3H Inference
Autumn 2016
Lecture 3
A closer look at the likelihood
Charis Chanialidis
River pollution revisited
It is usually not possible to count the number of bacteria in a sample of
river water; one can only determine whether or not any are present.
W
3H Inference
Autumn 2016
Lecture 7
Confidence regions
Charis Chanialidis
3.
Problems with more than one parameter
3.1 A two-parameter example (previous lecture)
3.2 Maximising likelihoods numerically (previous lecture)
3.3 The multiparameter case (previou
3H Inference
Autumn 2016
Lecture 11
Hypothesis tests on categorical data
Part I
Charis Chanialidis
5.
5.1
Hypothesis tests on categorical data
Tests on categorical data
5.2 Comparing several multinomials
5.3 Testing the independence of two categorical var
3H Inference
Autumn 2016
Lecture 9
Hypothesis tests
Part II
Charis Chanialidis
4.
Hypothesis tests
4.1 Some general principles (previous lecture)
4.2 The generalized likelihood ratio test (GLRT)
4.3 What to look for in a good testing procedure
Hypothesis
3H Inference
Autumn 2016
Lecture 19
The bootstrap technique
Charis Chanialidis
Sampling distributions
Sampling distributions
Let X = (X1 , . . . , Xn ) be a random sample from some
distribution F .
Sampling distributions
Let X = (X1 , . . . , Xn ) be a ra
3H Inference
Autumn 2016
Lecture 8
Hypothesis tests
Part I
Charis Chanialidis
4.
Hypothesis tests
4.1 Some general principles
4.2 The generalized likelihood ratio test (GLRT)
4.3 What to look for in a good testing procedure (next
lecture)
4.1 Some general
3H Inference
Autumn 2016
Lecture 10
Hypothesis tests
Part III
Charis Chanialidis
4.
Hypothesis tests
4.1 Some general principles (previous lectures)
4.2 The generalized likelihood ratio test (GLRT)
(previous lectures)
4.3 What to look for in a good testin
3H Inference
Autumn 2016
Lecture 8
Hypothesis tests
Part I
Charis Chanialidis
4.
Hypothesis tests
4.1 Some general principles
4.2 The generalized likelihood ratio test (GLRT)
4.3 What to look for in a good testing procedure (next
lecture)
2
4.1 Some gener
3H Inference
Autumn 2016
Lecture 19
The bootstrap technique
Charis Chanialidis
Sampling distributions
Let X = (X1 , . . . , Xn ) be a random sample from some
distribution F .
T = T (X) is a statistic (could be a vector of statistics).
Need to know about t
3H Inference
Autumn 2016
Lecture 5
Problems with more than one parameter
Part I
Charis Chanialidis
3.
Problems with more than one parameter
3.1 A two-parameter example
3.2 Maximising likelihoods numerically
3.3 The multiparameter case (next lecture)
3.4 C
3H Inference
Notes
Autumn 2016
Lecture 5
Problems with more than one parameter
Part I
Charis Chanialidis
Notes
3.
Problems with more than one parameter
3.1 A two-parameter example
3.2 Maximising likelihoods numerically
3.3 The multiparameter case (next le
3H Inference
Autumn 2016
Lecture 13
Properties of MLEs
Part I
Charis Chanialidis
6.
The properties of MLEs
6.1 Invariance
6.2 Results on expected log-likelihood
6.3 Consistency (next lecture)
6.4 Large sample distribution of MLEs (next lecture)
6.5 Suffic
3H Inference
Notes
Autumn 2016
Lecture 10
Hypothesis tests
Part III
Charis Chanialidis
Notes
4.
Hypothesis tests
4.1 Some general principles (previous lectures)
4.2 The generalized likelihood ratio test (GLRT)
(previous lectures)
4.3 What to look for in a
3H Inference
Autumn 2016
Lecture 2
Likelihood with continuous distributions
Charis Chanialidis
First, the genetics example
The n progeny in a breeding experiment are of three types, there
being xi of the ith type (i = 1, 2, 3). According to a genetic mode
3H Inference
Autumn 2016
Lecture 18
Flexible regression
Charis Chanialidis
Modelling the relationship between variables
Modelling the relationship between variables
Modelling the relationship between variables
Modelling the relationship between variables
3H Inference
Autumn 2016
Lecture 11
Hypothesis tests on categorical data
Part I
Charis Chanialidis
5.
