil.  Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory , 1997, ASTIN Bulletin.

[44]  D. McFadden.  Testing for Stochastic Dominance , 1989 .

[45]  G. Dionne,et al.  Optimal Financial Portfolio and Dependence of Risky Assets , 2000 .

[46]  A. Tchen.  Inequalities for distributions with given marginals , 1976 .

[47]  Michel Denuit,et al.  Premium calculation with dependent time-until-death random variables : the widow's pension , 1999 .

[48]  Antonio Forcina,et al.  Inference for Lorenz curve orderings , 1999 .

[49]  C. Gouriéroux,et al.  Likelihood Ratio Test, Wald Test, and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Parameters , 1982 .

[50]  G. Kimeldorf,et al.  Positive dependence orderings , 1987 .