Relationships between three dispersion measures used in flood frequency analysis

Abstract

Three dispersion measures of a random variable, i.e., the standard deviation, the mean deviation (MD) about the mean and the second L-moment, are analyzed in terms of their properties and mutual relationships. Emphasis is placed on the MD, as it is less recognized than two other dispersion measures. The relationships between the dispersion measures are derived for distributions commonly applied in flood frequency analysis (FFA). For distributions that are unbounded, there is a distribution-dependent constant value of the ratio of dispersion measures, or equivalently of respective coefficients of variation. For two-parameter distributions that are lower-bounded, the relationship between the coefficients of variation is also distribution dependent and is not linear. For lower-bounded three-parameter distributions, the dispersion measure ratios, or equivalently the ratios of coefficients of variation, depend on the coefficient of skewness and show a strong distributional dependence. For selected distributions, the three dispersion measures are compared both in terms of the robustness to the largest samples element and the accuracy of upper quantile estimation. The MD statistics may be highly competitive to the two other dispersion measure statistics if applied in FFA for parameters estimation.

Appendices

Appendix A

Algebraic bound of variation coefficients

Katsnelson and Kotz (1957) proved that for a set n (≥ 2) non-negative values x _i , not all equal, the coefficient of variation (CV=σ/μ) cannot exceed (n − 1)^1/2 attaining this value if and only if all but one of the x _i values are zero.

For such a set of values one gets the MD

$$d_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{x}}} = \frac{1}{n}{\left[{\frac{{{\left({n - 1} \right)}}}{n}x + {\left({x - \frac{1}{n}x} \right)}} \right]} = \frac{{2{\left({n - 1} \right)}}}{{n^{2}}}x$$

(24)

Hence the sampling upper algebraic bound of the coefficient of variation d − CV is

$$d - CV = {d_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{x}}}} / {\ifmmode\expandafter\bar\else\expandafter\=\fi{x}} = 2{\left({1 - 1 / n} \right)}$$

(25)

and the asymptotic value δ− CV=2. Proceeding in the same way one finds that for a distribution that takes only positive values, the estimate of the L-coefficient of variation does not have the algebraic bound dependent on the sample size and its value is in the range 0 ⩽ t ⩽ 1.

Appendix B

The δ_μ/σ ratio for selected distributions.

Taking the uniform distribution $f{\left(x \right)} = 1/{b;\;x \in {\left[{0,\,b} \right]}}$ one gets $\sigma = b/{(2{\sqrt 3})}$ and δ_μ=b / 4. Hence δ_μ/σ=0.866, which is greater than for the normal distribution (0.798).

For the binomial distribution, i.e., $P{\left({X = 0} \right)} = p,\quad \;P{\left({X = b} \right)} = q = 1 - p,$ we get $\sigma = b{\sqrt {q - q^{2}}};$ $CV = {\sqrt {1/{q - 1}}}$ and δ_μ=2bq(1 − q) ; δ− CV=2(1 − q). Hence ${\delta _{\mu}}/{\sigma = 2{\sqrt {q - q^{2}}}},$ which gets maximum equal one for q=0.5.