Theory and Methods of Statistics

Examples of Conditional Tests

Example 6.9.3

Let $\mathit{X}=\left(X_1,\dots ,X_m\right)$ and $\mathit{Y}=\left(Y_1,\dots ,Y_n\right)$ be independent random samples from Poisson(λ) and Poisson(μ), respectively. We want to test H₀: μ = aλ vs H₁: μ≠aλ where a > 0 is given. Here $(X,Y)=\left({\sum }_{i=1}^m{X}_i,{\sum }_{i=1}^n{Y}_i\right)$ is sufficient for (λ, μ) in (X, Y). As in Example 6.9.2, reparametrize and transform the data as:

$\begin{array}{l}\hfill \pi =n\mu /(m\lambda +n\mu ),\xi =m\lambda +n\mu ,\text{and}U=Y,T=X+Y,\end{array}$

to arrive at

$\begin{array}{l}\hfill f_{U,T}(u,t;\pi ,\xi )=\left(\genfrac{}{}{0.0pt}{}{t}{u}\right){\pi }^u{(1-\pi )}^{t-u}{\text{e}}^{-\xi }{\xi }^t/t!,\end{array}$

t = 0, 1, … and u = 0, 1, …,  ...