1. HYPOTHESIS TESTING Distributions(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.

2. Gamma Distribution More general form of the Chi-Squared distribution  : scale parameter  : shape parameter

3. Gamma Distribution  : scale parameter  : shape parameter

4. Beta Distribution

5. Lognormal Distribution Science Example

6.  Null Hypothesis (H 0 )  Test Statistic  Alternative Hypothesis (H A ) Hypothesis Testing

7. Variance MeanStandard Deviation NINO3 (90-150W, 5S-5N) Gaussian Series?

8. Histogram Gaussian?

9. Gaussian Distribution (cont) How do we invoke Gaussian Null hypothesis? Can we use P G alone? Z is a test statistic!

10. 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!

11. Gaussian Distribution (cont) Z is a test statistic! Two-Sided or Two-tailed test! p=0.05

12. 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?

13. 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

14. Easiest to work with the Log-Likelihood function: Method of Maximum Likelihood The most probable values of  and  are obtained by maximizing P with respect to these parameters

15. Easiest to work with the Log-Likelihood function: We want to maximize L relative to the two parameters of interest: Method of Maximum Likelihood

17. But we know, Maximum likelihood estimates are often biased estimates!

18. Central Limit Theorem What is the standard deviation in the mean ? Uncertainties of Gaussian distributed quantities add in quadrature

19. Central Limit Theorem What is the standard deviation in the mean ?

20. Chi-Squared

21.  2 ( =5) Chi-Squared

22. Reduced Chi-Squared

25. Histogram How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)? “Goodness of fit”

26. What is  2 (h i )? hihi How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)?

27. What is  2 (h i )? hihi How do we determine if the observed histogram is consistent with a particular distribution (e.g. Gaussian)?

28. Use reduced Chi-Squared distribution  2 (h i )= h i = N-2 (sigma estimated from data) = N-3 (mu and sigma estimated from data)