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第229行：
第229行： − + + − Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in − Markovich.<ref name="Markovich2007">{{cite book − | author=Markovich N.M. − | title=Nonparametric Analysis of Univariate Heavy-Tailed data: Research and Practice − | year=2007 − | series=Chitester: Wiley − | isbn=978-0-470-72359-3 − }}</ref> These are approaches based on variable bandwidth and long-tailed kernel estimators; on the preliminary data transform to a new random variable at finite or infinite intervals which is more convenient for the estimation and then inverse transform of the obtained density estimate; and "piecing-together approach" which provides a certain parametric model for the tail of the density and a non-parametric model to approximate the mode of the density. Nonparametric estimators require an appropriate selection of tuning (smoothing) parameters like a bandwidth of kernel estimators and the bin width of the histogram. The well known data-driven methods of such selection are a cross-validation and its modifications, methods based on the minimization of the mean squared error (MSE) and its asymptotic and their upper bounds.<ref name="WandJon1995">{{cite book − | author=Wand M.P., Jones M.C. − | title=Kernel smoothing − | year=1995 − | series=New York: Chapman and Hall − | isbn=978-0412552700 − }}</ref> A discrepancy method which uses well-known nonparametric statistics like Kolmogorov-Smirnov's, von Mises and Anderson-Darling's ones as a metric in the space of distribution functions (dfs) and quantiles of the later statistics as a known uncertainty or a discrepancy value can be found in.<ref name="Markovich2007"/> Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of re-samples selection, see e.g.<ref name="Hall1992">{{cite book − | author=Hall P. − | title=The Bootstrap and Edgeworth Expansion − | year=1992 − | series=Springer − | isbn=9780387945088 − }}</ref> 第271行：
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→Estimation of heavy-tailed density 重尾密度的估计
* 用于估计重尾指数的软件[http://www.cs.bu.edu/~crovella/aest.html aest]和C。<ref>{{Cite journal | last1 = Crovella | first1 = M. E. | last2 = Taqqu | first2 = M. S. | title = Estimating the Heavy Tail Index from Scaling Properties| journal = Methodology and Computing in Applied Probability | volume = 1 | pages = 55–79 | year = 1999 | doi = 10.1023/A:1010012224103 | url = http://www.cs.bu.edu/~crovella/paper-archive/aest.ps| pmid = | pmc = }}</ref>
* 用于估计重尾指数的软件[http://www.cs.bu.edu/~crovella/aest.html aest]和C。<ref>{{Cite journal | last1 = Crovella | first1 = M. E. | last2 = Taqqu | first2 = M. S. | title = Estimating the Heavy Tail Index from Scaling Properties| journal = Methodology and Computing in Applied Probability | volume = 1 | pages = 55–79 | year = 1999 | doi = 10.1023/A:1010012224103 | url = http://www.cs.bu.edu/~crovella/paper-archive/aest.ps| pmid = | pmc = }}</ref>
== Estimation of heavy-tailed density 重尾密度的估计 ==
== 重尾密度的估计 ==
Markovich中给出了估计重尾和超重尾概率密度函数的非参数方法。<ref name="Markovich2007">{{cite book
Markovich中给出了估计重尾和超重尾概率密度函数的非参数方法。<ref name="Markovich2007">{{cite book
| isbn=9780387945088
| isbn=9780387945088
}}</ref>
}}</ref>
== See also 其他参考资料 ==
== See also 其他参考资料 ==