THE UNIVERSITY OF NEW SOUTH WALES
DEPARTMENT OF STATISTICS
PRACTICE MID SESSION TEST – 2022 – Week 6
Time allowed: 135 minutes
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X = (X1, X2, . . . , Xn) be i.i.d. Poisson(θ) random variables with density function
f(x,θ)= x! , x=0,1,2,…, and θ>0.
a) The statistic T(X) = ni=1 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 hˆ 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 τ(θ) = 4θ2e−2θ. What is your decision when n = 5 and T = 10. You may use:
x12e−7xdx = 0.00317
Note: The continuous random variable X has a gamma density f with param- eters α > 0 and β > 0 if
f(x;α,β) = 1 xα−1e−x/β Γ(α)βα
Γ(α + 1) = αΓ(α) = α!
2. Let X1, X2, . . . , Xn be independent random variables, with a density e−(x−θ), x > θ,
f(x;θ) = 0 else
where θ ∈ 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
ne−n(x−θ), t > θ, fT (t) = 0 else
Hint: You may find the CDF first by using
P(X(1)
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.
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