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

1

12

1

1 1

1 1

Gamma Distribution: | , 0; , 0

Likelihood: , | ,

log Likelihood: ln ln 1 ln

Lognormal Distribution: ~

n

ii

y

nn n y

i ini i

n n

i ii i

f y y e y E Y V Y

L f y y e

l n n y y

X N

2

2 22 2

2 2

ln( )

222 22

2

2

2 2 2 122

1

1

, ~ Lognormal ,

1| , 0; , 0

2

ln1

Likelihood: , | , 2 exp2

X

y

n

n ini

i ni

ii

Y e

ef y y E Y e V Y e e

y

yL f y

y

2

2 12

1

lnlog Likelihood: ln 2 ln

2 2

n

i ni

ii

yn

y

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

2

22~ ~

2 2

22

~ ~

2 22 2

22

~ ~

2 22 2

Gamma Distribution:

6.3376.337Males: 5.665 35.896

1.058 1.058

5.8405.840Females: 8.456 49.381

0.831 0.831

M M

F F

Y Y

MMMM

Y Y

FFFF

Y Y

E Y V Y

YE YE Y YE Y

V Y V Y S S

YY

S S

YY

S S

Method of Moments Estimators - Lognormal

22 2

2 2

2 22

22

2 2

2 2

22 2

2 22 22~ ~

2 2

2~

Lognormal Distribution:

ln 12

1ln ln ln

2

1ln ln ln

2

Males:

Y Y

M

E Y e V Y e e

V Y E YV YE Y e e

E Y E Y

V Y E Y V Y E YE Y

E Y E Y

S Y S YY

Y Y

2 2~

2

2 22~ ~

2

1.058 6.337 1ln 0.02748 ln 6.337 0.02748 1.83266

26.337

0.831 5.840 1Females: ln 0.02005 ln 5.840 0.02005 1.75468

25.840

M

F F

Method of Moments Estimates / Graphs

Maximum Likelihood Estimators - Gamma

1

* * * *

1

1 1

1 1

* * * * *

1 1

*

Likelihood: , | , 0; , 0

log Likelihood: ln ln 1 ln

ln 1 ln ln ln ,

n

ii

nn n y

i ini i

n n

i ii i

n n

i ii i

L f y y e y

l n n y y

ne n e e y e y

lne

* * * *

* *

* * * *

* *

*

1

*1

22* 2

2 2*1

2 2

2 * **1

2 2

* * 2 * *^ ^ *1

* * 2^ ^

1

lnln

ln' ln '

n

ii

n

ii

n

ii

n

ii

t t

t t

xn e e e y x

x

lne e y

xlne n e e e y x

x

l le y ne

l l

* *^ ^

*

2

*2* * * ,

,

** ^^

1

Newton-Raphson Algorithm (iterate to convergence)

t t

t t

l

ll l

Maximum Likelihood Estimators - Lognormal

2

2 2 2 122

1

1

2

2 12

1

^1

2

ln1

Likelihood: , | , 2 exp2

lnlog Likelihood: ln 2 ln

2 2

ln ln0

n

n ini

i ni

ii

n

i ni

ii

n

i seti

yL f y

y

yn

l y

y yl

1

2 2 2^ ^ ^

2^1 1 1

2 2 4 2 4

ln 1 ln ln2

04 2 2 2

n

ii

n n n

i i iseti i i

n

y y yl n n

n

Maximum Likelihood Estimates

* *^ ^ ^ ^3.6031 1.7567

* *^ ^ ^ ^3.9370 2.1723

^

Gamma with Moment Estimates as Starting Values:

Males:

3.6031 36.7114 1.7567 5.7932

Females:

3.9370 51.2636 2.1723 8.7782

Lognormal:

M M M M

F F F F

M

e e

e e

2^

2^ ^

2664.776 39.64281.8327 0.027265

1454 1454

1833.889 20.18101.7549 0.019312

1045 1045

M

FF

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,∞)

2

2

1

# of cells df #of estimated parameters (2 for Gamma, Lognormal) k

i i

i i

n E nX k k

E n

Minimum Chi-Square Results

~ ~2 2

,

~ ~2 2

,

~ ~2 2

,

Males/Gamma: 34.87 5.51 7.71 0.05;11 2 9 16.919

Females/Gamma: 47.20 8.12 20.25 0.05;8 2 6 12.592

Males/Lognormal: 1.832 0.168 6.11 0.05;11 2 9 16.919

Females/Lo

M M M G

F F F G

MM M L

X

X

X

~ ~

2 2,gnormal: 1.750 0.144 13.24 0.05;8 2 6 12.592FF F LX

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.