• University of Waterloo

• STAT 431  

STAT 431/831 - Midterm Test Solution
This material is for the personal use of students enrolled in the Fall 2019 offering of Stat 431/831. Distribution or reproduction of these material for commercial or non-commercial means is strictly prohibited.
1. [15+5 marks] The probability mass function of negative binomial distribution can be written
as
f(y; r; p) = r + yy - 1(1 - p)rpy y = 0; 1; 2; : : :
where r > 0 and 0 < p < 1. Consider r as fixed and known.
(a) [3 marks] Let Y1; Y2; : : : ; Yn be independent identically distributed negative binomial
random variables, and we observe y1; y2; : : : ; yn. Find the MLE for p. You don’t need to
verify it corresponds to the maximum point.
Solution:
‘(p) =
nX i
=1 yi log p + r log(1 - p) + log r + yyii - 1
S(p) =
nX i
=1 ypi + 1--rp
S(^ p) = 0 =
P yi
p^ -
rn
1 - p^
p^
1 - p^ =
P yi
rn
p^ = y=r ¯
1 + ¯ y=r =
y¯
r + ¯ y
where ¯ y = X yi=n
(b) [4 marks] Show that this distribution is a member of the exponential family with
p.d.f./p.m.f. written as
f(y; θ; φ) = exp (yθa-(φb)(θ)) + c(y; φ) :
Specify the canonical parameter θ and functions a(·), b(·) and c(·; ·).
Solution: We can write the pmf as:
f(y; π; r) = exp y log p + r log(1 - p) + log r + yy - 1
1