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Neyman-Pearson Lemma

If there exists a critical region $C$ of size $\alpha$ and a Nonnegative constant $k$ such that

\begin{displaymath}
{\prod_{i=1}^n f(x_i\vert\theta_1)\over \prod_{i=1}^n f(x_i\vert\theta_0)} \geq k
\end{displaymath}

for points in $C$ and

\begin{displaymath}
{\prod_{i=1}^n f(x_i\vert\theta_1)\over \prod_{i=1}^n f(x_i\vert\theta_0)} \leq k
\end{displaymath}

for points not in $C$, then $C$ is a best critical region of size $\alpha$.


References

Hoel, P. G.; Port, S. C.; and Stone, C. J. ``Testing Hypotheses.'' Ch. 3 in Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 56-67, 1971.




© 1996-9 Eric W. Weisstein
1999-05-25