Some tests of chi-square type for continuous distributions

In testing the hypothesis that a sample $X_1,…,X_n$ is drawn from a d.f. $F(x,\theta)$ where $\theta\in R^s$ is an unspecified parameter, the following three test statistics are considered. 1. The $\chi^2$-statistic $X^2(\widehat\theta)$ with class boundaries fixed in advance and class probabilities $p_i(\widehat\theta)$ determined by an estimate $\widehat\theta$ (cf. [2]). 2. The $\chi^2$-statistic $X^2(\theta^*,\widehat\theta)$ with class boundaries $(a^*_{i-1},a^*_i)$ determined by $F(a^*_i,\theta^*)-F(a^*_{i-1},\theta^*)=p_i$, $p_1,…,p_k$ being prescribed probabilities and $\theta^*$ an estimate of $\theta$ (cf. [4]). 3. $Z^2(\widehat\theta)=n\sum p_i^{-1}[p_i-(F(Y_i,\widehat\theta)-F(Y_{i-1},\widehat\theta))]^2$, $Y_i$ being the sample $(p_1+…+p_i)$-quantile. It is proved, under certain regularity conditions, that $X^2(\theta^*,\widehat\theta)-X^2(\widehat\theta)\to0$ and $Z^2(\widehat\theta)-X^2(\widehat\theta)\to0$ provided $\theta^*$ is a consistent and $\widehat\theta$ a root $n$ consistent estimate and $p_i(\theta_0)=p_i$, $\theta_0$ being the true value of $\theta$. Therefore asymptotic results on $X^2(\widehat\theta)$ hold true for $X^2(\theta^*,\widehat\theta)$ and $Z^2(\widehat\theta)$. It is shown that the minimization of any of the three statistics gives estimates equivalent to the multinomial ML estimate, and that the use of the ML estimate based on the whole sample can decrease as well as increase the power.