Estimating the 1-parameter Benini distribution by maximum likelihood estimation.
benini1(y0 = stop("argument 'y0' must be specified"), lshape = "loglink", ishape = NULL, imethod = 1, zero = NULL, parallel = FALSE, type.fitted = c("percentiles", "Qlink"), percentiles = 50)
Positive scale parameter.
Parameter link function and extra argument of the parameter
b, which is the shape parameter.
Optional initial value for the shape parameter. The default is to compute the value internally.
The Benini distribution has a probability density function that can be written
f(y) = 2*s*exp(-s * [(log(y/y0))^2]) * log(y/y0) / y
for 0 < y0 < y, and shape s > 0. The cumulative distribution function for Y is
F(y) = 1 - exp(-s * [(log(y / y0))^2]).
Here, Newton-Raphson and Fisher scoring coincide. The median of Y is now returned as the fitted values, by default. This VGAM family function can handle a multiple responses, which is inputted as a matrix.
On fitting, the
extra slot has a component called
y0 which contains the value of the
An object of class
The object is used by modelling functions
Yet to do: the 2-parameter Benini distribution estimates another shape parameter a too. Hence, the code may change in the future.
T. W. Yee
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
y0 <- 1; nn <- 3000 bdata <- data.frame(y = rbenini(nn, y0 = y0, shape = exp(2))) fit <- vglm(y ~ 1, benini1(y0 = y0), data = bdata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) fit@extra$y0 c(head(fitted(fit), 1), with(bdata, median(y))) # Should be equal
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