Midterm exam STA355H1S
March 3–4, 2021
Instructions:
1. Solutions to problems 1–3 are to be submitted on Quercus (PDF files only) – the
deadline is 9am (EST) on March 4, 2021.
2. This is an open-book test so you are free to use material from the lectures and other
published sources; in the event that you make use of material that was not covered in
the lectures, you must cite your source or sources. Show all relevant work (including
R code and output, when appropriate) to receive full credit.
3. Obtaining or providing unauthorized assistance (which includes working in groups)
is considered an academic o↵ence as outlined in the University of Toronto’s Code of
Behaviour on Academic Matters.
1. Suppose that X1, · · · , Xn are independent Exponential random variables with pdf
f(x;) = exp(x) for x 0
where > 0 is an unknown parameter.
(a) [10 marks] Suppose that {kn} is a sequence of positive integers such that kn/n ! ⌧ 2
(0, 1) where
p
n(kn/n ⌧)! 0 as n!1. Consider estimators of of the form
bn(⌧) = a(⌧)
X(kn)
for some a(⌧).
(i) For ⌧ 2 (0, 1), find a(⌧) such that bn(⌧) p! as n!1.
(ii) For a(⌧) derived above, we will have
p
n(bn(⌧) ) d! N (0, 2(⌧))
Find an expression for 2(⌧) and find the value of ⌧ (to 3 decimal places) that minimizes
2(⌧).
(b) [10 marks] The file prob1.txt contains the 50 smallest values of a sample of 100 obser-
vations from an Exponential distribution with parameter .
(i) Compute the maximum likelihood estimate of based on these data. (Hint: Use spacings.)
(ii) Compute an estimate of the standard error of the estimate in (i).


2. Suppose that X1, · · · , Xn are independent positive random variables with common cdf
F (x). The Atkinson index of F is defined by
A(F ) = 1 1
EF (X)
exp [EF (ln(X))]
where EF (·) denotes expected value with respect to the distribution F .
(a) [10 marks] Suppose that G(x) = F (x b) for some b > 0.
(i) Show that A(G)  A(F ). (Hint: If X ⇠ F then X + b ⇠ G. You may find Jensen’s
inequality useful here.)
(ii) When is A(G) = A(F ) (for b > 0)?
(b) [10 marks] A sample of 200 observations from an unknown distribution F is given in a
file prob2.txt on Quercus. Compute an estimate of A(F ) and an estimate of its standard
error. Justify your method.
3. Suppose that X1, · · · , Xn are independent continuous random variables with density
f(x; ✓) = ✓(✓ + 1)x✓1(1 x) for 0  x  1
where ✓ > 0 is an unknown parameter.
(a) [7 marks] Find the MLE of ✓ and give an estimator of its standard error based on the
observed Fisher information.
(b) [13 marks] Define X¯n to be the sample mean of X1, · · · , Xn.
(i) Find b✓n = g(X¯n) so that b✓n p! ✓, that is, {b✓n} is a consistent sequence of estimators.
(ii) By the Delta Method,
p
n(b✓n ✓) d! N (0, 2(✓)). Find 2(✓).
(iii) Two estimators of the standard error of b✓n are
bse1(b✓n) = |g0(X¯)|Sp
n
and bse2(b✓n) = (b✓n)p
n
where S2 is the sample variance of X1, · · · , Xn and 2(✓) is as defined in part (ii).
When would we prefer bse1(b✓n) to bse2(b✓n)?