Case Study – Gumbel Distribution Rhone River – Maximum Daily Discharge (Annual) – 1826-1936 Data Source: E. Gumbel (1941). “The Return Period of Flood Flows,” The Annals of Mathematical Statistics, Vol. 12,#2, pp.163-190.

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

Gumbel Distribution

Maximum Likelihood Estimation (I)

Maximum Likelihood Estimation (II)

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))

Rhone River Discharge Data

Rhone River Discharge Data (2) Iteration History: Round alpha beta Delta <k? 2178.157 546.0046 N/A   1 2161.899 604.3112 3663.998 No 2 2156.052 631.3299 764.1949 3 2155.18 635.4366 17.62557 4 2155.163 635.5164 0.006657 5 8.99E-10 Yes

Rhone River Discharge (3)

Rhone River Discharge Data (4)