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Chapter 7 Dr Wayne Stewart
Calculate n
Must know this proof for the exam
Method of moments to create point estimates
The following will be an example of MLE
Learn by doing!!
Go through all examples
Maximum likelihood method
Important!!
Maximum likelihood estimators
3 ways we will learn to make MLE’s
x
10
5.2
2
NR approximation
2.98
3 ways to perform MLE
0
20
40
f(x)
60
80
100
Newton-Raphson Algorithm
Newton Raphson derivation
Other methods
Other point estimators not discussed in detail within this course
JACKNIFE
ROBUST
JACK-KNIFE method
JackknifeR
R package
• Leave one(or more) datum out then estimate
Function that creates a vector of stats
compare
BOOTSTRAP
Interval estimates
Pivotal method
• Follow the proof
Two ways of obtaining ci’s
R as calculator
t.test()
n-1 degrees freedom
Two interpretations of ci’s
1
2
names(cc) head(cc) ccnew←within(cc, ASPHALT ← factor(ASPHALT)) with(ccnew, boxplot(WATERLOSS ~ ASPHALT)) with(ccnew,var.test(WATERLOSS~ASPHALT)) with(ccnew, t.test(WATERLOSS ~ ASPHALT, mu = 0, var.equal = TRUE))
Welch 2 sample t-test
paired samples
Large sample ci for a proportion
ci for difference in proportions
ci for a population variance
The F pivotal statistic
Properties of the f statistic
𝐹𝛼 𝜈1 , 𝜈2 2
see proof
1
𝐹1−𝛼 𝜈2 , 𝜈1 2
ci for ratio of population variances
Calculate n for a half width
Check out all examples
Solution
We could use 𝑡𝛼 2
Not much difference The use of 𝑍𝛼 2
instead of 𝑡𝛼 is to 2
simplify the computation
PET EXAMPLE
R!!
Using the Bayesian paradigm
Bayes’ Rule
𝒑 𝜽 𝒇(𝒙|𝜽) 𝒑 𝜽𝒙 = 𝒑 𝒙
Use