Student Notes To Accompany MS4214: STATISTICAL INFERENCE

3. A certain type of plant cell may appear in any one of four versions. According to
a genetic theory, the four versions have the following probabilities of appearance:
1/2 + θ/4, (1 − θ)/4, (1 − θ)/4, θ/4,
where θ is a parameter not specified by the genetic theory (0 < θ < 1). If a
sample of n cells had observed frequencies (a, b, c, d) where n = a + b + c + d, find
the maximum likelihood estimate of θ, and an estimate of Var( ˆ θ).
4. Let X1, X2, . . . , Xn be a random sample from a population with probability den-
sity
f(x) =
(a) Show that E(X 2 ) = θ.
�
2
πθ exp
�
− x2
�
, x > 0.
2θ
(b) Find the maximum likelihood estimator (MLE), ˆ θ, of θ.
(c) Show that ˆ θ is an unbiased estimator of θ and that the Cramér-Rao lower
bound is attained. [You may assume Var(X 2 ) = 2θ 2 .]
(d) Suppose now that φ = √ θ is the parameter of interest. Without undertaking
further calculations, write down the MLE of φ and explain why it is a biased
estimator of φ.
5. A random sample X1, X2, . . . , Xn is available from a Poisson distribution with
mean θ. Define the parameter λ = e −θ .
(a) Find the maximum likelihood estimator (MLE), ˆ θ, of θ. Hence deduce the
MLE of ˆ λ, of λ.
(b) Find the variance of ˆ θ, and deduce the approximate variance of ˆ λ using the
delta method.
(c) An alternative estimator of λ is ˜ λ, defines as the observed proportion of zero
observations. Find the bias of ˜ λ and show that
Var( ˜ λ) = e−θ � 1 − e−θ� .
n
(d) Draw a rough sketch of the efficiency of ˜ λ relative to ˆ λ, and discuss its
properties.
6. Let X1, . . . , Xn be independent and identically distributed as N (µ, σ 2 ), where µ
and σ 2 are both unknown, and assume n = 2p + 1 is odd. Define the estimators
ˆµ = ¯ X = (X1 + · · · + Xn)/n, ˜µ = median(X1, . . . , Xn) = X(p+1), and
�
�
�
ˆσ = S = � n �
(Xi − ¯ X) 2 /(n − 1).
i=1
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