pareto_tail {tailplots}R Documentation

Estimate of tail functional t and confidence intervals for t and alpha

Description

This function computes the estimate of t and the associated confidence interval for t as well as alpha, the corresponding shape parameter under the assumption of a Pareto model according to Klar (2024). Three methods are implemented to compute the confidence intervals: a method based on the unbiased variance estimators of the underlying U-statistics and two resampling methods (jackknife and bootstrap).

Usage

pareto_tail(
  x,
  u,
  confint = FALSE,
  method = c("unbiased", "bootstrap", "jackknife"),
  R = 1000,
  conf.level = 0.95,
  alpha.max = 100
)

Arguments

x

a vector containing the sample data.

u

the threshold for the computation of t.

confint

a boolean value indicating whether the confidence interval should be computed.

method

the method used for computing the confidence intervals (options include unbiased variance estimator, jackknife, and bootstrap).

R

the number of the bootstrap replicates.

conf.level

the confidence level for the interval.

alpha.max

the upper limit of the interval to be searched for the root in an internal routine (the default value of 100 should be increased in case of error).

Details

In Klar (2024) the function

t_X(u) \;=\; \mathbb{E}\!\biggl[ \frac{\lvert X_1 - X_2 \rvert}{X_1 + X_2} \;\Big|\; \min\{X_1, X_2\} \,\ge u \biggr]

is proposed as a tool for detecting Pareto-type tails, where X_1, X_2, X are i.i.d. random variables from an absolutely continuous distribution supported on [x_m,\infty). Theorem 1 in Klar (2024) shows that t_X(u) is constant in u if and only if X has a Pareto distribution.

The estimator \hat{t}_n\bigl(X_{(k)}\bigr) can be computed recursively. For k = 2,\ldots,n-1,

\hat{t}_n\bigl(X_{(k)}\bigr) \;=\; \frac{n-k+2}{n-k}\,\hat{t}_n\bigl(X_{(k-1)}\bigr) \;-\; \frac{1}{\binom{\,n-k+1\,}{2}} \sum_{j=k}^{n} \frac{X_{(j)} - X_{(k-1)}}{X_{(j)} + X_{(k-1)}}\,,

which can be evaluated efficiently starting from \hat{t}_n\bigl(X_{(n-1)}\bigr) = \bigl(X_{(n)} - X_{(n-1)}\bigl)/\bigl(X_{(n)} + X_{(n-1)}\bigl), where X_{(k)} denotes the k-th order statistic.

Confidence intervals for t(u) based on the following methods for variance estimation are also provided:

A two-sided (1 - \gamma) confidence interval for the estimator \hat{t}_n(u) is :

\left[ \max\!\Bigl\{ \hat{t}_n(u) \;-\; z_{1 - \frac{\gamma}{2}} \,\frac{\hat{\sigma}_{u}}{ \sqrt{n\,U_n^{(2)}(u)} }, \;0 \Bigr\}, \, \min\!\Bigl\{ \hat{t}_n(u) \;+\; z_{1 - \frac{\gamma}{2}} \,\frac{\hat{\sigma}_{u}}{ \sqrt{n\,U_n^{(2)}(u)} }, \;1 \Bigr\} \right],

where z_{1 - \frac{\gamma}{2}} = \Phi^{-1}(1 - \tfrac{\gamma}{2}) is the appropriate quantile of the standard normal distribution, \hat{\sigma}_u is an estimator of the standard deviation of c\,\hat{t}_n(u), for a constant c specified in section 4.1. of Klar (2024), and U_n^{(2)}(u) is a U-statistic given by

U_n^{(2)}(u) \;=\; \frac{2}{n\,(n-1)} \sum_{i = 1}^n (n - i) 1\{X_{(i)} \,\ge\, u\}.

Value

A matrix containing:

threshold

The value of the threshold u.

t.estimate

Estimate of the tail functional t.

t.ci1

The lower bound of the confidence interval for t (if confint = TRUE).

t.ci2

The upper bound of the confidence interval for t (if confint = TRUE).

alpha

Estimate of the shape parameter under a Pareto model.

alpha.ci1

The lower bound of the confidence interval for alpha (if confint = TRUE).

alpha.ci2

The upper bound of the confidence interval for alpha (if confint = TRUE).

References

Klar, B. (2024). A Pareto tail plot without moment restrictions. The American Statistician. doi:10.1080/00031305.2024.2413081

Examples

x <- actuar::rpareto1(1e3, shape=1, min=1)
pareto_tail(x, round( quantile(x, c(0.1, 0.5, 0.75, 0.9, 0.95, 0.99)) ), confint = FALSE) 


[Package tailplots version 0.1.0 Index]