VGAM (version 0.9-2)

yulesimon: Yule-Simon Family Function

Description

Estimating the parameter of the Yule-Simon distribution.

Usage

yulesimon(link = "loge", irho = NULL, nsimEIM = 200, zero = NULL)

Arguments

link

Link function for the $\rho$ parameter. See Links for more choices and for general information.

irho

Optional initial value for the (positive) parameter. See CommonVGAMffArguments for more information. The default is to obtain an initial value internally. Use this argument if the defau

nsimEIM, zero

See CommonVGAMffArguments for more information.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Details

The probability function is $$f(y;\rho )=rho\ast beta(y,rho+1),$$ where the parameter $\rho>0$, $beta$ is the beta function, and $y=1,2,\ldots$. The function dyules computes this probability function. The mean of $Y$, which is returned as fitted values, is $\rho/(\rho-1)$ provided $\rho > 1$. The variance of $Y$ is $\rho^2/((\rho-1)^2 (\rho-2))$ provided $\rho > 2$.

The distribution was named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution. This family function can handle multiple responses.

References

Simon, H. A. (1955) On a class of skew distribution functions. Biometrika, 42, 425--440.

See Also

ryules.

Examples

ydata <- data.frame(x2 = runif(nn <- 1000))
ydata <- transform(ydata, y = ryules(nn, rho = exp(1.5 - x2)))
with(ydata, table(y))
fit <- vglm(y ~ x2, yulesimon, ydata, trace = TRUE)
coef(fit, matrix = TRUE)
summary(fit)