R: Pareto and Truncated Pareto Distribution Family Functions

paretoff {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

    paretoff(scale = NULL, lshape = "loge")
    truncpareto(lower, upper, lshape = "loge", ishape = NULL, imethod = 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.
`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 `min(y)` is used, where `y` is the response vector.
`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.
`imethod` | See `CommonVGAMffArguments` for information. If failure to converge occurs then try specifying 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;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)).

### 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 `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.

### Note

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 `NA`s. 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.

### Author(s)

T. W. Yee

### 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)

[Package VGAM version 1.0-3 Index]