hypothesis testing distributions(continued); maximum likelihood; parametric hypothesis tests...
TRANSCRIPT
HYPOTHESIS TESTINGDistributions(continued); Maximum Likelihood;
Parametric hypothesis tests (chi-squared goodness of fit, t-test, F-test)
LECTURE 2
Supplementary Readings:
Wilks, chapters 4,5;
Bevington, P.R., Robinson, D.K., Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, 1992.
Gamma Distribution
More general form of the Chi-Squared distribution
2/2)2/(
)2/exp(2/)2(
)(NN
xN
xxNP
)()/exp(
)1()/()(
xxx
NP
: scale parameter : shape parameter
22
Variance
?22 s
Mean
?11 Ni ix
Nx
Ni
xxN
si1
2
11
Standard Deviation
NINO3 (90-150W, 5S-5N)
Gaussian Series?
Gaussian Distribution (cont)
How do we invoke Gaussian Null hypothesis? Can we use PG alone?
Z is a test statistic!
Gaussian Distribution (cont)
Z is a test statistic!
A more readily applicable form of the Gaussian Null Hypothesis is provided by Integral of Gaussian Distribution
Two-Sided or Two-tailed test!
Central Limit Theorem
For a sum of a large number of arbitrary independent, identically distributed (IID) quantities, joint PDF approaches a Gaussian Distribution.
Consequence:
the distribution of a mean quantity is approximately Gaussian for large enough
sample size.
Why?
2
21exp
21
,...,1
ixN
iP
NP
Method of Maximum Likelihood
Most probable value for the statistic of interest is given by the peak value of the joint probability distribution.
The most probable values of and are obtained by maximizing P with respect to these parameters
Consider Gaussian distribution
Easiest to work with the Log-Likelihood function:
2
212lnln),(
2
ixNNL
Method of Maximum Likelihood
2
21exp
21
,...,1
ixN
iP
NP
The most probable values of and are obtained by maximizing P with respect to these parameters
Easiest to work with the Log-Likelihood function:
2
212lnln),(
2
ixNNL
We want to maximize L relative to the two parameters of interest:
0),(
L 0),(
L
Method of Maximum Likelihood
Central Limit Theorem
2
21exp
21
,...,1
ixN
iP
NP
xxN
N
ii
1
1What is the standard deviation in the mean ?
Uncertainties of Gaussian distributed quantities add in quadrature
Central Limit Theorem
2
21exp
21
,...,1
ixN
iP
NP
xxN
N
ii
1
1What is the standard deviation in the mean ?
2222 ...21 Nxxxx
22 Nx
NN xx
1
Histogram
How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)?
“Goodness of fit”
Ni h
ih
ig
1
2
)(2
What is 2(hi)? hi
How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)?
What is 2(hi)? hi
How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)?
ihN
i ih
ig /)
1( 22