Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
15
Solutions
log p
2(x )
=
1 + (x )2
I ( ) =
1
2(x )
1 + (x )2
2
1
x2
dx =
2 )3
(1 + x
2
1
4
dx =
2
1 + (x )
f (, x1 , . . . , xn ) = n exp[
n
log f (, x1 , . . . , xn )
=
=
E [] =
It is biased and t =
V ar (t) = (n 1)2 2
0
n1
n
xi ]
xi
i
n
xi
n n
n
ex
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
20
Analysis of Variance
Suppose we have a eld trial in which various types of treatments have been
tried on dierent subjects and the eects recorded as an observation x for each
individual. There are ni individuals with treatment i and the observations
fro
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
17
Paired ttest
Sometimes we have to deal with data that come in linked pairs. For example
initial weight and nal weight for a grwoth harmone. We have observations
(xi , yi ). They are very higly correlated. So if we want to test that the
harmone has no
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Final Take Home xamination.
Due before end of term.
Q1. From a normal population with unknown mean and variance a sample of size 17 is
drawn and the sample mean is found to be 0.13 while the sample standard deviation is
0.25. Are the observations consiste
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
1
Parametric Models.
To begin with on a space X we have a family P of probability distributions. In
practice X will be either a countable set of points cfw_x and P specied by the
individual probabilities p(, x) for x X . They will of course satisfy
p(, x)
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
22
Nonparametric Methods.
In parametric models one assumes apriori that the distributions have a specic form with one or more unknown parameters and one tries to nd the
best or atleast reasonably ecient procedures that answer specifc questions
regardng th
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Feb, 24, 2000
Q 1. Consider the problem of testing the simple hypothesis that the true density is f0
against f1 based on a single observation x. For any > 0 construct the UMP test of size
according to NeymanPearson lemma. Determine the power as a functi
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
23
Two stage procedures.
Suppose we are interested in constructing a 95% condence interval for the
mean of a normal populaton with an unknown variance 2 , We want to
make sure that the width of the interval is no more than 0.1. In other words
we want to c
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Feb 17, 2000
Q 1. Let x1 , . . . , xn be n independent observations from a normal poulation with both
mean and variance equal to an unknown parameter . Is there a sucient statistic? What
is the MLE? What is the CramrRao lower bound? Is it reached asympto
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
12
Testing of Hypotheses.
The simplest kind of a testing of hypothesis is when we have two possible
alternate models and based on the sample have to make a choice between
them.
Suppose f0 (x) and f1 (x) are two possible densities on R and we have an
obser
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Jan 27,00 Q1. Find the CramerRao lower bound for the estimation of the location parameter from the Cauchy Distribution p(, x) = 1 1 1 + (x ) 2
Q2. Find the maximum likelihood estimator based on a sample of size n from the exponential distribution f (, x)
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Problems: March 30, 2000
1. Show that for any collection (X1 , . . . , Xd ) of random variables the covariance matrix
C = cfw_ci,j dened by ci,j = Cov Xi Xj = E Xi Xj E Xi E Xj , which is symmetric, is
always positive semidenite. Show that if the rank of
Advanced Topics in Analysis (Calculus of Variations)
MATH GA 2660

Spring 2010
Feb 10, 2000
Q 1, Suppose we have observed the number of heads x in n independent tosses of a coin. It
is known that the probability of heads in a single toss is either 1 or 3 . Can you construct
4
4
an estimator fn (x) which is unbiased at these two valu