pareto1 {VGAM} | R Documentation

## 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

    pareto1(lshape = "loge", earg=list(), location=NULL)
    tpareto1(lower, upper, lshape = "loge", earg=list(), ishape=NULL,
             method.init=1)

### Arguments

`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`.

### 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) = 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)).

### Value

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

### Warning

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.

### Note

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 `NA`s. 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.

### Author(s)

T. W. Yee

### References

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.

### See Also

`Pareto`, `Tpareto`, `paretoIV`, `gpd`.

### Examples

    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
    head(fitted(fit))
    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
    head(fitted(fit2))
    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)

[Package VGAM version 0.7-9 Index]