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Anonymous

5 questions total

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Homework Sheet 3
Basic Questions
1. Let X be a random sample from a binomial distribution with n = 100 and
p unknown.
(a) Show that the maximum likelihood estimate for p is unbiassed.
(b) The variance of X is 100p(1 − p). Find the bias of the maximum
likelihood estimate for this variance.
2. Show that X+1
n+2 is a biased estimator of the binomial parameter θ. Is this
estimator asymptotically unbiased?
3. Let Ymin be the smallest order statistic in a random sample of size n
drawn from the uniform pdf, fY (y; θ) = 1/θ, 0 ≤ y ≤ θ. Find an unbiased
estimator for θ based on Ymin .
4. A random sample of size 2, Y1 and Y2 , is drawn from the pdf fY (y; θ) =
2yθ2 , 0 < y < 1/θ. What must c equal if the statistic c(Y1 + 2Y2 ) is to be
an unbiased estimator for 1/θ?
5. Let X1 , . . . , Xn be a random sample from a N (0, σ 2 ) distribution. We
want to
the standard deviation σ. Find the constant c so that
Pestimate
n
Y = c i=1 |Xi | is an unbiased estimator for σ and determine its efficiency
(efficiency is defined as 1/var(Y )).
1
...

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