Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that...

Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that the maximum likelihood estimator (MLE) of a is ˆ a = max(Xi).
a) Let Y = max(Xi). Use the fact that Y ≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y.
b) Find the probability density function of Y from cdf.
c) Use the obtained pdf to show that MLE for a (ˆ a = max(Xi)) is biased.
d) Say I would like to consider another estimator for a, I will call it ˆ b = 2 ¯ X. Is it unbiased estimator of a (show)? How you can explain someone without calculations why ˆ b = 2 ¯ X is a reasonable estimator of a?
e) Based on the result in (c), I will propose to use unbiased estimator for a instead of ˆ a = max(Xi), say ˆ c = n+1 n max(Xi). Given that the relative eﬃciency of any two unbiased estimators ˆ b,ˆ c is the ratio of their variances
V ar(ˆ b) V ar(ˆ c)
,
explain which of these two unbiased estimators is more eﬃcient. You can obtain the V ar(ˆ c) = V ar(n+1 n max(Xi)) from V ar(ˆ a) = Var(Y ). The variance of the Y = max(Xi) is
Var(Y ) =
n/( (n + 1)^2(n + 2))*a^2

Expert Solution

orchestra answered 2 years ago

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