DEPARTMENT OF STATISTICS
PRACTICE MIDTERM TEST - 2024 - Week 6
MATH5905
Time allowed: 135 minutes
1. Let X = (X1 , X2 ,..., Xn ) bei.i.d. Poisson(θ) random variables with density function
a) The statistic T(X) =P Xi is complete and sufficient for θ. Provide justifi- cation for why this statement is true.
b) Derive the UMVUE of h(θ) = e-kθ where k = 1, 2,..., n is a known integer. You must justify each step in your answer. Hint: Use the interpretation that P(X1 = 0) = e- θ and therefore P(X1 = 0,...,Xk = 0) = P(X1 = 0)k = e-kθ.
c) Calculate the Cramer-Rao lower bound for the minimal variance of an unbiased estimator of h(θ) = e-kθ.
d) Show that there does not exist an integer k for which the variance of the UMVUE of h(θ) attains this bound.
e) Determine the MLE h(ˆ) of h(θ).
f) Suppose that n = 5, T = 10 and k = 1 compute the numerical values of the UMVUE in part (b) and the MLE in part (e). Comment on these values.
g) Consider testing H0 : θ 2 versus H1 : θ > 2 with a 0-1 loss in Bayesian setting with the prior ⌧ (θ) = 4θ2 e-2✓. What is your decision when n = 5 and T = 10. You may use:
Note: The continuous random variable X has a gamma density f with param- eters a > 0 and β > 0 if
and
Γ(a +1) = aΓ(a) = a!
2. Let X1 , X2 ,..., Xn be independent random variables, with a density
where θ 2 R1 is an unknown parameter. Let T = min{X1 ,..., Xn } = X(1) be the minimal of the n observations.
a) Show that T is a sufficient statistic for the parameter θ.
b) Show that the density of T is
Hint: You may find the CDF first by using
P(X(1) < x) = 1 − P(X1 > x \ X2 > x ··· \ Xn > x).
c) Find the maximum likelihood estimator of θ and provide justification.
d) Show that the MLE is a biased estimator. Hint: You might want to consider using a substitution and then utilize the density of an exponential distribution
when computing the integral.
e) Show that T = X(1) is complete for θ.
f) Hence determine the UMVUE of θ.