Student Notes To Accompany MS4214: STATISTICAL INFERENCE

5. Let X1, . . . , Xn be iid with density
fX(x|θ) = 1
exp [−x/θ]
θ
for 0 ≤ x < ∞. Let Y1, y . . . , Ym be iid with density
for 0 ≤ y < ∞.
fY (y|θ, λ) = λ
exp [−λy/θ]
θ
(a) Write down the likelihood and log-likelihood functions L(θ, λ) and ℓ(θ, λ).
(b) Derive ( ˆ θ, ˆ λ) the MLE of (θ, λ). Calculate the information matrix I = I(θ, λ).
(c) Show that ˆ θ is unbiased but that ˆ λ is biased. Suggest an alternative ˜ λ to ˆ λ
which is unbiased. Show that ˆ θ has efficiency 1. Calculate the efficiency of
˜λ and show that as m → ∞ the efficiency converges to 1.
(d) Suppose n = 11 and the average of the data values x1, x2, . . . , x11 is 2.0.
Suppose m = 5 and the average of the data values y1, y2, . . . , y5 is 4.0.
Calculate the maximum likelihood estimate ( ˆ θ, ˆ λ). Evaluate the information
matrix at the point ( ˆ θ, ˆ λ) and show that both of its eigenvalues are positive.
Calculate ˜ λ – the unbiased alternative to ˆ λ derived in (c).
6. Let X1, . . . , Xn be iid observations from a Poisson distribution with mean θ. Let
Y1, . . . , Ym be iid observations from a Poisson distribution with mean λθ.
(a) Write down the likelihood log-likelihood functions L(θ, λ) and ℓ(θ, λ).
(b) Derive ( ˆ θ, ˆ λ) the MLE of (θ, λ). Calculate the information matrix I = I(θ, λ).
(c) Suppose n = 10 and the average of the data values x1, x2, . . . , x10 is 2.0.
Suppose m = 5 and the average of the data values y1, y2, . . . , y5 is 4.0.
Calculate the maximum likelihood estimate ( ˆ θ, ˆ λ). Evaluate the information
matrix at the point ( ˆ θ, ˆ λ) and show that both of its eigenvalues are positive.
7. Let Y be the number of particles emitted by a radioactive source in 1 minute. Y
is thought to have a Poisson distribution whose mean θ is given by exp[α + βt]
where t is the temperature of the source. The numbers of particles y1, y2, . . . , yn
emitted in n 1 minute periods are observed; the temperature of the source for
the ith period was ti. Assume that Y1, . . . , Yn are independent with Yi having a
Poisson distribution with mean θi = exp[α + βti]. Derive an expression for the
log likelihood ℓ(α, β). Suppose that you were required to find the MLE (ˆα, ˆ β).
Clearly describe how you would perform this task. Your account should include
the derivation of the likelihood equations and a detailed account of how these
equations would be solved including how initial values for the iterative procedure
involved would be obtained.
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