Statistics | Part IB, 2001

What is a simple hypothesis? Define the terms size and power for a test of one simple hypothesis against another.

State, without proof, the Neyman-Pearson lemma.

Let XX be a single random variable, with distribution FF. Consider testing the null hypothesis H0:FH_{0}: F is standard normal, N(0,1)N(0,1), against the alternative hypothesis H1:FH_{1}: F is double exponential, with density 14ex/2,xR\frac{1}{4} e^{-|x| / 2}, x \in \mathbb{R}.

Find the test of size α,α<14\alpha, \alpha<\frac{1}{4}, which maximises power, and show that the power is et/2e^{-t / 2}, where Φ(t)=1α/2\Phi(t)=1-\alpha / 2 and Φ\Phi is the distribution function of N(0,1)N(0,1).

[Hint: if XN(0,1),P(X>1)=0.3174.]X \sim N(0,1), P(|X|>1)=0.3174 .]

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