PRACTICE MID SESSION TEST
MATH5905
Time allowed: 135 minutes
1. Let X = (X1, X2, . . . , Xn) be i.i.d. Poisson(θ) random variables with density function
f(x, θ) =
e−θθx
x!
, x = 0, 1, 2, . . . , and θ > 0.
a) The statistic T (X) =
∑n
i=1Xi 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: ∫ 2
0
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
Γ(α)βα
xα−1e−x/β
and
Γ(α + 1) = αΓ(α) = α!
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2. Let X1, X2, . . . , Xn be independent random variables, with a density
f(x; θ) =
{
e−(x−θ), 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
fT (t) =
{
ne−n(x−θ), t > θ,
0 else
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.
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