1) [14 pts] You are given the following probability density function:
fX (x) = 3x2 ,
where 0 x 1. (a) Derive the cdf FX (x). (b) Compute P (X > 0.5). (c) Compute P (X > 0.5 X >
0.25). (d) Derive the inverse cdf F−1(q). (e) Derive E[X]. (f) Derive V [X]. (g) Fisher’s moment
coefficient of skewness is
" X − µ 3#
where µ = E[X] and σ = V [X]. Compute the skewness of the pdf given above. (Hint: expand this out
and plug in previous results when possible.)
2) [14 pts] You sample n iid data whose distribution, you assume, is
fX (x) = θxθ−1 ,
where 0 ≤ x ≤ 1 and θ > 0. You propose
θˆ = X1 + Xn
n
as your estimator of θ. Now, no one said you proposed a good estimator! Let’s characterize it. (a) What is the bias of your estimator? (It will help to compute E[X] first.) (b) What is the asymptotic bias of your estimator? (c) Is your estimator consistent? (d) What is the variance of this estimator? (You may use the fact that
θ
V [X] = (θ + 2)(θ + 1)2
when answering this question.) (e) What is the Cramer Rao Lower Bound on the variance of an unbiased estimator, given the pdf above? (See the bottom of page 78 of MPSI, but use the form of I(θ) given on page 80.) (f) Is your estimator’s variance smaller than the CRLB? Answer this by assuming, e.g., θ = 1 and see how your variance and the CRLB compare. Is it problematic if your variance is smaller than the CRLB? Why or why not?
3) [14 pts] The probability density function for the half-normal distribution is
fX (x) =
for x > 0 and σ > 0. For this distribution,
2 exp
πσ2
x2
− 2σ2 ,
E[X] = σ 2
π
and V [X] = σ2 1 − 2 .
Assume we have a set of n iid half-normal random variables X1, . . . , Xn . (a) Derive the maximum likelihood estimate for σ2. (You need not compute a second derivative to confirm that the extremum is a maximum point.) (b) Derive the MLE for σ. (c) Derive the asymptotic variance of σ^2M LE .
4) [14 pts] Assume the same conditions as for Q3 above. (a) Use the factorization criterion to identify the sufficient statistic U . (b) Derive the MVUE for σ2.
5) [14 pts] The Gompertz distribution has the pdf
fX (x) = αβ exp α + βx − αeβx ,
where α > 0, β > 0, and x 0. Assume that β = 1 and that you have a sample of n iid data drawn from this distribution. (a) Derive a sufficient statistic for α. (You will find that you would not be able to derive an MVUE for α given this sufficient statistic!) (b) Derive the MLE for α. (c) Derive the asymptotic variance of αˆMLE.
6) [14 pts] One form of the probability mass function for the negative binomial distribution is
p (x) = x − 1 ps(1 p)x−s ,
s − 1
where x [s, s + 1, . . . , ), 0 < p < 1 is the success probability, and s > 0 is the number of successes necessary for the negative binomial experiment to stop (e.g., “we flip a coin until we observe s = 4 heads”). For this distribution, E[X] = s/p. We sample n iid data from this distribution. (a) Derive the sufficient statistic for p. (b) Derive the MVUE for p.
7) [16 pts] Let {X1, . . . , Xn} be n iid samples from an exponential distribution:
f (x) = 1 exp − x ,
where x 0 and β > 0. (a) Derive a sufficient statistic for β. (b) Derive the MVUE for β. (c) Derive the MVUE for β2. Note that you cannot assume an invariance property here! One approach is to take the MVUE for β, square it, and determine its expected value, and work from there to isolate β2.
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