1,721,006 research outputs found
A note on the behaviour of a kernel-smoothed kernel density estimator
No full textKernel density estimators have been studied in great detail. In this note a new family of kernels, depending on a parameter c, is obtained by kernel-smoothing an initial kernel density estimator. Under certain conditions, we show that nonparametric density estimators based on such kernels outperform the initial estimator in terms of minimized asymptotic mean integrated squared error and in kernel efficiency.We thank the associate editor and the two reviewers for their valuable comments (leading e.g. to Example 3) and good suggestions to improve the original manuscript.Janssen, P (reprint author), Hasselt Univ, Ctr Stat, Agr Laan, B-3590 Diepenbeek, Belgium.
[email protected]; [email protected]; [email protected]
The bootstrap methodology : a critical review
No full textSedert die ontstaan van die skoenlusmetodologie, het dit beide 'n kragtige versameling oplossings gebied vir die praktiese gebruiker van statistiek , sowel as 'n ryk bron van teoretiese en metodologiese oplossings vir probleme in statistiek. In hierdie artikel word 'n oorsig gegee van onlangse ontwikkelings in die nie-parametriese skoenlusmetodologie, waar gekonsentreer word op basiese idees en toepassings instede van teoretiese oorweginge. Onderwerpe van belang sluit in statistiese fout, vertrouensintervalle, die dubbel-skoenlus, skoenluskalibrering, skoenlus parsiele aanneemlikheid, toepassing van die skoenlusmetode op gekompliseerde datastelle, die wilde skoenlus en die aangepaste skoenlusmetode. Die genoemde onderwerpe word bespreek onder onafhanklikheidsaannames van die data, 'n Onlangse belangrike ontwikkeling in die skoenlusmetodologie behels die toepassing van die skoenlus op afhanklike data. Onderwerpe wat onder hierdie aanname bespreek word, sluit die bewegende blokskoenlusmetode in sowel as die outoregressiewe sif-skoenlusEver since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological solutions for problems in statistics. In this paper, a survey of some recent developments in the non-parametric bootstrap methodology is given, concentrating on basic ideas and applications rather than theoretical considerations. Topics include statistical error, confidence intervals, double bootstrapping, bootstrap calibration, bootstrap partial likelihood, bootstrapping complicated data sets, wild bootstrap and the modified bootstrap. The above topics will be discussed under the assumption of independent data. A major development of bootstrap methods recently has been their application to dependent data. Topics that will be discussed under this heading include the moving block bootstrap and the autoregressive sieve bootstra
A note on Brownian areas and arcsine laws
Full text linkFirstly, we provide simple elementary proofs to derive the exact distributions of the areas under functions of a Brownian motion process and a Brownian bridge process. In the latter case, a solution is therefore provided to a question raised recently in the Mathematics community on StackExchange (http://math.stackexchange.com/questions/1006101). These random areas often occur in statistical applications and play an important role in, for example, financial mathematics. Comparisons are made between the variances of the two random areas, deriving interesting results that appear to be new in the statistical literature. Some illustrative examples are provided. Secondly, we derive a new arcsine law for a standard Brownian bridge proces
On a generalization of a theorem by Euler
No full textIn this paper a natural generalization of a theorem by Euler in 1744 is presented. Extensive searches failed to locate this result in existing literature or in well known mathematical websites such as MathWorld (http://mathworld.wolfram.com), nor could it be derived by using software for analytical computation like Maple. The obtained identity is fascinating and surprisingly simple, and it paves the way for interesting application
A note on the asymptotic behavior of the Bernstein estimator of the copula density
No full textCopulas and their corresponding densities are functions of a multivariate joint distribution and the one-dimensional marginals. Bernstein estimators have been used as smooth nonparametric estimators for copulas and copula densities. The purpose of this note is to study the asymptotic distributional behavior of the Bernstein estimator of a copula density. Compared to the existing results, our general theorem does not assume known marginals. This makes our theorem applicable for real data.The work was supported by the IAP Research Network P7/13 of the Belgian State (Belgian Science Policy). The second author thanks the National Research Foundation of South Africa for financial support. The third author acknowledges support from research grant MTM 2008-03129 of the Spanish Ministerio de Ciencia e Innovacion. He is also an extraordinary professor at the North-West University, Potchefstroom, South Africa
Large sample behavior of the Bernstein copula estimator
No full textBernstein polynomial estimators have been used as smooth estimators for density functions and distribution functions. The idea of using them for copula estimation has been given in Sancetta and Satchell (2004). In the present paper we study the asymptotic properties of this estimator: almost sure consistency rates and asymptotic normality. We also obtain explicit expressions for the asymptotic bias and asymptotic variance and show the improvement of the asymptotic mean squared error compared to that of the classical empirical copula estimator. A small simulation study illustrates this superior behavior in small samples. (C) 2011 Elsevier B.V. All rights reserved.This work was supported by the IAP Research Network P6/03 of the Belgian State (Belgian Science Policy).
