Probability of correct selection from lognormal and convective diffusion models based on the likelihood ratio

Abstract

The objective of the paper is to show that the use of a discrimination procedure for selecting a flood frequency model without the knowledge of its performance for the considered underlying distributions may lead to erroneous conclusions. The problem considered is one of choosing between lognormal (LN) and convective diffusion (CD) distributions for a given random sample of flood observations. The probability density functions of these distributions are similarly shaped in the range of the main probability mass and the discrepancies grow with the increase in the value of the coefficient of variation (C _V ). This problem was addressed using the likelihood ratio (LR) procedure. Simulation experiments were performed to determine the probability of correct selection (PCS) for the LR method. Pseudo-random samples were generated for several combinations of sample sizes and the coefficient of variation values from each of the two distributions. Surprisingly, the PCS of the LN model was twice smaller than that of the CD model, rarely exceeding 50%. The results obtained from simulation were analyzed and compared both with those obtained using real data and with the results obtained from another selection procedure known as the QK method. The results from the QK are just the opposite to that of the LR procedure.

Appendices

Appendix A: Pertinent characteristics of the CD and LN distributions

Some relations of the competing distributions used in this study are summarized in Table A.1.

Table A.1 Moment characteristics, quantile and ML-solution of CD and LN

Sample average log likelihood function Eq. (7):

The CD distribution

$$\hat{\Lambda}_{(CD)}^{(N)} = \ln \alpha - \ln \sqrt{\pi} - \frac{3}{2} \hat{\hbox{E}} (\ln X) - \alpha^{2} \hat{\hbox{E}} (X^{-1}) + 2 \beta - \left(\frac{\beta}{\alpha}\right)^{2} \hat{\hbox{E}} (\hbox{X}) $$

(A.1)

the LN distribution

$$\hat{\Lambda}_{(LN)}^{(N)} = - \hat{\hbox{E}} (\ln X) - \ln \sigma - \ln \sqrt{2\pi} - \frac{1}{2\sigma^{2}} \hat{\hbox{E}} (\ln \hbox{X} - \mu)^{2}$$

(A.2)

where

$$\hat{\hbox{E}} (Z) = \frac{1}{N} \sum\limits_{i = 1}^{N} z_{i} $$

The logarithms of the selection statistics S _i [Eq. (9)] for the CD and the LN models are of the form:

$$\begin{aligned} \ln \hbox{S}_{(CD)}^{(N)} &= \ln 2 + \frac{\hbox{N}}{2} \ln \left(\frac{\beta}{\pi}\right) + 2 N \beta - \frac{3\hbox{N}}{2} \hat{\hbox{E}} (\ln \hbox{X})\\ &\quad + \frac{1}{4} \left\{ \ln \left[\hat{\hbox{E}} (\hbox{X})\right] - \ln \left[\hat{\hbox{E}} (\hbox{X}^{-1}) \right] \right\}\\ &\quad + \ln \left\{K_{-N/2} \left[ 2N\beta \sqrt{\hat{\hbox{E}} (\hbox{X}) \hat{\hbox{E}} (\hbox{X}^{-1})}\right] \right\}\\ \end{aligned}$$

(A.3)

where K _ν (z) represents the modified Bessel function of second kind.

$$\ln \hbox{S}_{(LN)}^{(N)} = (1 - N) \ln \left(\sigma \sqrt{2\pi}\right) - \frac{1}{2} \ln (\hbox{N}) - \hbox{N} \cdot \hat{\hbox{E}}(\ln \hbox{X}) - \frac{N}{2\sigma^{2}}\left[\hat{\hbox{E}} (\ln^{2} \hbox{X}) - \hat{\hbox{E}}^{2} (\ln \hbox{X}) \right] $$

(A.4)

As stated in Introduction the CD model can be derived from the Halphen type A distribution. To show it let us reparameterize the Halphen type A probability density function (e.g., Perrault et. al., 1999a):

$$f_{A} (x) = \frac{1}{2\left(\frac{\alpha^{2}}{\beta}\right)^{v} K_{v} (2\beta)} x^{v-1} \exp \left[- \left(\frac{\beta^{2}}{\alpha^{2}}x + \frac{\alpha^{2}}{\beta x}\right) \right],\quad x > 0$$

(A.5)

For large values of the argument z the modified Bessel function of the second kind of order v can be approximated by the first term of the expansion:

$$K_{v} (z) = \sqrt{\frac{\pi}{2z}}e^{-z} \left[1 + \frac{u- 1}{8z} + \frac{(u - 1)(u - 9)}{2! (8z)^{2}}+\cdots\right]$$

(A.6)

where u=4v ². Substituting it into Eq. (A.5) and putting v=− 1/12, one gets the CD density function [Eq.(10)]. Note from Table A.1 that large values of the argument z (z=2β) correspond to small values of the C _V .

Appendix B: Distribution of (M ^(N) (CD|LN), M ^(N) (LN|LN)) variable

The $\hat{\hbox{M}}^{(N)} (\hbox{CD}|\hbox{LN})$ and $\hat{\hbox{M}}^{(N)} (\hbox{LN}|\hbox{LN})$ variables are highly correlated so the scatter points diagram $\left((\hat{\hbox{M}}_{s}^{(N)} (\hbox{CD}|\hbox{LN}),\; \hat{\hbox{M}}_{s}^{(N)} (\hbox{LN}|\hbox{LN})),\;s = 1,\ldots,S\right)$ cannot be instructive as it can be hardly distinguished from the straight line

$$\hbox{M}^{(N)} (\hbox{CD}|\hbox{LN}) = \hbox{M}^{(N)} (\hbox{LN}|\hbox{LN})$$

(B.1)

Therefore the whole area of the possible occurrence of the estimates was divided into seven sub-areas (Fig. B.1a, b). The diagonal corresponds to the equality of the maximum log L of the true and false distributions [Eq. B.1)]. Having the set of the S elements, one can assign each element to the respective areas and then get the rate of occurrence. Our particular interest is in the cases where despite the fact that (Fig. B.1a)

$$Median \hbox{M}^{(N)} (\hbox{LN|LN}) > Median \hbox{M}^{(N)} (\hbox{CD}|\hbox{LN})$$

(B.2)

the PCS is still lower than 0.5 (Table B.1). The PCS values of the LN model (the last column of Table B.1) are as in Table 1.

Fig. B.1

a, b Scheme of respective areas arrangement

Table B.1 Rates of occurrence in each distinguished areas for selected the (C _V , N) combinations

Looking at Fig. B.1 one learn that in the limiting cases PCS = 0 if p(F)=p(D)=0.5 (i.e., with all other rates equal to zero), or PCS = 1 if p(A)=p(B)=0.5 (i.e., with all other rates equal to zero). It leads to the conclusion that (B.2) is not informative in respect to PCS and the key is hidden in the form of log L of the both competing distributions. From rate array of all (C _V ,N) combinations (not shown), certain regularities of the results are noted:(1) p(A) ≈ p(C) and they are large values if PCS is large, i.e., they grow with sample size and with the C _V value; (2) p(D) ≈ p(F and they are large values if PCS is small, i.e., they decrease with sample size and with the C _V value; (3) p(B) ≈ p(E) and they are always small values, growing with the sample size and with the C _V value; (4) p(G) ≈ 0