Statistical Properties of the Number of Positive Sums

Abstract

Let $X_1, \cdots, X_n$ be independent random variables having a common, continuous distribution function (df) $F$ and define \begin{align*}\tag{1.1a}S_k &= S_k(X) = \sum^k_{j = 1} X_j,\quad k = 1, \cdots, n, \\ \tag{1.1b}S_V &= S_V(X) = \sum_{j\varepsilon V} X_j,\quad \varnothing \neq V \subseteq \{1, \cdots, n\}, \\ \tag{1.1c}M_n &= M_n(X) = \sum^n_{k = 1} e(S_k), \\ \tag{1.1d}N_n &= N_n(X) = \sum_{V \neq \varnothing} e(S_v),\\ \end{align*} where $X = (X_1, \cdots, X_n), e$ is the indicator function of $(0, \infty)$, and $\varnothing$ denotes the empty set. Recently, Kraft and van Eeden [9] have pointed out that since \begin{align*}\tag{1.2a}P(M_n = k) &= 4^{-n}\binom{2k}{k}\binom{2n - 2k}{n - k},\quad k = 0, \cdots, n, \\ \tag{1.2b}P(N_n = k) &= 2^{-n},\quad k = 0, \cdots, 2^n - 1, \\ \end{align*} if $F$ is symmetric about zero (in the sense that $F(x) = 1 - F(-x), -\infty < x < \infty)$, both $M_n$ and $N_n$ may be used to test the hypothesis $H_0$ which specifies that $F$ is symmetric about zero. They also considered the consistency of such tests. The present paper gives some further sufficient conditions for the consistency of tests based on $M_n$ and $N_n$ and computes a measure of their asymptotic relative efficiency with respect to each other. The latter, of course, involves finding the asymptotic distributions of $M_n$ and $N_n$ under a sequence of local alternatives. In a final section the asymptotic properties of some confidence intervals and point-estimates based on $M_n$ and $N_n$ are considered. The alternatives of interest specify that $X_1$ is stochastically larger than a symmetric random variable in the sense that \begin{equation*}\tag{1.3}F(x) \leqq G(x),\quad -\infty < x < \infty, F \neq G,\end{equation*} for some $G$ which is symmetric about zero. The tests considered will be denoted by $\varphi_n$ and $\delta_n$ and reject for large values of $M_n$ and $N_n$ respectively. It should be noted that (1.2a) does not require the continuity of $F$; in fact, none of our results in Sections 2 and 3 which concern only $M_n$ or $\varphi_{nd}$ do.