pareto_tail {tailplots} | R Documentation |
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
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. |
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
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 |
t.ci2 |
The upper bound of the confidence interval for t (if |
alpha |
Estimate of the shape parameter under a Pareto 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
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)