5.1
Hypothesis tests on categorical data
Tests on categorical data
5.2 Comparing several multinomials
5.3 Testing the independence of two categorical var
3H Inference
Notes
Autumn 2016
Lecture 12
Hypothesis tests on categorical data
Part II
Charis Chanialidis
Notes
5.
5.1
Hypothesis tests on categorical data
Tests on categorical data (previous lecture)
5.2 Comparing several multinomials
5.3 Testing the ind
3H Inference
Notes
Autumn 2016
Lecture 8
Hypothesis tests
Part I
Charis Chanialidis
Notes
4.
Hypothesis tests
4.1 Some general principles
4.2 The generalized likelihood ratio test (GLRT)
4.3 What to look for in a good testing procedure (next
lecture)
Note
3H Inference
Autumn 2016
Lecture 14
Properties of MLEs
Part II
Charis Chanialidis
6.
The properties of MLEs
6.1 Invariance (previous lecture)
6.2 Results on expected log-likelihood (previous lecture)
6.3 Consistency
6.4 Large sample distribution of MLEs
6
3H Inference
Autumn 2016
Lecture 1
Introduction to Statistical Inference
Charis Chanialidis
River pollution
Some laboratory tests are run on samples of river water in order to
determine whether the water is safe for swimming.
River pollution
Some laborato
3H Inference
Autumn 2016
Lecture 3
A closer look at the likelihood
Charis Chanialidis
River pollution revisited
It is usually not possible to count the number of bacteria in a sample of
river water; one can only determine whether or not any are present.
R
3H Inference
Autumn 2016
Lecture 14
Properties of MLEs
Part II
Charis Chanialidis
6.
The properties of MLEs
6.1 Invariance (previous lecture)
6.2 Results on expected log-likelihood (previous lecture)
6.3 Consistency
6.4 Large sample distribution of MLEs
6
Exercises 5: More on hypothesis testing
Inference 3
These questions relate to lectures 11-15.
1. A clinical trial was conducted to compare four drugs used to treat hypertension. Each patient was randomly allocated to receive one of the four drugs during t
3H degree exam - May 2011 - Inference questions
Solutions
1. (a) The likelihood function is
P
L() = n xi e /xi ! = i xi en / i xi !
i=1
The log-likelihood function is
l() = ( n xi ) log n + K
i=1
The derivatives are
n
l () = ( xi )/ n
i=1
l () = ( n xi )/
3H degree exam - May 2012 - Inference questions
Solutions
1. (b) The likelihood function is
P
L() = i cfw_3 x2 exi /2 = 3n e i xi K
i
and the log-likelihood is
l() = 3n log()
i
xi + log(K)
The rst derivative is
l () = 3n/ i xi
Setting this to 0 requires
3H degree exam - May 2013 - Inference questions
Solutions
1. (a) Note to external: these are standard manipulations.
Model:
X1 , X2 , . . . , Xn independent, with each Xi Ex()
Data:
x1 , x2 , . . . , xn
Likelihood:
n
i=1
L() =
exi = n e
n
i=1
xi
Log-like
3H degree exam - May 2010 - Inference questions
Solutions
l. (a) The likelihood function is
12(0) = H3=1em?l
The log-likelihood function is
1(0) 2 2;;1{Iog a + (o: l) logmi} = nlogcx + (o: 1) log 3:1-
The derivatives are
rmo=wa+21uea
PM) = n/rr2
T
3H degree exam - May 2007 - Inference questions
Solutions
1. Note to external: parts (a)(c) are largely standard manipulations. Part (d) is unseen
although similar two-parameter problems have been discussed.
(a)
Model:
X1 , X2 , . . . , Xn independent, wi
3H Inference
Autumn 2016
Lecture 12
Hypothesis tests on categorical data
Part II
Charis Chanialidis
5.
5.1
Hypothesis tests on categorical data
Tests on categorical data (previous lecture)
5.2 Comparing several multinomials
5.3 Testing the independence of
3H Inference
Autumn 2016
Lecture 4
Interval estimates
Charis Chanialidis
Likelihood intervals
We can get a confidence interval from the result that
l(0 )] 21 (c) c,
Pcfw_2[l()
Likelihood intervals
We can get a confidence interval from the result that
l(
3H Inference
Autumn 2016
Lecture 7
Confidence regions
Charis Chanialidis
3.
Problems with more than one parameter
3.1 A two-parameter example (previous lecture)
3.2 Maximising likelihoods numerically (previous lecture)
3.3 The multiparameter case (previou