Minimum Chi-Squared Estimation Using Independent Statistics

Abstract

In a multinomial situation with observed proportions $Q_i (i = 1, \cdots, r + 1)$ and corresponding expected proportions $p_i$, it has been shown by Chapman [1] that for large samples $Y_i = \ln Q_i - \ln Q_{i + 1},\quad i = 1, \cdots, r,$ and $Y_j$ are independent for $j \neq i - 1, i, i + 1$. In this note the efficiency obtained when estimating the parameters of a distribution from these mutually independent odd (or even) $Y_i$'s is examined in the case of the geometric and Poisson distributions and it is shown that the resulting estimators are inefficient.