| pareto1 {VGAM} | R Documentation |
Estimates one of the parameters of the Pareto(I) distribution by maximum likelihood estimation. Also includes the upper truncated Pareto(I) distribution.
pareto1(lshape = "loge", earg=list(), location=NULL)
tpareto1(lower, upper, lshape = "loge", earg=list(), ishape=NULL,
method.init=1)
lshape |
Parameter link function applied to the parameter k.
See Links for more choices.
A log link is the default because k is positive.
|
earg |
List. Extra argument for the link.
See earg in Links for general information.
|
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 NULL means a value is obtained internally.
If failure to converge occurs try specifying a value, e.g., 1 or 2.
|
location |
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 min(y) where y is the response
vector.
|
method.init |
An integer with value 1 or 2 which
specifies the initialization method. If failure to converge occurs
try the other value, or else specify a value for ishape.
|
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) = k * alpha^k / y^(k+1)
for 0< alpha < y and k>0. The alpha is known as the location parameter, and k is known as 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 pareto1 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 pareto1,
if the estimate of k is less than or equal to unity
then the fitted values will be NAs.
Also, pareto1 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
tpareto 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
Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third 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,
Tpareto,
paretoIV,
gpd.
alpha = 2; k = exp(3) y = rpareto(n=1000, location=alpha, shape=k) fit = vglm(y ~ 1, pareto1, trace=TRUE) fit@extra # The estimate of alpha is here fitted(fit)[1:5] mean(y) coef(fit, matrix=TRUE) summary(fit) # Standard errors are incorrect!! # Here, alpha is assumed known fit2 = vglm(y ~ 1, pareto1(location=alpha), trace=TRUE, crit="c") fit2@extra # alpha stored here fitted(fit2)[1:5] mean(y) coef(fit2, matrix=TRUE) summary(fit2) # Standard errors are ok # Upper truncated Pareto distribution lower = 2; upper = 8; k = exp(2) y = rtpareto(n=100, lower=lower, upper=upper, shape=k) fit3 = vglm(y ~ 1, tpareto1(lower, upper), trace=TRUE, cri="c") coef(fit3, matrix=TRUE) c(fit3@misc$lower, fit3@misc$upper)