lnorm_tail: Estimate of tail functional s and confidence intervals for s...

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

Estimate of tail functional s and confidence intervals for s and sigma

Description

This function computes the estimate of s and the associated confidence interval for s as well as the standard deviation sigma on the log-scale of the lognormal distribution. 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

lnorm_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 s.
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

The function s, defined by

s(u) = \mathbb{E}\!\left[ \frac{|X_{1} - X_{2}|}{X_{1} + X_{2}} \;\middle|\; X_{1}\,X_{2} > u \right],

where X_1,X_2 are independent and identically distributed (i.i.d.) positive random variables, takes a constant value if and only if X_1 follows a lognormal distribution. Thus, s can be used to detect distributions with lognormal tails. The characterization of the lognormal distribution is based on the work of Mosimann (1970). This function estimates s(u) using U-statistics, similarly as in Iwashita and Klar (2024).

Value

A matrix containing:

threshold	The value of the threshold u.
s.estimate	Estimate of the tail functional s.
s.ci1	The lower bound of the confidence interval for s (if confint = TRUE).
s.ci2	The upper bound of the confidence interval for s (if confint = TRUE).
sigma	Estimate of the scale parameter under a lognormal model.
sigma.ci1	The lower bound of the confidence interval for sigma (if confint = TRUE).
sigma.ci2	The upper bound of the confidence interval for sigma (if confint = TRUE).

References

Mosimann, J. E. (1970). Size allometry: size and shape variables with characterizations of the lognormal and generalized gamma distributions. Journal of the American Statistical Association, 65(330):930–945. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.2307/2284599")}

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")}

Examples

x = rlnorm(1e3, 2, 2)
u = round( quantile(x, 0.98) )
lnorm_tail(x, u, confint = FALSE)

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