An Extended Inverse
Gaussian Model
Iman Mabrouk
Statistical and Actuarial Sciences Department
The University of Western Ontario
Supervised by:
Dr. Serge Provost
Outline
• Introduction
• Parameter Effect
• Some Statistical Functions
• Numerical Examples
1 Introduction
• The proposed distribution referred to
as an Extended Inverse Gaussian
Model (EIG) has density function
where
• The EIG model is a generalization of a
distribution introduced by Jørgensen
(1982) whose density function is
where
is a Generalization of the
Inverse Gaussian model (GIG) where the
density of Inverse Gaussian model is
Reduced Models
1) A reduced model called the Reduced
Extended Inverse Gaussian(REIG) is
obtained by omitting
. The density
function of REIG model is
Reduced Models
2) Omitting
yields a
Reparameterized Generalized Gamma
model (RGG)
where the density of the generalized
gamma model is
and it can be obtained from
letting
by
2 Parameter Effect
2.1 The Extended Inverse Gaussian Model (EIG)
2)
1
2.2 The Reduced Extended
Inverse Gaussian Model (REIG)
3 Some Statistical Functions
3.1 The Extended Gaussian Model (EIG)
3.2 The Reduced Extended
Inverse Gaussian Model (REIG)
4 Numerical Examples
4.1 Goodness-of-fit statistics
4.2 Maximum Flood Levels
4.3 Snowfall Precipitations
4.1 Goodness-of-fit statistics:
i. The Anderson-Darling statistic
where
and the
are the
ordered observations.
ii. The Cramér-von Mises statistic
4.2 Maximum Flood Levels
Maximum Flood Levels in millions of cubic
feet per second for the Susquehanna River
at Harrisburg, Pennsylvania over 20 fouryear periods.
0.654
0.740
0.297
0.613
0.416
0.423
0.402
0.338
0.379
0.379
0.315
0.3235
0.269
0.449
0.418
0.412
0.494
0.392
0.484
0.265
Maximum Likelihood Estimates
Estimates of the Parameters and Goodnessof-Fit Statistics for the Flood Data
Weibull
-1
11
Gamma
Inverse Gaussian - 2.5
Lognormal (-.898,.269)
339.1
RGG
-16.57
GIG
-10
REIG
-9.95
EIG
3.53
352
1
1
0
.8213 .1400
0
.4433 .0712
1.03 2.82 .3514 .0558
.3470 .0540
.036
9600 0
.3390 .0560
1
1.73 5.73 .2861 .0449
2.3
0
.311 .2567 .0436
2.24
.09
.34
.2551 .0437
Notes
 The EIG model and its reduced version
provide a better fit than that resulting from
the other models
 The EIG and REIG models fit the data
nearly equally well in this case. Note that
 The sample size is small
 Only scant data is available in the tails of
the distribution
CDF (solid line) and empirical CDF (dots) for
the flood data set using EIG model
4.3 Snowfall Data
Snow precipitation in centimeters over 63
consecutive years in the city of Buffalo.
25
39.8
39.9
40.1
53.5
54.7
55.5
55.9
69.3
70.9
71.4
71.5
78.2
78.4
79
71.3
83.6
83.6
48.8
85.5
90.9
97
98.3
101.4
110.5
110.5
113.7
114.5
46.7
58
71.8
79.6
87.4
102.4
115.6
49.1
60.3
72.9
80.7
88.7
103.9
120.5
49.6
63.6
74.4
82.4
89.6
104.5
120.7
51.2
65.4
76.2
82.4
89.8
105.2
124.7
51.6
66.1
77.8
83
89.9
110
126.4
Maximum Likelihood Estimates
Estimates of the Parameters and Goodnessof-Fit Statistics for the Snowfall Data
Inverse Gaussian -2.5
1
.055
353.3 .8676 .1504
.7752 .1284
Lognormal (-.898,.269)
REIG
53.66
.167 0
650.2 .7417 .0886
Gamma
GIG
Weibull
RGG
EIG
8
1
.125
0
7.97
1
.122
.0025 .4291 .0532
-1
3.83
0
.2964 .0454
-.289
3.63
0
.2817 .0428
-0.056 3.14
.4840 .0792
1.743 .2625 .0403
CDF (solid line) and empirical CDF (dots)
for the snowfall data set using EIG Model
Conclusion
The Extended Inverse Gaussian
model provides more flexibility
for fitting various types of data
sets
References
1. D'Agostino, R. B. and Stephens, M. A. (1986). Goodness-of-fit
Techniques. CRC Press, Boca Raton.
2. Dumonceaux, R., Antle, C. (1973). Discrimination between the
log-normal and the Weibull distributions, Technometrics, 15(4),
923-926.
3. Jorgensen, B. (1982). Lecture Notes in Statistics, eds. D.
Brillinger, S. Fienberg, J. Gani, J. Hartigan, J. Kiefer, and K.
Krickeberg, Springer-Verlag, New York.
4. Mathai, A.M. and Saxena, R.K. (1978). The H-function with
applications in statistics and other discplines. Wiley, New York.
5. Seshadri, V. (1993). The Inverse Gaussian Distribution: A Case
Study in Exponential Families. Oxford Science Publication,
Oxford.