Limiting Distributions of Some Variations of the Chi-Square Statistic

Abstract

The interarrival times past a point at the intersection of cars that queued up during the stop phase were considered by the authors of [1]. Denoting by $\{T_i\}, i = 1,2, \cdots$, the process of these consecutive interarrival times or time headways, an examination of experimental data indicated that the random variables $\{T_{n_o+i}, i = 1,2, \cdots\}$ were independently identically distributed. For $n_0 = 2$ the $\chi^2$-test for homogeneity showed this hypothesis to be tenable. Therefore the data were pooled to estimate the parameters of a suitable density function. As a final step, each of the subpopulations was tested, using the $\chi^2$-statistic, against the population whose parameters were determined using the pooled data. Two methods of estimation, namely, the modified minimum $\chi^2$ and the method of maximum likelihood, were considered. In each case, this paper shows that these $\chi^2$-statistics do not have the usual limiting $\chi^2$-distribution, but are stochastically larger than would be expected under the $\chi^2$-theory. More generally and precisely, let $\Pi_i, i = 1,2, \cdots, k$ populations and let the mathematical form of $\Pi_i$ be known except for the unknowns $\mathbf{\alpha}_i = (\alpha_{1i}, \alpha_{2i}, \cdots, \alpha_{si})$. Assume that the hypothesis $H_0:\Pi_1 = \Pi_2 = \cdots = \Pi_k$ is true, and also that a random sample of size $n_i$ from the $i$th population $(i = 1,2, \cdots, k)$ is available. The authors propose to treat in a subsequent communication a more complicated case in which the samples are tested separately, initially, and the null hypothesis merely fails to be rejected (so that it is not known whether or not it is true). Let $\mathbf{\alpha}^{(N)}$ and $\mathbf{\alpha}^{\ast(N)}$ be respectively the modified minimum $\chi^2$ and maximum likelihood estimates of $\mathbf{\alpha} = \mathbf{\alpha}_1 = \cdots = \mathbf{\alpha}_k$ based on the pooled sample of size $N = n_1 + n_2 + \cdots + n_k$. Let $\hat{\Pi}$ and $\Pi^\ast$ denote respectively $\Pi(\hat{\alpha}^{(N)})$ and $\Pi(\mathbf{\alpha}^{\ast(N)})$. The hypothesis $H_{01}:\Pi_i = \hat{\Pi}$ and $H_{02}:\Pi_i = \Pi^\ast$ are considered for any specified $i$. The following two theorems for the case $k = 2$ are proven.