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### unbiased and consistent example

If $X \sim Uniform (0, \theta)$, then the PDF and CDF of $X$ are given by \end{align} I think this is the biggest problem for graduate students. Explanation 3. STAT 801: Mathematical Statistics Unbiased Tests De nition: A test ˚ of 0 against 1 is unbiased level if it has level and, for every 2 1 we have ˇ( ) : When testing a point null hypothesis like = 0 this requires that the power function be minimized at 0 which will mean that if ˇ is di erentiable then ˇ0( 0) = 0 Example: N( ;1): data X = (X1;:::;Xn). Let $X_1$, $X_2$, $X_3$, $...$, $X_n$ be a random sample from a $Uniform(0,\theta)$ distribution, where $\theta$ is unknown. On the obvious side since you get the wrong estimate and, which is even more troubling, you are more confident about your wrong estimate (low std around estimate). Relative e ciency: If ^ 1 and ^ 2 are both unbiased estimators of a parameter we say that ^ 1 is relatively more e cient if var(^ 1)

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