Intermediate Statistical Theory Worksheet

  

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Intermediate Statistical Theory
Homework Sheet 4
Basic Questions
1. Show that the sample proportion
is a minimum variance unbiased
estimator of the binomial parameter θ (Hint: Treat as the mean of a
random sample of size n from a Bernoulli population with the parameter θ.)
2. If X¯1 is the mean of a random sample of size n from a normal population
with the mean µ and the variance
is the mean of a random sample of
size n from a normal population with the mean µ and the variance , and
the two samples are independent, show that
(a) ωX¯1 + (1 − ω)X¯2, where 0 ≤ ω ≤ 1, is an unbiased estimator of µ; (b) the
variance of this estimator is a minimum when
.
(c) find the efficiency of the estimator of part (a) with ω = 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 θ.
(b) Show that
is a minimum-variance estimator for θ.
4. Let Xn denote a random variable with mean µ and variance b/np, where p >
0, µ and b are constants (not functions of n). prove that Xn converges in
probability to µ. (use Chebyshev’s inequality).
5. An estimator θˆn is said to be squared-error consistent for θ if limn→∞E[(θˆn−
θ)2] = 0.
(a) Show that any squared-error consistent θˆn is asymptotically unbiased.
(b) Show that any squared-error consistent θˆn is consistent.
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