Stat 411 Lecture Notes Point estimation
Ryan Martin Spring 2012
1
Introduction
The statistical inference problem starts with the identification of a population of interest, about which something is unknown. For example, before introducing a law that home
Stat 411 Homework 01
Solutions
1. Since the Xi 's are independent, E(Yn ) = E(X1 ) E(Xn ) = cn . By Markov's inequality, P(Yn > ) -1 E(Yn ) = -1 cn 0, n , since c (0, 1). Since > 0 is arbitrary, Yn 0. 2. (a) The maximum of a list of numbers less than x if
Valuations
"They must know something that I don't" => Dangerous
Don't be a lemming
-> Don't just blindly follow
Valuation gives you a lifevest
-> Slows you down from selling / buying
-> Gives a rational foundation
There is always a bias before valuation
I
Stat 411 Homework 06
Due: Friday 10/19
1. Let X1 , X2 be iid with PDF f (x) = (1/)ex/ , x > 0.
(a) Let Y1 = X1 and Y2 = X1 + X2 . Find the joint PDF fY1 ,Y2 (y1 , y2 ), the marginal
PDF fY2 (y2 ) and, nally, the conditional PDF fY1 |Y2 (y1 | y2 ). (Though
Stat 411 Homework 05
Solutions
1. (a) The likelihood function looks like
n
n/2 (1/22 )
f (Xi ) 2
L(1 , 2 ) =
e
n
2
i=1 (log Xi 1 )
.
i=1
Taking a natural logarithm gives
(1 , 2 ) = const
1
n log 2
2
22
n
(log Xi 1 )2 .
i=1
Dierentiating with respect to 1
Stat 411 Homework 06
Solutions
1. (a) First need to nd the joint PDF of Y1 = X1 and Y2 = X1 + X2 . The Jacobian
of the (inverse) transformation is 1, so the joint PDF of (Y1 , Y2 ) is given by
fY1 ,Y2 (y1 , y2 ) = fX1 ,X2 (y1 , y2 y1 ) = (1/2 )ey2 / ,
0 <
Stat 411 Homework 07
Due: Friday 10/26
1. Let X1 , . . . , Xn be iid N(, 1). Find the MVUE of = 2 .
2. Problem 7.5.3 in HMC67.
3. Suppose X has PDF/PMF f (x) = ex+S (x)+q() ; this is a special case of the (regular)
exponential family with p() = and K (x)
Stat 411 Homework 08
Due: Wednesday 10/31
iid
2
1. Let X1 , . . . , Xn N(1 , 2 ), where 2 > 0 is the variance. Find the MVUE of 1 .
2. Problem 7.7.10 in HMC67. (Hint: Use the exponential family form.)
3. Let X1 , . . . , Xn be iid Unif (0, ), a scale para
Stat 411 Homework 07
Solutions
1. The distribution N(, 1) is a regular one-parameter exponential family problem
with K (x) = x. Therefore, T = n=1 Xi is a complete sucient statistic for and,
i
2
consequently, the MVUE of is X = T /n. It is easy to check t
Stat 411 Homework 09
Due: Friday 11/16
Some of the problems ask for plots; youre welcome (actually encouraged) to use a computer to produce these plots.
1. Problem 5.5.1 in HMC6 = Problem 4.5.1 in HMC7.1
iid
2. Let X1 , . . . , Xn Unif (0, ). The goal is
Stat 411 Homework 09
Solutions
1. Problem 5.5.1 in HMC6 = Problem 4.5.1 in HMC7. For testing H0 : = 0
vs. H1 : > 0 , the power function in equation (5.5.12/4.5.12) is
n(0 )
pow() = z
,
where z satises (z ) = , and both 0 and are xed and known constants.
