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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) Ãâ€°XÃ‚Â¯1 + (1 Ã¢Ë†â€™ Ãâ€°)XÃ‚Â¯2, where 0 Ã¢â€°Â¤ Ãâ€° 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 Ãâ€° = 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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