Determining Bounds on Expected Values of Certain Functions

Abstract

Let $\mathfrak{F}$ be the collection of cumulative distribution functions on $(- \infty, \infty)$ and $\mathfrak{F}_{\lbrack a, b\rbrack}$ that subset of $\mathfrak{F}$ all of whose elements have $F(a - 0) = 0$ and $F(b) = 1$. Let $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)} (\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack})$ be the class of cumulative distribution functions on $(- \infty, \infty) (\lbrack a, b\rbrack)$ whose first $k$ moments are $\mu_1, \mu_2, \cdots, \mu_k$ respectively. We will suppose throughout that $\mu_1, \mu_2, \cdots, \mu_k$ is a legitimate moment sequence, i.e., that there exists a cumulative distribution function $F(x) \varepsilon \mathfrak{F} (\mathfrak{F}_{\lbrack a, b\rbrack})$ whose first $k$ moments are $\mu_1, \mu_2, \cdots, \mu_k$. Let $g(x)$ be a continuous and bounded function on $\lbrack a, b\rbrack$. Then, we wish to determine $F^\ast (x) \varepsilon \mathscr{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack}$ with \begin{equation*}\tag{1}\int^b_a g(x) dF^\ast (x) = \min (\max)_{F\varepsilon \mathfrak{F}^{{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack}}} \int^b_a g(x) dF(x).\end{equation*} Any $F^\ast (x)$ satisfying (1) will be called an extremal distribution with respect to $g(x)$. Let $\mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$ be the set of continuous, bounded, and monotonic functions on $\lbrack a, b\rbrack$, whose first $k$ derivatives exist and are monotonic in $(a, b)$. In addition, we further require that $\mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$ contain only functions not linearly dependent on the monomials $1, x, x^2, \cdots, x^k$. This paper characterizes the extremal distributions for $g(x) \varepsilon \mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$. The results are extended to $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack 0, \infty)}$ and $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}$, in that we investigate determining $\inf (\sup)_F \varepsilon\mathscr{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack 0, \infty)} \int^\infty_0 g(x) dF(x) \text{and} \inf (\sup)_F \varepsilon \mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)} \int^\infty_{-\infty} g(x) dF(x).$ These results are then applied to the computation of bounds on the moment generating function, knowing the first $k$ moments, in some specific cases. The methodology is largely a straightforward extension of results in an earlier paper by the author [1].