gamma_tail: Estimate of tail functional g and confidence intervals for g...

View source: R/gamma_tailplot.R

gamma_tailR Documentation

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

Description

This function computes the estimate of g and the associated confidence interval for g as well as alpha, the corresponding shape parameter under the assumption of a gamma model, according to Iwashita and 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

gamma_tail(
  x,
  d,
  confint = FALSE,
  method = c("unbiased", "bootstrap", "jackknife"),
  R = 1000,
  conf.level = 0.95
)

Arguments

x

a vector containing the sample data.

d

the threshold for the computation of g.

confint

a boolean value indicating whether a 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

The function g introduced by Asmussen and Lehtomaa (2017) is used to distinguish between log-concave and log-convex tail behavior. It is defined as:

g(d) = E\left[ \frac{|X_1 - X_2|}{X_1 + X_2} \bigg| X_1 + X_2 > d \right]

where X_1, X_2 are independent and identically distributed (i.i.d.) positive random variables. For gamma distributions, g takes a constant value, making it a useful tool for detecting gamma-tailed distributions.

This function estimates g(d) using U-statistics. The estimator \hat{g}(d) is given by:

\hat{g}(d) = \frac{ U^{(1)}_n (d) }{ U^{(2)}_n (d) }, \quad d > 0,

where

U^{(1)}_n (d) = \frac{2}{n(n-1)} \sum_{1 \leq i < j \leq n} \frac{|X_i - X_j|}{X_i + X_j} 1(X_i + X_j > d),

U^{(2)}_n (d) = \frac{2}{n(n-1)} \sum_{1 \leq i < j \leq n} 1(X_i + X_j > d).

Confidence intervals for g(d), based on the following variance estimation methods, are also provided:

  • Unbiased Variance Estimator

  • Bootstrap Resampling

  • Jackknife Resampling

The (1-\gamma) confidence interval for \hat{g}_{n}(d) is given by:

\left[ \max\!\Bigl\{ \hat{g}_{n}(d)\;-\; z_{1 - \gamma/2} \,\frac{\hat{\sigma}_{d}}{ \sqrt{n\,U^{(2)}_{n}(d)} }, \;0 \Bigr\}, \;\; \min\!\Bigl\{ \hat{g}_{n}(d)\;+\; z_{1 - \gamma/2} \,\frac{\hat{\sigma}_{d}}{ \sqrt{n\,U^{(2)}_{n}(d)} }, \;1 \Bigr\} \right].

Here, z_{1 - \gamma/2} = \Phi^{-1}(1 - \tfrac{\gamma}{2}) is the appropriate quantile of the standard normal distribution and \hat{\sigma}_d is an estimator of the standard deviation based on one of the methods above.

Value

A matrix containing:

threshold

The value of the threshold d.

g.estimate

Estimate of g.

g.ci1

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

g.ci2

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

alpha

Estimate of the shape parameter under a gamma 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

Iwashita, T. & Klar, B. (2024). A gamma tail statistic and its asymptotics. Statistica Neerlandica 78:2, 264-280. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.1111/stan.12316")}

Asmussen, S. & Lehtomaa, J. (2017). Distinguishing Log-Concavity from Heavy Tails. Risks 2017, 5, 10. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.3390/risks5010010")}

Examples

x <- rgamma(100, shape = 2, scale = 1)
gamma_tail(x, d = 2, confint = FALSE, method = "unbiased", R = 1000)


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