1. Case Study – Gumbel Distribution
Rhone River – Maximum Daily Discharge (Annual) –
Data Source: E. Gumbel (1941). “The Return Period of Flood Flows,”
The Annals of Mathematical Statistics, Vol. 12,#2, pp

2. Data Years 1826-1936: maximum daily discharge of river obtained. order
899 23
1992 45
2240 67
2586 89
3067 2
1172 24 46
2258 68
2594 90
3126 3
1231 25
2006 47
2281 69 91
3179 4
1272 26 48
2296 70 92
3214 5 27
2013 49
2327 71
2602 93
3250 6
1432 28
2050 50
2342 72
2626 94
3266 7 29 51
2358 73
2627 95
3293 8
1439 30
2072 52
2381 74
2643 96
3310 9
1444 31
2094 53
2420 75
2675 97 10
1502 32
2101 54
2444 76 98
3354 11
1541 33
2115 55
2452 77
2773 99
3426 12
1560 34
2145 56
2467 78
100
3444 13
1639 35 57
2475 79
101 14
1706 36
2153 58 80
2839
102
3480 15
1780 37
2160 59 81
2856
103
3606 16
1829 38
2168 60
2491 82
2881
104
3625 17
1850 39
2175 61
2514 83
105
3708 18
1857 40
2206 62 84
2965
106
3801 19
1913 41 63 85
3007
107
3810 20 42 64 86
3050
108
3905 21
1934 43
2221 65
2538 87
3058
109
4096 22
1955 44
2236 66
2554 88
110
4105
111
4390

3. Gumbel Distribution

4. Maximum Likelihood Estimation (I)

5. Maximum Likelihood Estimation (II)

6. Maximum Likelihood Estimation (III)
Step 1, choose starting values, say: a(0) and b(0) are assigned the method of moment estimators
Step 2: Choose a tolerance level for the change is estimates, say k=10-6
Step 3: Obtain H-1(a(0), b(0)) and f(a(0), b(0))
Step 4: Obtain a(1) and b(1) from Newton-Raphson algorithm
Step 5: Check to see if (a(1)- a(0))2+ (b(1)- b(0))2 <k.
If Yes, Stop.
If No, Return to Step 3, obtaining H-1(a(1), b(1)), f(a(1), b(1))

7. Rhone River Discharge Data

8. Rhone River Discharge Data (2)
Iteration History:
Round
alpha
beta
Delta
<k?
N/A 1 No 2 3 4 5
8.99E-10
Yes

9. Rhone River Discharge (3)

10. Rhone River Discharge Data (4)