Algorithm AS 99: Fitting Johnson Curves by Moments

Fitting human exposure data with the Johnson SB distribution

“…This procedure formed the basis of an algorithm presented by Hill et al (1976), which used a Newton-Raphson iterative scheme to estimate gamma and delta. This algorithm calculates the first six moments of the distribution by an approximate numerical integration technique.…”

“…This work presents a variation on an existing algorithm (Hill et al, 1976) for estimating all four parameters of the S B distribution based upon the method-of-moments outlined by Johnson and Kitchen (1971). Instead of a table look-up process, a computer program has been written to estimate gamma and delta given sample values of the coefficients of skewness and kurtosis.…”

“…The program uses a Newton iteration subroutine (the IMSL subroutine DNEQNF), which calculates a finite difference approximation for the Jacobian. Unlike (Hill et al, 1976) this requires only the first four moments, which are calculated according to the analytic series formulas presented in Flynn (2005). This method of estimation is compared with the percentile methods of Slifker and Shapiro (1980), Mage (1980), and the quantile method of Wheeler (1980).…”

Coastal waters monitoring data: frequency distributions of the principal water quality variables

“…They may also be estimated from the moments. Applied Statistics algorithm 99 (Hill et al 1976), has been translated into C for this implementation and is currently available in the R SuppDist. We should add that the appropriate translation function g() is uniquely identified by the third and fourth order moments of the sample distribution of x, i.e.…”

“…curves will increasingly present as two ascending branches at the extremities, until they are represented by two simple ordinates. In the paper of Hill et al (1976), these limit-distributions are classified in the S T family, where T represents "two-ordinates" and the related Johnson parameters, ξ  and λ, are fixed depending on the values of the x at which ordinates occur, δ is set to the proportion of values at λ, and γ is set, arbitrarily, to zero.…”