\documentclass[10pt]{book}
\usepackage{array}
\textwidth=31.9pc
\textheight=46.5pc
\oddsidemargin=1pc
\evensidemargin=1pc
\headsep=15pt
%\headheight=.2cm
\topmargin=.6cm
\parindent=1.7pc
\parskip=0pt

\def\author{\centerline}

\setcounter{page}{1}
\begin{document}
\pagestyle{empty}

\footnotesize
\noindent Statistica Sinica {\bf 13}(2003), 903-926

\vspace{3pc} \normalsize
\centerline{\large\bf ESTIMATION OF DISTRIBUTION FUNCTIONS UNDER}
\vspace{2pt}
\centerline{\large\bf SECOND ORDER STOCHASTIC DOMINANCE}
\vspace{.25cm}
\author{Javier Rojo Jim\'{e}nez and Hammou El Barmi}
\vspace{.4cm}
\centerline{\it Rice University and City University  of New York}
\vspace{.55cm}

\begin{quotation}
\noindent {\it 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.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Bonds, expected utility, investment portofolio, stocks, weak convergence.\par
\end{quotation}
\end{document}
