STATISTICAL THEORY FOR CS
EXAM 2017 A
PAVEL CHIGANSKY
Problem 1. Let Z1 , Z2 , ... be i.i.d. Ber(θ), θ ∈ (0, 1) r.v.’s. We have seen
that there is no unbiased estimate of the odds θ/(1 − θ), based on Z1 , ..., Zn
for any fixed n ≥ 1. Let
X1 = min{n : Zn = 0},
be the number of tosses till the first 0 occurs. As is well known, X1 has
Geo(1 − θ) distribution1:
Pθ (X1 = k) = θk−1 (1 − θ),
k ∈ {1, 2, ...}.
(1) Find the MLE of θ on the basis of X1 .
(2) Find the MLE of η(θ) := θ/(1 − θ).
Hint: η(θ) is a one-to-one function of θ ∈ (0, 1).
(3) Show that MLE η̂ is unbiased. Explain the “contradiction” with the
non-existence claim above.
(4) Is the obtained estimate above UMVUE ? Calculate its MSE risk.
(5) Calculate the C-R lower bound for the MSE risk of unbiased estimates of θ/(1 − θ). Is C-R bound attained ?
(6) Encouraged by his progress, the statistician suggests to count the
tosses till m ≥ 1 zeros occur:
n
n X
o
Xm = min n :
(1 − Zi ) = m .
i=1
Argue that the p.m.f. of Xm is given by:
k−1
pXm (k; θ) = Pθ (Xm = k) =
θk−m (1 − θ)m ,
m−1
k≥m
(7) Find the MLE of η(θ) = θ/(1 − θ) on the basis of Xm .
(8) Do MLEs from the previous question form a consistent sequence of
estimates of θ/(1 − θ) ?
P
Hint: Show that Xm = m
j=1 ξi , where ξi are i.i.d. Geo(1 − θ)
r.v.’s and apply the LLN.
(9) Find the asymptotic rate and the asymptotic error distribution of
the MLE’s found above
1E =
θ
1
1−θ
and varθ (X1 ) =
θ
(1−θ)2
1
2
PAVEL CHIGANSKY
Problem 2.(Change Detection)
A plant produces bottles of mineral water. Production is monitored by
a statistician, who samples the water acidity daily. She suspects that the
production line stopped to work properly within the last n days and suggests
to perform a test.
Let (X1 , ..., Xn ) denote the acidity of the water measured on each one of
the days and let θ ∈ {1, ..., n} denote the day index at which the change
have occurred. The statistician wants to test the hypothesis:
H0 : X1 , ..., Xn are i.i.d. N (0, 1) r.v.’s
against the alternatives
(
X1 , ..., Xθ−1 ∼ N (0, 1)
H1 : X1 , ..., Xn are independent,
Xθ , ..., Xn ∼ N (a, 1)
θ ∈ {1, ..., n}
where a > 0 is a known constant.
(1) Find the likelihood function of the data under the alternative
(2) For n = 1, find the most powerful level α test
(3) Find the α level test, using the test statistic X̄n and calculate its
power function.
(4) Prove that the GLRT test with a critical value c rejects H0 iff
n
X
max
(Xi − a/2) > c.
θ∈{1,...,n}
i=θ
(5) Find the GLRT statistic, if a is unknown and is to be considered as
a parameter as well.
Department of Statistics, The Hebrew University, Mount Scopus, Jerusalem
91905, Israel
E-mail address: pchiga@mscc.huji.ac.il