Stat 411 Homework 11
Solutions
1. (a) The exponential distribution is an exponential family and, therefore, has monon
tone likelihood ratio property in T =
i=1 Xi . So, in this case, the mostpowerful test will reject H0 : = 2 in favor of H1 : = 1 i T is l
Stat 411 Homework 12
Solutions
1. (a) Following the work in Example 6.3.1 in HMC67, we nd that the likelihood
ratio test of H0 : = 0 versus H1 : = 0 rejects H0 i T = (2/0 ) n=1 Xi is
i
less than some cuto c1 or more than some other cuto c2 , with c1 < c2
Stat 411 Homework 11
Due: Friday 12/07
iid
1. Let X1 , . . . , Xn N(0, ), where > 0 is the variance. Consider testing H0 : = 0
versus H1 : = 0
(a) Show that the likelihood ratio test can be expressed in terms of T =
n
i=1
Xi2 /0 .
(b) Find the distributio
Stat 411 Homework 12
NOT DUE!
iid
1. Let X1 , . . . , Xn Exp(), with PDF f (x) = (1/)ex/ .
(a) Find the size- likelihood ratio test for H0 : = 0 versus H1 : = 0 . (Hint:
See Example 6.3.1 on page 333.)
(b) Find a 100(1 )% condence interval for by invertin
Stat 411 Homework 05
Due: Friday 10/12
1. Let X1 , . . . , Xn be an iid sample from a log-normal distribution with PDF
1
2
f (x) =
e(log x1 ) /22 ,
x 22
x > 0,
= (1 , 2 ) R R+ .
(a) Find the MLE of = (1 , 2 ) .
(b) Calculate the Fisher information matri
Stat 411 Homework 04
Solutions
1. Problem 6.2.7 in HMC7. The PDF for the Gamma(4, ) distribution is
f ( x) =
1 3 x/
xe
,
64
x > 0,
> 0.
(a) For the Fisher information, we rst need second derivative of log-PDF:
x
4
2
2
2x
= 2 3.
log f (x) = 2 const 4 log
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Stat 411 Homework 04
Solutions
1. Problem 6.2.7 in HMC7. The PDF for the Gamma(4, ) distribution is
f ( x) =
1 3 x/
xe
,
64
x > 0,
> 0.
(a) For the Fisher information, we rst need second derivative of log-PDF:
x
4
2
2
2x
= 2 3.
log f (x) = 2 const 4 log
Stat 411 Homework 03
Solutions
1. Problem 6.1.2 in HMC7.
(a) Let X1 , . . . , Xn be iid with PDF f (x) = x1 , x (0, 1) (a beta distribution).
The likelihood function is
n
n
Xi1
L() =
=
1
n
i=1
Xi
.
i=1
Taking log and then derivative gives
0=
log L() =
n l
Stat 411 Homework 01
Solutions
1. Since the Xi s are independent, E(Yn ) = E(X1 ) E(Xn ) = cn . By Markovs inequality,
P(Yn > ) 1 E(Yn ) = 1 cn 0, n ,
p
since c (0, 1). Since > 0 is arbitrary, Yn 0.
2. (a) Let Mn = X(n) . The maximum of a list of numbers
Stat 411 Review problems for Exam 2
1. (a) Geometric distribution: L() = n (1 )
imal) sucient statistic for .
n
i=1
Solutions
Xi n
n
i=1
, so T =
(b) Truncated scale exponential distribution: L() = e(1/)
so T = n=1 Xi is a (minimal) sucient statistic for
Stat 411 Homework 01
Due: Friday 09/07
1. Let cfw_Xn : n 1 be a sequence of positive independent random variables with
E(Xn ) = c (0, 1) for each n. Let Yn = X1 X2 Xn , the product of the Xi s. Use
Markovs inequality to prove that Yn 0 in probability.
2.
Stat 411 Homework 02
Due: Friday 09/14
1. Let X1 , . . . , Xn be iid Unif (0, ), where > 0 is unknown.
(a) Let n = X(n) , the sample maximum. Find the CDF of n . (Hint: This is
similar to Problem #2 on Homework 01.)
(b) Show that n is a consistent estimat