1. Goodness-of-Fit Tests Applications
Ch07
Goodness-of-Fit Tests Applications

2. Objective of this chapter：
CHAPTER CONTENTS
CHAPTER CONTENTS
7.1 Introduction
7.2 The Chi-Square Tests for Count Data
7.3 Goodness-of-Fit Tests to Identify the Probability Distribution
7.4 Applications: Parametric Analysis
7.5 Exercises
7.6 Chapter Summary
7.7 Computer Examples
Projects for Chapter
Objective of this chapter：
To determine if a given set of data follows a particular probability distribution.

3. Karl Pearson ( )
In 1893, Pearson coined the term “standard deviation.”

4. Phenomenon of interest
7.1 Introduction
Phenomenon of interest
the amount of carbon dioxide, CO2, in the atmosphere on a daily basis
the sizes of cancerous breast tumors
the monthly average rainfall in the State of Florida
the average monthly unemployment rate in the United States
the hourly wind forces of a hurricane
Etc.
What are the probabilistic behaviors of these phenomena？
i.e., what are the probability density functions, pdf, that characterizes these phenomenon of interest？

5. Statistical tests (methods) for determining
how good the data fits a particular probability distribution：
Pearson’s chi-square test
Kolmogorov-Smirnov test
Anderson-Darling test
Shapiro-Wilk test
P-P plots
Q-Q plots
Nonparametric (probability distribution-free) analysis (Ch. 12)

6. 7.2 The Chi-Square Tests for Count Data
How likely is it that an observed probability distribution is due to chance？
2 Test

7. 7.2.1 TESTING THE PARAMETERS OF A MULTINOMIAL DISTRIBUTION: GOODNESS-OF-FIT TEST
The 2-goodness-of-fit test.

9. 7.2.2 CONTINGENCY TABLE: TEST FOR INDEPENDENCE
Objective: To test for dependencies between the rows and columns in a contingency table.

12. 7.3 Goodness-of-Fit Tests to Identify the Probability Distribution

13. Null hypothesis: X ~ pdf
=> Expected values Ei
Observation: (Xi) Oi

14. 7.3.1 PEARSON’S CHI-SQUARE TEST
Observed:
Expected: O3 O4 O5 O6 O1 O2 3 e3 2 e2 1 e1 5 e5 6 e6
Prob. = 4
Freq. =e4
I I I I I I6 (Ik)

16. 7.3.2 THE KOLMOGOROV-SMIRNOV TEST: (ONE POPULATION)
H0 : The true probability distribution that follows the given data, F(x), is actually the assumed distribution F0(x)
Ha : The actual cumulative distribution, F(x) is not F0(x),
Test statistic: D = Max F0(x) – Fn(x)
where F0(x): the hypothesized cdf
Fn(x): the empirical distribution function
Fn(x) = (Xi – X)/n
Critical value: D (from the Kolmogorov-Smirnov tables)

18. 7.3.3 THE ANDERSON-DARLING TEST
H0 : The given data follow a specific probability distribution
Ha : The given data do not follow the specified probability distribution:
Test statistic: A ( A2 =  n  s )
Critical value: A (from the Anderson+Darling tables)

20. 7.3.4 SHAPIRO-WILK NORMALITY TEST

21. 7.3.5 THE P-P PLOTS AND Q-Q PLOTS
To determine if a given random sample of data follows or is drawn from a well-known probability distribution

22. Th P-P plot compares
the empirical cdfs (of the given data) with
the assumed true cdfs

23. Q-Q Plot the quantiles of the empirical distribution of the given data
versus
the quantiles of the assumed true pdf that we are testing.

24. 7.4 Applications: Parametric Analysis

25. 7.5 Exercises

29. 6.6 Chapter Summary

30. 6.7 Computer Examples

31. Projects for Chapter 7