The Annals of Mathematical Statistics

Determining Bounds on Expected Values of Certain Functions

Bernard Harris

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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].

Article information

Source
Ann. Math. Statist., Volume 33, Number 4 (1962), 1454-1457.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177704377

Digital Object Identifier
doi:10.1214/aoms/1177704377

Mathematical Reviews number (MathSciNet)
MR141182

Zentralblatt MATH identifier
0114.34804

JSTOR
links.jstor.org

Citation

Harris, Bernard. Determining Bounds on Expected Values of Certain Functions. Ann. Math. Statist. 33 (1962), no. 4, 1454--1457. doi:10.1214/aoms/1177704377. https://projecteuclid.org/euclid.aoms/1177704377


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