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,
  alpha.max = 100
)

Arguments

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:

References

Iwashita, T. & Klar, B. (2024). A gamma tail statistic and its asymptotics. Statistica Neerlandica 78:2, 264-280. doi:10.1111/stan.12316

Asmussen, S. & Lehtomaa, J. (2017). Distinguishing Log-Concavity from Heavy Tails. Risks 2017, 5, 10. doi:10.3390/risks5010010

Examples

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

Plot the estimated g and the corresponding confidence intervals

Description

This function produces a tail plot for the estimate \hat{g} over a range of thresholds for a given sample, including confidence intervals computed by one of three methods (unbiased, bootstrap or jackknife). The function also allows a choice between original and log scale.

Usage

gamma_tailplot(
  x,
  method = c("unbiased", "bootstrap", "jackknife"),
  R = 1000,
  conf.level = 0.95,
  ci.points = 101,
  xscale = "o"
)

Arguments

Details

For more details about the estimator \hat{g} and the computation of the confidence intervals see gamma_tail.

Value

A plot showing the estimated g(d) versus threshold d, optionally on a logarithmic x-axis and including confidence intervals.

References

Iwashita, T. & Klar, B. (2024). A gamma tail statistic and its asymptotics. Statistica Neerlandica 78:2, 264-280. doi:10.1111/stan.12316

Examples

x <- rgamma(2e2, 0.5, 0.2)
gamma_tailplot(x, method="unbiased", xscale="o")

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

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:

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) 

Plot the estimated t and the corresponding confidence intervals

Description

This function produces a tail plot for the estimate \hat{t} over a range of thresholds for a given sample, including confidence intervals computed by one of three methods (unbiased, bootstrap or jackknife). The function also allows a choice between original and log scale.

Usage

pareto_tailplot(
  x,
  method = c("unbiased", "bootstrap", "jackknife"),
  R = 1000,
  conf.level = 0.95,
  ci.points = 101,
  xscale = "b"
)

Arguments

Details

For more details about the estimator \hat{t} and the computation of the confidence intervals see pareto_tail.

Value

A plot showing the estimated t(u) versus threshold u, optionally on a logarithmic x-axis and including confidence intervals. Note that on the right side of the plot, one can observe the corresponding alpha values, which indicate the shape parameter of the Pareto distribution associated with the estimated t-values.

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_tailplot(x, method="unbiased", xscale="o")