pareto_tail: Estimate of tail functional t and confidence intervals for t...

In tailplots: Estimators and Plots for Gamma and Pareto Tail Detection

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
)

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.

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:

• Unbiased variance estimator

• Bootstrap resampling

• Jackknife resampling

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/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) 

tailplots documentation built on Sept. 9, 2025, 5:52 p.m.