VGAM (version 1.1-14)

paretoff: Pareto and Truncated Pareto Distribution Family Functions

Description

Estimates one of the parameters of the Pareto(I) distribution by maximum likelihood estimation. Also includes the upper truncated Pareto(I) distribution.

Usage

paretoff(scale = NULL, lshape = "loglink")
truncpareto(lower, upper, lshape = "loglink", ishape = NULL, imethod = 1)

Value

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

Details

A random variable \(Y\) has a Pareto distribution if $$P[Y>y] = C / y^{k}$$ for some positive \(k\) and \(C\). This model is important in many applications due to the power law probability tail, especially for large values of \(y\).

The Pareto distribution, which is used a lot in economics, has a probability density function that can be written $$f(y;\alpha,k) = k \alpha^k / y^{k+1}$$ for \(0 < \alpha < y\) and \(0<k\). The \(\alpha\) is called the scale parameter, and it is either assumed known or else min(y) is used. The parameter \(k\) is called the shape parameter. The mean of \(Y\) is \(\alpha k/(k-1)\) provided \(k > 1\). Its variance is \(\alpha^2 k /((k-1)^2 (k-2))\) provided \(k > 2\).

The upper truncated Pareto distribution has a probability density function that can be written $$f(y) = k \alpha^k / [y^{k+1} (1-(\alpha/U)^k)]$$ for \(0 < \alpha < y < U < \infty\) and \(k>0\). Possibly, better names for \(k\) are the index and tail parameters. Here, \(\alpha\) and \(U\) are known. The mean of \(Y\) is \(k \alpha^k (U^{1-k}-\alpha^{1-k}) / [(1-k)(1-(\alpha/U)^k)]\).

References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

Aban, I. B., Meerschaert, M. M. and Panorska, A. K. (2006). Parameter estimation for the truncated Pareto distribution, Journal of the American Statistical Association, 101(473), 270--277.

See Also

Pareto, Truncpareto, paretoIV, gpd, benini1.

Examples

alpha <- 2; kay <- exp(3)
pdata <- data.frame(y = rpareto(n = 1000, scale = alpha, shape = kay))
fit <- vglm(y ~ 1, paretoff, data = pdata, trace = TRUE)
fit@extra  # The estimate of alpha is here
head(fitted(fit))
with(pdata, mean(y))
coef(fit, matrix = TRUE)
summary(fit)  # Standard errors are incorrect!!

# Here, alpha is assumed known
fit2 <- vglm(y ~ 1, paretoff(scale = alpha), data = pdata, trace = TRUE)
fit2@extra  # alpha stored here
head(fitted(fit2))
coef(fit2, matrix = TRUE)
summary(fit2)  # Standard errors are okay

# Upper truncated Pareto distribution
lower <- 2; upper <- 8; kay <- exp(2)
pdata3 <- data.frame(y = rtruncpareto(n = 100, lower = lower,
                                      upper = upper, shape = kay))
fit3 <- vglm(y ~ 1, truncpareto(lower, upper), data = pdata3, trace = TRUE)
coef(fit3, matrix = TRUE)
c(fit3@misc$lower, fit3@misc$upper)