Wavelet density estimation for weighted data

2015

Nonparametric estimation of densities defined over non-negative observations using asymmetric kernels is of special interest as it has potential to remove the spill-over effect at the boundary. One important problem in this context is the selection of the smoothing parameter. The purpose of the this note is to review some recent work on the application of Bayes criterion for this purpose and investigate its application in the context of length biased data.

2012

The density estimation problem under bias and multiplicative censoring is considered. Adopting the wavelet approach, we construct a linear nonadaptive estimator and a nonlinear adaptive estimator. The adaptive one belongs to the family of the hard thresholding estimators. We evaluate their performances by determining upper bounds of the mean integrated squared error over a wide range of functions. Sharp upper bounds are obtained.

2019

The estimation of a quantile density function in the biased nonparametric regression model is investigated. We propose and develop a new waveletbased methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls. The performance of the proposed estimator is investigated by a numerical study.

IEEE Transactions on Information Theory, 2002

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the Wavelet Transform Modulus Maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of Fractional Auto-Regressive Integrated Moving Average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cut-off scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cut-off scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

1997

In these notes we present the main uses of wavelets in statistics. Advances in the area are occuring rapidly, with applications in several elds. There are several books about the mathematical aspects of wavelets, like those by Daubechies(1992) and Chui(1992), and others in areas like signal processing and image compression. The content of these notes borrows heavily from recent works, most of them referenced in the bibliography section. We decided to not include the proofs of the theorems, which the interested reader may nd in the original papers. I would like to thank the organizers of the Third International Conference on Statistical Data Analysis Based on the L 1 ?Norm and Related Methods for the invitation to give a tutorial on wavelets. This gave me the opportunity to prepare this text. Comments and suggestions are welcome and can be sent to pam@ime.usp.br. I would like to thank Chang Chiann for several conversations and David Brillinger for the continuous support. A version in portuguese was presented as a short course in the Seventh Time Series and Econometrics School, in Canela, RS, Brazil and is part of a book to be published by the

Statistics & Probability Letters, 2002

Linear functions on spacings-instead of linear functions on order statistics-are considered, in order to simplify the form of best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the scale parameter in the classical location-scale family. Also, a su cient condition for the non-negativity of the scale estimator is presented and, moreover, necessary and su cient conditions for the BLUE (and the BLIE) to be a constant multiple of the sample range are derived. Finally, a modiÿcation of this approach is applied in order to simplify the derivations of both the location and the scale estimators in the Uniform Type-II Censored model.

Computational Statistics & Data Analysis, 1999

A technique is suggested for reducing the order of bias of kernel estimators by weighting the contributions that di erent data values make to the estimator. The method is developed initially in the context of density estimation, where, unlike the "variable kernel" method proposed by Abramson, our approach does not involve using di erent bandwidths at di erent data values. Rather, it is a weighted-bootstrap version of the standard uniform-bootstrap method that is used to construct traditional kernel density estimators. The reduction in bias is achieved by biasing the bootstrap appropriately, in a global rather than local way. Our technique has a variety of di erent forms, each of which reduces the order of bias from the square to the fourth power of bandwidth, but does not alter the order of variance. It has immediate application to nonparametric regression, where it allows bias to be reduced without prejudicing the one sign of an estimator.

International Journal of Scientific Research in Mathematical and Statistical Sciences, 2018

In this paper, we proposed a new probability model called as Size Biased Ailamujia Distribution (SBAD). Structural properties of this new distribution are discussed comprehensively. Parameter estimation along with simulation study is also discussed thoroughly. Finally, the model is examined with real life data sets for significance purpose.

Journal of statistical theory and practice, 2018

This paper considers estimation of the density function in the context of biased data using thresholded wavelet estimator. We adapt the method of estimating the square root of the density function as advocated in Pinheiro and Vidakovic (Comput Stat Data Anal 25:399-415, 1997) in the iid case. The density estimator is obtained by squaring the resulting estimator that as a result guarantees non-negativity. It is shown that the resulting estimator achieves the optimal 2 convergence in Besov spaces B s pq (M) for p ≥ 2 whereas for p ∈ [1, 2) there is a logarithmic penalty attached to the optimal order. Finally, a simulation study shows that the resulting estimator may be favored over the usual wavelet estimator, with a proper choice of the preliminary estimator of the biased density.

Applied Mathematical Sciences

In this paper, a wavelet estimator is proposed for baseline intensity function in a Cox Proportional Hazards model. The estimator has a linear form and is derived from the functional wavelet approximation method under a censoring scheme. Finally, asymptotic properties of the proposed estimator are discussed, namely, the unbiasedness of the estimator, its consistency and the optimal value of the resolution.