Statistica Sinica 13(2003), 903-926

ESTIMATION OF DISTRIBUTION FUNCTIONS UNDER

SECOND ORDER STOCHASTIC DOMINANCE

Javier Rojo Jiménez and Hammou El Barmi

Rice University and City University of New York

Abstract: The concept of stochastic ordering as introduced by Lehmann (1955) plays a major role in the theory and practice of statistics, and a large body of existing statistical work concerns itself with the problem of estimating distribution functions $F$ and $G$ under the constraint that $F(x)\leq G(x)$ for all $x$. Nevertheless in economic theory, the weaker concept of second order stochastic dominance plays a prominent role in the general framework of analyzing choice under uncertainty by considering maximization of expected utilities. More specifically, an investment portfolio B with random return $Y$ dominates an investment portfolio A with random return $X$ if and only if $E(U(Y)) \geq E(U(X))$ for all increasing and concave utility functions $U$. This condition can be seen to be equivalent to the condition that the distribution functions of $X$ and $Y$ are ordered according to the second order stochastic dominance requirement. Here, a family of strongly uniformly consistent estimators for the survival functions under a second order stochastic dominance constraint is proposed in the one-sample and the two-sample problems. In the one-sample problem the new family of estimators dominate the empirical distribution function with respect to a certain class of loss functions. The asymptotic distributions of the estimators are explored and the new estimators are compared, via simulations, in terms of Mean Squared Error (MSE) with the empirical distribution. The case of right-censored data is also considered. Stocks and bonds data from 1810$-$1989 are used to illustrate the estimators.

Key words and phrases: Bonds, expected utility, investment portofolio, stocks, weak convergence.