AMA529 Statistical Inference
Tutorial 4
1. (Ex.6.3.1) Consider the decision rule (6.3.5) derived in Example 6.3.1 of HMC.
Obtain the distribution of the test statistic under a general alternative and use it to
obtain the power function of the test. If com
AMA 529 Statistical Inference
Lecture 4
Goodness of Fit Tests and interval estimation
1. Goodness of Fit Test (Refer Section 4.7 of HMC)
H 0 : pi pio , i, i 1, 2,
, k 1; are specified, where
k
p
io
i 1
1.
H1 : all alternatives.
If the hypothesis H 0 is t
AMA 529 Statistical Inference
Lecture 5
Maximum Likelihood Estimation
Introduction
Given a R.S. from a distribution with pdf f ( x, ) , say the R.S. is X1 ,
, X n , our
question is how to estimate from the R.S. In most of our discussions we let be a
scala
AMA 529 Statistical Inference
Lecture 8
Completeness and Exponential Family
Introduction:
Recall we start with a search of MVUE, then we find the idea of sufficiency is
important. Loosely speaking, an unbiased estimator which is a function of sufficient
s
AMA 529 Statistical Inference
Lecture 9
Most Powerful Test and Uniformly Most Powerful Test
Preliminaries:
A statistical hypothesis is an assertion (or a statement) about the distribution of
one or more rvs.
Example: X ~ N ( ,1) , then we may consider
H0
AMA 529 Statistical Inference
Lecture 6
Maximum Likelihood Tests and Measure of Quality of Estimators
1. Max likelihood tests
We are given the test problem: H 0 : 0 vs H1 : 0 .
Recall in Thm 6.1.1 we have Lim P[ L(0 ) L( )] 1 if H 0 is true, so in general
AMA 529 Statistical Inference
Lecture 12
Bayesian Procedures
Bayesian estimation
Preliminary results:
(a) Let W be a r.v. Find c that minimizes E W c , i.e. find the constant with the
2
minimum mean-squared error.
Solution: f (c) E W c
2
f (c) 2E W c 0
Tutorial 1
Question 1. Let Wn denote a random variable with mean and variance b
, where
np
p > 0, , and b are constants (not function of n ). Prove that Wn converges in probability
to .
Hint: Use the Chebyshevs inequality.
Question 2. Let X 1 , , X n be i
AMA 529 Statistical Inference
Lecture 11
Introduction to Bayesian statistics
1. Prior and Posterior distributions
In classical statistics, given a probability density f x, , is a parameter. In
other words, is not a random variable. But in Bayesian statist
AMA529 Statistical Inference
Tutorial 5
1. (Ex.7.2.2) Prove that the sum of the observations of a random sample of size n
from a Poisson distribution having parameter , 0 < < , is a sufficient statistic
for .
2. (Ex.7.2.3) Show that the nth order statisti
AMA 529 Statistical Inference
Lecture 7
Sufficient Statistic and Its Properties
Introduction:
Note that a statistic is a form of data reduction. The simplest example is the
sample mean X . We basically want to use a single number x , to represent the
samp
AMA 529 Statistical Inference
Lecture 2
Introduction
This lecture is about elementary limit theory for sequence of r.v.s, which is important
in the understanding of the properties of estimators.
Review
Chebyshevs inequality
If X is a r.v. with finite mea
AMA 529 Statistical Inference
Lecture 3
Introduction
This lecture wraps up the standard asymptotic theory for obtaining approximate
distributions for estimators based on large samples.
1. Bounded in probability
Motivation: Recall in basic calculus we have
AMA529 Statistical Inference
Tutorial 9
1. (Ex.11.3.2) Let X1 , X 2 , , X 10 be a random sample of size n = 10 from a
gamma distribution with = 3 and = 1 . Suppose we believe that has a
gamma distribution with = 10 and = 2 .
(a)
Find the posterior distrib
AMA529 Statistical Inference
Tutorial 6
1. (Ex.7.4.1) If az 2 + bz + c = 0 for more than two values of z , then a = b = c = 0 .
Use this result to show that the family cfw_b ( 2, ) : 0 < < 1 is complete.
2. (Ex.7.4.2) Show that each of the following famil
AMA529 Statistical Inference
Tutorial 7
1. (Ex.8.1.4) Let X1 , X 2 ,
distribution N ( 0,
2
).
, X 10 be a random sample of size 10 from a normal
Find a best critical region of size = 0.05 for testing
H 0 : 2 = 1 against H1 : 2 = 2 . Is this a best critic
AMA529 Statistical Inference
Tutorial 8
1. (Ex.8.3.6) Let X1 , X 2 , , X n be a random sample from the normal distribution
N ( ,1) . Show that the likelihood ratio principle for testing H 0 : = , where
is specified, against H1 : leads to the inequality x
Show that the Rao-Cramer lower bound is 2/n, where n is the size of a random sample
from this Cauchy distribution. What is the asymptotic distribution of
n ( ), if is
the mle of ?
6. (Ex. 6.2.10) Let X1, X2, , Xn be a random sample from a N (0, ) distribu
AMA529 Statistical Inference
Assignment 1
1. Find the maximum likelihood estimator (MLE) of the unknown parameter based on a random
sample X1 , , X n from each of the following distributions:
(a) X i
(b) X i
(c) X i
(d) X i
BIN (1, p) .
GEO( p) .
N (0, )
AMA529 Statistical Inference
Assignment 2
1. Consider a random sample of size n from a two-parameter exponential
distribution, X i
Exp(1, ). Namely, f ( x, , )
x
exp
, x .
1
Show that S= Y1 , the first order statistic, is sufficient for . (Hint: Yo
IBLE OF UP STANDARHISQP_NORMAL DISTRIBUTION
. .......-..._...-...-._.__'
Th0 t0b1e gives the probability
P = Pr (2 > z)
where Z ~ N(0,1J.
.5000 .4960. .4920 .4360 I.4840 .4801 .4761 .4721 .4601 .4641
.4602 .4562 .4522 .4463 .4443 .4404 .43
AMA529 Statistical Inference
Tutorial 2
1. Let the random variable Z n have a Poisson distribution with parameter = n . Show
that the limiting distribution of the random variable Yn = ( Z n n )
n is normal with
mean zero and variance 1.
2. Use Stirlings f
AMA 529 Statistical Inference
Lecture 1
Introduction and motivation
Estimation of parameters (in a distribution or statistical model) and testing of hypothesis
are the two cornerstones of modern statistics.
Example: Linear regression
Yi = + xi + i ,
i NID
AMA 529 Statistical Inference
Lecture 10
Generalized Likelihood Ratio Test
1. GLR test
Recall the Neyman Pearson Theorem gives a method of constructing statistical
tests. The resulting tests are often uniformly most powerful (UMP) for a one-sided
test. A