Két pszichológiai populáció sztochasztikus egyenlőségének ellenőrzésére alkalmas statisztikai próbák összehasonlító vizsgálata

A jelen tanulmányban a sztochasztikus egyenlőség ellenőrzésére alkalmas hat statisztikai próbát hasonlítottunk össze számítógépes szimulációval az érvényesség és a hatékonyság kritériuma szempontjából. Két populációt akkor mondunk sztochasztikusan egyenlőnek valamely X változó tekintetében, ha véletlenszerűen kiválasztva egy-egy X-értéket a két populációból, az elsőből kiválasztott érték ugyanakkora eséllyel lesz nagyobb a második kiválasztottnál, mint kisebb.A szimulációban széles tartományban variáltuk az eloszlások ferdeségét és csúcsosságát, valamint a szórásheterogenitás mértékét. Vizsgáltunk kicsi és közepes nagyságú, illetve egyenlő és különböző elemszámú mintákat. A szimulációba a korábban már mások által is javasolt próbák (rang t, rang Welch, Fligner-Policello, Cliff) mellett elméleti megfontolások alapján két új próbát (FPW és FPCW) is bevontunk.A szimulációs vizsgálatok arra az érdekes eredményre vezettek, hogy az újonnan javasolt két próba, FPW és FPCW az érvényesség tekintetében sokkal megbízhatóbb eljárásnak bizonyult, mint a többiek, miközben az erő tekintetében nem tapasztaltunk számottevő különbséget közöttük. Különösen FPW jeleskedett azzal, hogy I. fajta hibája sosem tért el számottevően a névleges szinttől. Közepes nagyságú minták esetén FPCW FPW-vel egyenértékű eljárás benyomását keltette.In the current paper six statistical tests of stochastic equality are to be compared by a Monte Carlo simulation with respect to Type I error and power. Two populations are said to be stochastically equal with respect to a variable X, if for any two independently and randomly drawn observations X1 and X2 from the two populations P(X1 ≯ X2) = P(X1 < X2).In the simulation the skewness and kurtosis levels as well as the extent of variance heterogeneity of the two parent distributions were varied across a wide range. The sample sizes applied were either small or moderate, and equal or unequal. The involved tests of stochastic equality were as follows: rank t test, rank Welch test, Fligner-Policello test, Cliff's modified Fligner-Policello test as well as two modifications of the last two tests, denoted FPW and FPCW, that utilized adjusted degrees of freedom.An interesting result obtained is that the two newly introduced test variants, FPW and FPCW, proved to be substantially more accurate with regard to their Type I error rates than the others, whereas they kept a similar power level. Specifically, the estimated Type I error of FPW at .05 nominal level always fell in the range of .043-.063 even if the variance ratio of the two distributions was as large as 1:16. The same ranges were .049-.068 for FPCW, but .029-.160 for the rank t test, .049-.096 for the rank Welch test, .035-.075 for the Fligner-Policello test, and .040-.078 for Cliff's test.

Multivariate Convex Orderings, Dependence, and Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Multivariate Convex Orderings, Dependence, and Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Characterization of life distributions under some generalized stochastic orderings

In this paper we investigate the characterizations of life distributions under four stochastic orderings, &lt; p , &lt; (p), &lt; (p) and &lt; L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings &lt; p and &lt; (p) and their moment generating functions under orderings &lt; p and &lt; (p). Under ordering &lt; L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings &lt; (p), &lt; (p) and &lt; L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

Characterization of life distributions under some generalized stochastic orderings

In this paper we investigate the characterizations of life distributions under four stochastic orderings, < p, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings < p and < (p) and their moment generating functions under orderings < p and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.