smoothtail-package(smoothtail)
smoothtail-package()所属R语言包：smoothtail
Smooth Estimation of GPD Shape Parameter
GPD形状参数的平滑估计
译者：生物统计家园网 机器人LoveR
描述----------Description----------
Given independent and identically distributed observations X_1 < … < X_n from a Generalized Pareto distribution with shape parameter γ \in [-1,0], this package offers three methods to compute estimates of γ. The estimates are based on the principle of replacing the order statistics X_{(1)}, …, X_{(n)} of the sample by quantiles \hat X_{(1)}, …, \hat X_{(n)} of the distribution function \hat F_n based on the log–concave density estimator \hat f_n. This procedure is justified by the fact that the GPD density is
鉴于独立同分布的观测X_1 < … < X_n从广义帕累托分布形状参数γ \in [-1,0]的，这个包提供了三种方法计算估计γ。的估计是基于更换次序统计量的原则，X_{(1)}, …, X_{(n)}的样品位数\hat X_{(1)}, …, \hat X_{(n)}的分布函数\hat F_n的基础上对数凹密度估计\hat f_n。本程序是合理的的的GPD密度是一个事实，即
Details
详细信息----------Details----------
Package:
包装方式：
</td><td align="left"> smoothtail
</ TD> <TD ALIGN="LEFT"> smoothtail
Type:
类型：
</td><td align="left"> Package
</ TD> <TD ALIGN="LEFT">包装
Version:
版本：
</td><td align="left"> 2.0.1
</ TD> <TD ALIGN="LEFT"> 2.0.1
Date:
日期：
</td><td align="left"> 2011-11-29
</ TD> <TD ALIGN="LEFT"> 2011-11-29
License:
许可：
</td><td align="left"> GPL (>=2)
</ TD> <TD ALIGN="LEFT"> GPL（> = 2）
Use this package to estimate the shape parameter γ of a Generalized Pareto Distribution (GPD). In extreme value theory, γ is denoted tail index. We offer three new estimators, all based on the fact that the density function of the GPD is log–concave if γ \in [-1,0], see Mueller and Rufibach (2009). The functions for estimation of the tail index are:
用这个包来估计形状参数γ的广义Pareto分布（GPD）。在极值理论，γ表示尾部指数。我们提供了三种新的估计，所有的GPD的密度函数是对数凹的事实，如果γ \in [-1,0]，看到穆勒和Rufibach的（2009）的基础上。功能的尾部指数的估计是：
pickands <br> falk<br> falkMVUE<br> generalizedPick
pickands参考falk参考falkMVUE参考generalizedPick
This package depends on the package logcondens for estimation of a log–concave density: all the above functions take as first argument a dlc object as generated by logConDens in logcondens.
这个软件包依赖于包logcondens的log凹密度：估计所有上述功能需要一个dlc对象作为第一个参数所产生的logConDens中logcondens。
Additionally, functions for density, distribution function, quantile function and random number generation for a GPD with location parameter 0, shape parameter γ and scale parameter σ are provided:
此外，功能密度，分布函数，分位数函数和随机数生成的GPD的位置参数，形状参数γ和规模参数σ提供：
dgpd<br> pgpd<br> qgpd<br> rgpd.
dgpd参考pgpd参考qgpd参考rgpd。
Let us shortly clarify what we mean with log–concave density estimation. Suppose we are given an ordered sample Y_1 < … < Y_n of i.i.d. random variables having density function f, where f = \exp \varphi for a concave function \varphi : [-∞, ∞) \to R. Following the development in Duembgen and Rufibach (2009), it is then possible to get an estimator \hat f_n = \exp \hat \varphi_n of f via the maximizer \hat \varphi_n of
让我们在短期内澄清我们的意思log凹密度估计。假定我们有一个有序的样品Y_1 < … < Y_n独立同分布随机变量的密度函数f，其中f = \exp \varphi的凹函数\varphi : [-∞, ∞) \to R。继发展在Duembgen和Rufibach（2009年），它是那么可能得到的估计\hat f_n = \exp \hat \varphi_nf通过最大化\hat \varphi_n
over all concave functions \varphi. It turns out that \hat \varphi_n is piecewise linear, with knots only at (some of the) observation points. Therefore, the infinite-dimensional optimization problem of finding the function \hat \varphi_n boils down to a finite dimensional problem of finding the vector (\hat \varphi_n(Y_1),…,\hat \varphi(Y_n)). How to solve this problem is described in Rufibach (2006, 2007) and in a more general setting in Duembgen, Huesler, and Rufibach (2010). The distribution function based on \hat f_n is defined as
以上所有凹函数\varphi。事实证明，这\hat \varphi_n是分段线性的，只在一些观测点与结。因此，无限维的优化问题，发现的功能\hat \varphi_n归结为一个有限维问题寻找向量(\hat \varphi_n(Y_1),…,\hat \varphi(Y_n))。如何解决这个问题的描述Rufibach（2006年，2007年），在一般设置在Duembgen，Huesler，并Rufibach的（2010）。分布函数的基础上\hat f_n被定义为
for x a real number. The definition of \hat F_n is justified by the fact that \hat F_n(Y_1) = 0.
