Package 'tailplots'

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. 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

x	a vector containing the sample data.
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.
ci.points	the number of thresholds used in the calculation of the confidence intervals.
xscale	the scale of the x-axis (options include "o" = original, "l" = log scale, "b" = both).

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 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. doi:10.2307/2284599

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

---

Plot the estimated s and the corresponding confidence intervals

Description

This function produces a tail plot for the estimate $\hat{s}$ 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

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

Arguments

x	a vector containing the sample data.
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.
ci.points	the number of thresholds used in the calculation of the confidence intervals.
xscale	the scale of the x-axis (options include "o" = original, "l" = log scale, "b" = both).

Value

A plot showing the estimated $s(u)$ 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 sigma values, which indicate the the standard deviation on the log-scale of the lognormal distribution associated with the estimated s-values.

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. doi:10.2307/2284599

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 = rlnorm(2e2, 2, 2)
lnorm_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
)

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. 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

x	a vector containing the sample data.
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.
ci.points	the number of thresholds used in the calculation of the confidence intervals.
xscale	the scale of the x-axis (options include "o" = original, "l" = log scale, "b" = both).

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")

---