The second author thanks the National Research Foundation of South Africa for financial support. The third author acknowledges support from research Grant MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovacion
Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals
No full textBernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.The authors thank Mr. Charl Pretorius for his important help with the simulation section. They also thank the editor, associate editor and two referees for their valuable remarks and suggestions. The work was supported by the IAP Research Network P7/13 of the Belgian State (Belgian Science Policy). J. Swanepoel thanks the National Research Foundation of South Africa for financial support. N. Veraverbeke is also extraordinary professor at the North-West University, Potchefstroom, South Africa
Smooth copula-based estimation of the conditional density function with a single covariate
No full textSome recent papers deal with smooth nonparametric estimators for copula functions and copula derivatives. These papers contain results on copula-based Bernstein estimators for conditional distribution functions and related functionals such as regression and quantile functions. The focus in the present paper is on new copula-based smooth Bernstein estimators for the conditional density. Our approach avoids going through separate density estimation of numerator and denominator. Our estimator is defined as a smoother of the copula-based Bernstein estimator of the conditional distribution function. We establish asymptotic properties of bias and variance and discuss the asymptotic mean squared error in terms of the smoothing parameters. We also obtain the asymptotic normality of the new estimator. In a simulation study we show the good performance of the new estimator in comparison with other estimators proposed in the literature.The authors thank Dr. Charl Pretorius for his important help with the simulations. They also thank the Editor, Associate Editor and a referee for their valuable comments and suggestions. The work was supported by the IAP Research Network P7/13 of the Belgian State (Belgian Science Policy). The second author thanks the National Science Foundation of South Africa for financial support (grant number 81038). The third author is also extraordinary professor at the North-West University, Potchefstroom, South Africa
A semi-parametric method for transforming data to normality
No full textA non-parametric transformation function is introduced to transform data to any continuous distribution. When transformation of data to normality is desired, the use of a suitable parametric pre-transformation function improves the performance of the proposed non-parametric transformation function. The resulting semi-parametric transformation function is shown empirically, via a Monte Carlo study, to perform at least as well as any parametric transformation currently available in the literature
A nonparametric point estimation technique using the m-out-of-n bootstrap
No full textWe investigate a method which can be used to improve an existing point estimator by a modification of the estimator and by using the m-out-of-n bootstrap. The estimation method used, known as bootstrap robust aggregating (or BRAGGing) in the literature, will be applied in general to the estimators that satisfy the smooth function model (for example, a mean, a variance, a ratio of means or variances, or a correlation coefficient), and then specifically to an estimator for the population mean. BRAGGing estimators based on both a naive and corrected version of them-out-of-n bootstrap will be considered. We conclude with proposed data-based choices of the resample size, m, as well as Monte-Carlo studies illustrating the performance of the estimators when estimating the population mean for various distribution
- …
