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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
Gamma Distribution
)()/exp(
)1()/()(
xxx
NP
: scale parameter : shape parameter
Beta Distribution
1)1(1)()()()(
qxpxqpqpx
iP
Lognormal Distribution
Science
Example
Null Hypothesis (H0)
Test Statistic
Alternative Hypothesis (HA)
Hypothesis Testing
Variance
?22 s
Mean
?11 Ni ix
Nx
Ni
xxN
si1
2
11
Standard Deviation
NINO3 (90-150W, 5S-5N)
Gaussian Series?
Histogram
Gaussian?
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!
Gaussian Distribution (cont)
Z is a test statistic!
Two-Sided or Two-tailed test!
p=0.05
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
2
212lnln),(
2
ixNNL
0),(
L 0
21
i
x
xxN
N
ii
1
1
0),(
L 01 2
3
i
xN
21 ix
N
21 ix
N
Ni
xxN
si1
2
11
But we know,
Maximum likelihood estimates are often biased estimates!
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
Ni
ix
1
22
Chi-Squared
2
21exp
21
,...,1
ixN
iP
NP
2/2)2/(
)2/exp(2/)2(
)(2
NN
xN
xxNP
2/2)2/(
)2/exp(2/)2(
)(2
NN
xN
xxNP
N2
N222
2(=5)
Ni
ix
1
22
Chi-Squared
Reduced Chi-Squared
vv /22
Reduced Chi-Squared
Reduced Chi-Squared
Reduced Chi-Squared
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
Use reduced Chi-Squared distribution
ihN
i ih
ig /)
1( 22
2(hi)= hi
=N-2 (sigma estimated from data)
=N-3 (mu and sigma estimated from data)
v/2
