Bootstrap and Jackknife Calculations in R
Version 6 April 2004
These notes work through a simple example to show how one can program R to do both jackknife and bootstrap sampling. We start with bootstrapping.
Bootstrap Calculations
R has a number of nice
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Chapter 5. Hypothesis Testing
1
Nested Hypotheses
In this chapter we provide a theoretical discussion on testing of statistical hypotheses.
Neyman and Pearson (1933) presented Neyman-Pearson Fundamental Lemma which unfolded the various complex problems
1
Chapter 3. Asymptotic Methods
1
Modes of Convergence of A Sequence of Random Variables
Due to the diculty of making exact calculation, we make use of asymptotic
results. For example, we experience the approximation of probabilities for
computing signica
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Chapter 2. Order Statistics
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The Order Statistics
For a sample of independent observations X1 , X2 , . . . , Xn on a distribution F , the ordered
sample values
X(1) X(2) X(n) ,
or, in more explicit notation,
X(1:n) X(2:n) X(n:n) ,
are called the order
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Chapter 4. Method of Maximum Likelihood
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Introduction
Many statistical procedures are based on statistical models which specify under
which conditions the data are generated. Usually the assumption is made that
the set of observations x1 , . . . , xn i
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Chapter 1. Bootstrap Method
1
1.1
Introduction
The Practice of Statistics
Statistics is the science of learning from experience, especially experience that arrives a little
bit at a time. Most people are not natural-born statisticians. Left to our own d
AN INTRODUCTION TO
EXTREME ORDER
STATISTICS AND
ACTUARIAL
APPLICATIONS
2004 ERM Symposium, Chicago
April 26, 2004
Sessions CS 1E, 2E, 3E:
Extreme Value Forum
H. N. Nagaraja
(hnn@stat.ohio-state.edu)
Ohio State University, Columbus
Hour 1: ORDER STATISTICS
Large Sample Theory
Homework 5: Maximum Likelihood Estimate, Testing, Asymptotic Distribution
Due Date: January 12th
1. Consider the classical Gaussian linear model Yi = i + i , 1 i n, where i = zT
i
and i are i.i.d. Gaussian with mean 0 and variance 2 .
Large Sample Theory
Homework 4: Methods of Estimation, Asymptotic Distribution, Probability and Conditioning
Due Date: December 1st
1. The Weibull distribution (after the Swedish physicist Waloddi Weibull, who proposed
the distribution in 1939 for the bre
Large Sample Theory
Homework 3: Probability and Conditioning
Due Date: November 10th
1. Let X be a random variable with range cfw_0, 1, 2, . . .. Show that if E (X ) < , then
P (X n).
E (X ) =
n=1
2. Let X be a random variable having a c.d.f. F (x). Show
Large Sample Theory
romework PX yrder ttistis
hue hteX ytoer PUth
IF how tht if X hs et @r; sA distriutionDthen
r @r C @k IAA
k a I; P;
@r C sA @r C s C @k IAA
rs
V ar@X A a
:
2
@r C sA @r C s C IA
E @X k A a
PF vet X1 ; : : : ; Xn e smple from uniform U
Large Sample Theory
Homework 1: Bootstrap Method, CLT
Due Date: October 3rd, 2004
1. Suppose that someone collects a random sample of size 4 of a particular measurement. The observed values are cfw_2, 4, 9, 12.
(a) Find the bootstrap mean and variance of
Topic 3: Tests in Parametric Models
Hypothesis Testing By Likelihood Methods
Let H0 denote a null hypothesis to be tested. Typically, we may represent H0
as a specied family F0 of distributions for the data.
For any test procedure T , we shall denote by
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Analysis of Categorical Data: Log-linear analysis
14. Analysing Categorical Data
Log-linear analysis
In clinical investigations we often have response and explanatory variables that are both
categorical. For example ill / not ill as response variable
136
Poisson Regression Analysis
13. Poisson Regression Analysis
We have so far considered situations where the outcome variable is numeric and Normally distributed, or binary. In clinical work one often encounters situations where the outcome variable is
128
Survival Analysis
12. Survival Analysis
In survival analysis we are interested in the time interval between entry into the study and an
event. The outcome of interest is time to an event. Survival analysis was originally developed
for studying time fr
Research methods II
113
11. Analysis of Case-control Studies
Logistic Regression
This chapter builds upon and further develops the concepts and strategies described in Ch.6 of
Mother and Child Health: Research methods.
We have so far considered situations
Research Methods II
99
10. Analysis of Longitudinal Studies
Repeat-measures analysis
This chapter builds on the concepts and methods described in Chapters 7 and 8 of Mother and
Child Health: Research methods.
In repeat-measures designs each subject is obs
Research Methods II
89
9. Analysis of Intervention Studies - III Factorial Designs
In Chapters 7 and 8 the explanatory categorical variables were referred to as treatments or factors. These terms are often used interchangeably. So in a way we have already
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Two-way ANOVA
8. Analysis of Intervention Studies II
Two-way Analysis of Variance
In the data file about depression the subjects were divided by one factor viz. their mental state
(healthy; non-melancholic depressed; melancholic depressed). Many clinic
Research methods II
63
7. Analysis of Intervention Studies I
One-way Analysis of Variance (ANOVA)
This chapter and two more that follow build on Chapter 8 of Mother and Child Health:
Research methods wherein the principles and the basic designs for interv
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Analysis of Cross-Sectional Studies
6. Analysis of Cross-Sectional Studies
Cross-sectional study designs and their variations have been described in the foundation
text - Mother and Child Health: Research Methods in Chapter 5. This chapter builds on th
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Regression Diagnostics
5. Regression Diagnostics
In the preceding chapters the broad principles of multiple linear regression analysis have been
described. The main features of the computer output have been presented, and what specific
features to look
Multiple Regression in Practice
30
4. Multiple Regression in Practice
The preceding chapters have helped define the broad principles on which regression analysis
is based. What features one should look for in the computer output and their interpretation h
Multiple Regression Analysis
16
3. Multiple Regression Analysis
The concepts and principles developed in dealing with simple linear regression (i.e. one
explanatory variable) may be extended to deal with several explanatory variables.
We begin with an exa
Research methods - II
3
2. Simple Linear Regression
Simple linear regression is a technique in parametric statistics that is commonly
used for analyzing mean response of a variable Y which changes according to the
magnitude of an intervention variable X.
Research Methods - II
1
1. About Multivariate Methods
In most studies there are one or more outcome (or response) variables and several explanatory
variables together with a variety of variables, which add a characteristic particularity to the
situation.