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IÃ¢â‚¬â„¢m stuck on a Statistics question and need an explanation.

Intermediate Statistical Theory
Homework Sheet 4
Basic Questions
1. Show that the sample proportion
is a minimum variance unbiased
estimator of the binomial parameter ÃŽÂ¸ (Hint: Treat as the mean of a
random sample of size n from a Bernoulli population with the parameter ÃŽÂ¸.)
2. If XÃ‚Â¯1 is the mean of a random sample of size n from a normal population
with the mean Ã‚Âµ and the variance
is the mean of a random sample of
size n from a normal population with the mean Ã‚Âµ and the variance , and
the two samples are independent, show that
(a) Ãâ€°XÃ‚Â¯1 + (1 Ã¢Ë†â€™ Ãâ€°)XÃ‚Â¯2, where 0 Ã¢â€°Â¤ Ãâ€° Ã¢â€°Â¤ 1, is an unbiased estimator of Ã‚Âµ; (b) the
variance of this estimator is a minimum when
.
(c) find the efficiency of the estimator of part (a) with Ãâ€° = 1/2 relative to this
estimator with
.
3. Let Y1,Y2,…,Yn be a random sample of size n from the pdf
(a) Show that
is an unbiased estimator for ÃŽÂ¸.
(b) Show that
is a minimum-variance estimator for ÃŽÂ¸.
4. Let Xn denote a random variable with mean Ã‚Âµ and variance b/np, where p >
0, Ã‚Âµ and b are constants (not functions of n). prove that Xn converges in
probability to Ã‚Âµ. (use ChebyshevÃ¢â‚¬â„¢s inequality).
5. An estimator ÃŽÂ¸Ã‹â€ n is said to be squared-error consistent for ÃŽÂ¸ if limnÃ¢â€ â€™Ã¢Ë†Å¾E[(ÃŽÂ¸Ã‹â€ nÃ¢Ë†â€™
ÃŽÂ¸)2] = 0.
(a) Show that any squared-error consistent ÃŽÂ¸Ã‹â€ n is asymptotically unbiased.
(b) Show that any squared-error consistent ÃŽÂ¸Ã‹â€ n is consistent.
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