Pareto and Truncated Pareto Distribution Family Functions
Estimates one of the parameters of the Pareto(I) distribution by maximum likelihood estimation. Also includes the upper truncated Pareto(I) distribution.
paretoff(scale = NULL, lshape = "loglink") truncpareto(lower, upper, lshape = "loglink", ishape = NULL, imethod = 1)
lshape |
Parameter link function applied to the parameter k.
See |
scale |
Numeric.
The parameter alpha below.
If the user inputs a number then it is assumed known with this value.
The default means it is estimated by maximum likelihood
estimation, which means |
lower, upper |
Numeric. Lower and upper limits for the truncated Pareto distribution. Each must be positive and of length 1. They are called alpha and U below. |
ishape |
Numeric.
Optional initial value for the shape parameter.
A |
imethod |
See |
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-(α/U)^k)]
for 0< alpha < y < U < Inf 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 * lower^k * (U^(1-k)-alpha^(1-k)) / ((1-k) * (1-(alpha/U)^k)).
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
The usual or unbounded Pareto distribution has two
parameters (called alpha and k here)
but the family function paretoff estimates only
k using iteratively reweighted least squares.
The MLE of the alpha parameter lies on the
boundary and is min(y) where y is the
response. Consequently, using the default argument
values, the standard errors are incorrect when one does a
summary on the fitted object. If the user inputs
a value for alpha then it is assumed known with
this value and then summary on the fitted object
should be correct. Numerical problems may occur for small
k, e.g., k < 1.
Outside of economics, the Pareto distribution is known as the Bradford distribution.
For paretoff,
if the estimate of k is less than or equal to unity
then the fitted values will be NAs.
Also, paretoff fits the Pareto(I) distribution.
See paretoIV for the more general Pareto(IV/III/II)
distributions, but there is a slight change in notation: s = k
and b = alpha.
In some applications the Pareto law is truncated by a
natural upper bound on the probability tail.
The upper truncated Pareto distribution has three parameters (called
alpha, U and k here) but the family function
truncpareto() estimates only k.
With known lower and upper limits, the ML estimator of k has
the usual properties of MLEs.
Aban (2006) discusses other inferential details.
T. W. Yee
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.
Pareto,
Truncpareto,
paretoIV,
gpd,
benini1.
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.