/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

This is an update of, and a supplement to, a 1986 survey paper by the author on basic
properties of strong mixing conditions.

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Basic properties of strong mixing conditions. A survey and some open questions

Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas.

Statistics of extremes

Part 1 The IID case - functional laws of small numbers: functional laws of small numbers - bounds for the functional laws of small numbers, applications extreme value theory - Von Mises conditions, the peaks over threshold method, initial estimation of the class index estimation of conditional curves - Poisson process approach, the nonparametric case, the semiparametric case, extension to several points, a nearest neighbour alternative, optimal accuracy of estimators multivariate maxima - limiting distributions, representations and dependence functions, Max-Stable stochastic processes multivariate extremes - strong approximation of exceedances, further concepts of extremes, extreme value analysis with XTREMES - artificial data, real grouped data, real continuous data. Part 2 Non IID observations: introduction to the non IID case - definitions, stationary random sequences, independent random sequences, nonstationary random sequences extremes of random sequences - introduction and general theory, stationary sequences, independent sequences, nonstationary sequences extremes of Gaussian processes - stationary Gaussian processes, nonstationary Gaussian processes, empirial characteristic functions extensions for rare events - rare events of random sequences, the point process of exceedances, application to peaks over threshold, application to rare events, triangular arrays of rare events, multivariate extremes of non IID sequences statistics of extremes - application to ecological data, frost data appendix - user's guide to XTREMES - getting started, preparations, becoming acquainted with XTREMES, hot keys, window options and help menus XTREMES in action - implemented distributions, generating and reading data, nonparametric estimation, estimation in the alpha-mode, estimation in the beta-mode, simulations, important facilities and advice.

Laws of Small Numbers: Extremes and Rare Events

Mass spectrometry based quantification of peptides can be performed using the iTRAQ™ reagent in conjunction with mass spectrometry. This technology yields information about the relative abundance of single peptides. A method for the calculation of reliable quantification information is required in order to obtain biologically relevant data at the protein expression level. A method comprising sound error estimation and statistical methods is presented that allows precise abundance analysis plus error calculation at the peptide as well as at the protein level. This yields the relevant information that is required for quantitative proteomics. Comparing the performance of our method named Quant with existing approaches the error estimation is reliable and offers information for precise bioinformatic models. Quant is shown to generate results that are consistent with those produced by ProQuant™, thus validating both systems. Moreover, the results are consistent with that of Mascot™ 2.2. The MATLAB® scripts of Quant are freely available via http://www.protein-ms.de
 and http://sourceforge.net/projects/protms/
 , each under the GNU Lesser General Public License. The software Quant demonstrates improvements in protein quantification using iTRAQ™. Precise quantification data can be obtained at the protein level when using error propagation and adequate visualization. Quant integrates both and additionally provides the possibility to obtain more reliable results by calculation of wise quality measures. Peak area integration has been replaced by sum of intensities, yielding more reliable quantification results. Additionally, Quant allows the combination of quantitative information obtained by iTRAQ™ with peptide and protein identifications from popular tandem MS identification tools. Hence Quant is a useful tool for the proteomics community and may help improving analysis of proteomic experimental data. In addition, we have shown that a lognormal distribution fits the data of mass spectrometry based relative peptide quantification.

/pdf/precise-protein-quantification-based-on-peptide-392nlt2q12.pdf

Precise protein quantification based on peptide quantification using iTRAQ

The problem is investigated whether a given kernel type estimator of a distribution function at a single point has asymptotically better performance than the empirical estimator. A representation of the relative deficiency of the empirical distribution function with respect to a kernel type estimator is established which gives a complete solution to this problem. The problem of finding optimal kernels is studied in detail.

Relative efficiency and deficiency of kernel type estimators of smooth distribution functions

In this paper the asymptotic relative deficiency of the sample $q$-quantile with respect to kernel type estimators of the $q$-quantile is evaluated. The comparison is based on the mean square errors of the estimators. The result suggests a purely analytic measure of performance within the class of kernels. It is notable that a similar situation occurs when kernel estimators of a distribution function are studied.

/pdf/relative-deficiency-of-kernel-type-estimators-of-quantiles-1cgl2pqpfx.pdf

Relative Deficiency of Kernel Type Estimators of Quantiles

Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.

/pdf/asymptotic-normality-of-the-kernel-quantile-estimator-4ise17q8jf.pdf

Michael Falk

Papers

Laws of Small Numbers: Extremes and Rare Events

Precise protein quantification based on peptide quantification using iTRAQ

Relative efficiency and deficiency of kernel type estimators of smooth distribution functions

Relative Deficiency of Kernel Type Estimators of Quantiles

Asymptotic Normality of the Kernel Quantile Estimator