Model Estimation and Comparison Gamma and Lognormal Distributions

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Model Estimation and Comparison Gamma and Lognormal Distributions 2015 Washington, D.C. Rock ‘n’ Roll Marathon Velocities

Data Description / Distributions Miles per Hour for 2499 people completing the marathon (1454 Males, 1045 Females) Males: Mean=6.337, SD=1.058, Min=4.288, Max=10.289 Females: Mean=5.840, SD=0.831, Min=4.278, Max=8.963

Gamma and Lognormal Distributions

Method of Moments Estimators - Gamma Obtain the Sample Mean and Variance and Use them to obtain estimates of parameters

Method of Moments Estimators - Lognormal

Method of Moments Estimates / Graphs

Maximum Likelihood Estimators - Gamma

Maximum Likelihood Estimators - Lognormal

Maximum Likelihood Estimates

Maximum Likelihood Estimates / Graphs

Minimum Chi-Square Estimator Slice Range of Y (mile per hour) values into a set of non-overlapping sub-ranges Create a grid of parameter values for each distribution (Gamma and Lognormal) Obtain the Pearson Chi-Square statistic for each set of parameter values and choose the values that minimize the Chi-Square statistic Ranges for this example: Males: (0,4.75],(4.75,5.25],…,(8.75,9.25] ,(9.25,∞) Females: (0,4.75],(4.75,5.25],…,(7.25,7.75] ,(7.75,∞)

Minimum Chi-Square Results For both Males and Females, the Lognormal appears to fit better than the Gamma (smaller minimum chi-square statistic). However, for Females, the chi-square statistic exceeds the critical value, rejecting the null hypothesis that the distribution is appropriate.