STA 3000F (Fall, 2013)
Homework 2, Complete
(October 11; due November 1)
1. Non-uniqueness of ancillary statistics. Suppose that (Y1 , Z1 ), . . . (Yn , Zn )
are independent and identically distributed and follow a bivariate normal distribution with E(Yi
STA3000: Likelihood asymptotics with nuisance parameters
Assume we have a sample Y = (Y1 , . . . , Yn ), where the Yi are independent, identically
distributed with density f (yi ; ). Refer to an earlier handout for the denitions and
orders of magnitude of
Second Term Exam STA 3000Y1Y
Friday April 15, 2011
Instructions: Answer all 4 questions in the exam booklets. Be as precise as possible in your answers, stating clearly the theorems and assumptions you are using.
Questions are of equal value.
1. Given a m
STA 3000F (Fall, 2013)
Comments on Homework 2
1. Non-uniqueness of ancillary statistics. Suppose that (Y1 , Z1 ), . . . (Yn , Zn )
are independent and identically distributed and follow a bivariate normal distribution with E(Yi ) = E(Zi ) = 0, var(Yi ) =
STA3000 Suciency and Ancillarity
Suciency
Denition:
A statistic S = s(Y ) is sucient for , in the family of models f (y; ); , if and
only if f (y|s) is free of .
Factorization Theorem: A statistic S is sucient for (in the family of distributions
cfw_f (y;
STA 3000F (Fall, 2013)
Homework 4
due December 6
1. CH 4.7 Assume y1 , . . . , yn is an independent sample from the exponential distribution with density exp(y), y 0, > 0.
(a) Obtain the uniformly most powerful test of H0 : = 0 against alternatives < 0 ,
STA 3000F (Fall, 2013)
Homework 3
due November 22
1. Prole log-likelihood. Suppose Y = (Y1 , . . . , Yn ) is a vector of independent, identically distributed random variables from the density
f (y; , ), where R is the parameter of interest and R is
a nuis
STA3000: Pivotal quantities based on prole log-likelihoods
The asymptotic theory outlined in the nuisance parameter notes leads to the following three pivotal quantities, in the case that = (, ) and R:
rp () = sign( )[2cfw_ p ()
re () = ( )jp ()1/2 ,
ru
STA 3000F (Fall, 2013)
Homework 1.(September 27; due October 11)
1. CH 2.2 Suppose that random variables Yr follow the rst order autoregressive process
Yr = + (Yr1 ) + r ,
where 1 , . . . , n are i.i.d. N(0, 2 ) and | < 1. Write down the likelihood for da