gamma_tail {tailplots} | R 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,
alpha.max = 100
)
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. |
alpha.max |
the upper limit of the interval to be searched for the root in an internal routine (the default value of 100 should be increased in case of error). |
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 |
g.ci2 |
The upper bound of the confidence interval for g (if |
alpha |
Estimate of the shape parameter under a gamma model. |
alpha.ci1 |
The lower bound of the confidence interval for alpha (if |
alpha.ci2 |
The upper bound of the confidence interval for alpha (if |
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)