TR-9 : Some studies on plotting position formulae for Gumbell EV-1 distribution

Abstract

Probability plots are used in flood frequency analysis to fit the probability distributions to given flood series, to identify the outliers and to assess goodness of fit. If the objective of such a plot is to determine distribution parameters, then one must use unbiased plotting positions.

Gumbel extreme value type 1 distribution is one of the commonly used distributions in India for flood frequency analysis. At present many plotting position formulae are in use for Gumbel EV-1 distribution. These formulae provide different results, particularly at the tails of the distribution.

In the present study various plotting position formulae have been compared with the unbiased plotting positions (the expected value of reduced variates) on the basis of seven statistical criteria for Gumbel EV 1 distribution. Unbiased plotting positions have been obtained from synthetically generated EV 1 reduced variates for different sample sizes. The results indicate that the plotting positions given by Gringorton formula P (X>,x ) = (i - 0. 44)/(N + 0. 12) are more close to the unbiased plotting positions (expected value of reduced variates) for all the sample sizes (10-100) on the basis of following four criteria

i) total sum of squares of error, ii) absolute error in highest reduced variate, iii) sum of squares of error in top 3 reduced variates and iv) sum of squares of error in top 6 reduced variates

2. The performance of cunnane and Hazen plotting position formulae is better than all other formulae (except Gringorton) on the basis of above 4 criteria.

3. The plotting positions given by Tukey formulae P(X:>x) = (i - 1/3)/(N+1/3) are more close to the unbiased plotting positions in the lower tail, on the basis of

i) absolute error in lowest reduced variate, ii) sum of squares of error in bottom 3 reduced variates and iii) sum of squares of error in bottom 6 reduced variates.

This clearly indicates the suitability of Gringorton plotting position formula for flood frequency analysis using Gumbel EV-1 distribution, which involves extrapolation in the upper tail region.