Case Study – Gumbel Distribution

1. 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.

2. Data • Years 1826-1936: maximum daily discharge of river obtained.

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:

9. Rhone River Discharge (3)

10. Rhone River Discharge Data (4)