When we test a simple null hypothesis against a composite alternative, a critical region is said to be unbiased if the corresponding power function takes on its minimum value at the value of the parameter assumed under the null hypothesis. In other words, a critical region is unbiased if the probability of rejecting the null hypothesis is least when the null hypothesis is true. Given a single observation of the random variable X having the density

Where €“1 ‰¤ θ ‰¤ 1, show that the critical region x ‰¤ α provides an unbiased critical region of size α for testing the null hypothesis θ = 0 against the alternative hypothesis θ ‰  0.

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