6533b7cefe1ef96bd125787b

RESEARCH PRODUCT

Testing for selectivity in the dependence of random variables on external factors

subject

description

Random variables AA and BB, whose joint distribution depends on factors (x,y)(x,y), are selectively influenced by xx and yy, respectively, if AA and BB can be represented as functions of, respectively, (x,SA,C)(x,SA,C) and (y,SB,C)(y,SB,C), where SA,SB,CSA,SB,C are stochastically independent and do not depend on (x,y)(x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of AA may depend on yy. But parameters characterizing stochastic interdependence of AA and BB, such as their mixed moments, are generally functions of both xx and yy. We derive two simple necessary conditions for selective dependence of (A,B)(A,B) on (x,y)(x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x′x,x′ and y,y′y,y′, sxy≤sxy′+sx′y′+sx′y,sxy≤sxy′+sx′y′+sx′y, where sxy=E[|f(Axy,x)−g(Bxy,y)|p]1/p with arbitrary f,gf,g, and p≥1p≥1, and (Axy,Bxy)(Axy,Bxy) denoting (A,B)(A,B) at specific values of x,yx,y. For p=2(f(Axy,x),g(Bxy,y))p=2(f(Axy,x),g(Bxy,y)) this condition is superseded by a more restrictive one: |ρxyρxy′−ρx′yρx′y′|≤1−ρxy21−ρxy′2+1−ρx′y21−ρx′y′2, where ρxyρxy is the correlation between f(Axy,x)f(Axy,x) and g(Bxy,y)g(Bxy,y). For bivariate normally distributed (f(Axy,x),g(Bxy,y))(f(Axy,x),g(Bxy,y)) this condition, if satisfied on a 2×2 subset {x,x′}×{y,y′}{x,x′}×{y,y′}, is also sufficient for a selective dependence of (A,B)(A,B) on (x,y)(x,y) confined to this subset.