x一个实数。 \hat F_n的事实，\hat F_n(Y_1) = 0是有道理的。
（作者）----------Author(s)----------
Kaspar Rufibach (maintainer), <a href="mailto:kaspar.rufibach@gmail.com">kaspar.rufibach@gmail.com</a> , <br> <a href="http://www.kasparrufibach.ch">http://www.kasparrufibach.ch</a>
Samuel Mueller, <a href="mailto:s.mueller@maths.usyd.edu.au">s.mueller@maths.usyd.edu.au</a>, <br> <a href="http://www.maths.usyd.edu.au/ut/people?who=S_Mueller">http://www.maths.usyd.edu.au/ut/people?who=S_Mueller</a>
Kaspar Rufibach acknowledges support by the Swiss National Science Foundation SNF, <a href="http://www.snf.ch">http://www.snf.ch</a>
参考文献----------References----------
Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.
Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.
Smooth tail index estimation. J. Stat. Comput. Simul., 79, 1155–1167.
On the max–domain of attraction of distributions with log–concave densities. Statist. Probab. Lett., 78, 1440–1444.
PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. <br> Available at http://www.stub.unibe.ch/download/eldiss/06rufibach_k.pdf.
Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul., 77, 561–574.
参见----------See Also----------
Package logcondens.
包装logcondens。
实例----------Examples----------
# generate ordered random sample from GPD[形成有序的随机抽样GPD]
set.seed(1977)
n <- 20
gam <- -0.75
x <- rgpd(n, gam)
# compute known endpoint[计算已知端点]
omega <- -1 / gam
# estimate log-concave density, i.e. generate dlc object[估计数凹密度，即产生DLC对象]
est <- logConDens(x, smoothed = FALSE, print = FALSE, gam = NULL, xs = NULL)
# plot distribution functions[图分布函数]
s <- seq(0.01, max(x), by = 0.01)
plot(0, 0, type = 'n', ylim = c(0, 1), xlim = range(c(x, s))); rug(x)
lines(s, pgpd(s, gam), type = 'l', col = 2)
lines(x, 1:n / n, type = 's', col = 3)
lines(x, est$Fhat, type = 'l', col = 4)
legend(1, 0.4, c('true', 'empirical', 'estimated'), col = c(2 : 4), lty = 1)
# compute tail index estimators for all sensible indices k[计算尾部指数估计，所有敏感指数K表]
falk.logcon <- falk(est)
falkMVUE.logcon <- falkMVUE(est, omega)
pick.logcon <- pickands(est)
genPick.logcon <- generalizedPick(est, c = 0.75, gam0 = -1/3)
# plot smoothed and unsmoothed estimators versus number of order statistics[图平滑和平滑的估计与顺序统计数]
plot(0, 0, type = 'n', xlim = c(0,n), ylim = c(-1, 0.2))
lines(1:n, pick.logcon[, 2], col = 1); lines(1:n, pick.logcon[, 3], col = 1, lty = 2)
lines(1:n, falk.logcon[, 2], col = 2); lines(1:n, falk.logcon[, 3], col = 2, lty = 2)
lines(1:n, falkMVUE.logcon[,2], col = 3); lines(1:n, falkMVUE.logcon[,3], col = 3,
lty = 2)
lines(1:n, genPick.logcon[, 2], col = 4); lines(1:n, genPick.logcon[, 3], col = 4,
lty = 2)
abline(h = gam, lty = 3)
legend(11, 0.2, c("Pickands", "Falk", "Falk MVUE", "Generalized Pickands'"),
lty = 1, col = 1:8)
转载请注明:出自 生物统计家园网(http://www.biostatistic.net)。
注：
注1：为了方便大家学习，本文档为生物统计家园网机器人LoveR翻译而成，仅供个人R语言学习参考使用，生物统计家园保留版权。
注2：由于是机器人自动翻译，难免有不准确之处，使用时仔细对照中、英文内容进行反复理解，可以帮助R语言的学习。
注3：如遇到不准确之处，请在本贴的后面进行回帖，我们会逐渐进行修订。
|
发表于 2012-9-30 10:34:13
