Density estimation with dependent observations

International Journal of Statistics and Probability

In this paper, we construct a new wavelet estimator of density for the component of a finite mixture under positive quadrant dependence. Our sample is extracted from almost periodically correlated processes. To evaluate our estimator we will determine a convergence speed from an upper bound for the mean integrated squared error (MISE). Our result is compared to the independent case which provides an optimal convergence rate.

Scandinavian Journal of Statistics, 2018

We consider a process that is observed as a mixture of two random distributions, where the mixing probability is an unknown function of time. The setup is built upon a wavelet-based mixture regression. Two linear wavelet estimators are proposed. Furthermore, we consider three regularizing procedures for each of the two wavelet methods. We also discuss regularity conditions under which the consistency of the wavelet methods is attained and derive rates of convergence for the proposed estimators. A Monte Carlo simulation study is conducted to illustrate the performance of the estimators. Various scenarios for the mixing probability function are used in the simulations, in addition to a range of sample sizes and resolution levels. We apply the proposed methods to a data set consisting of array Comparative Genomic Hybridization from glioblastoma cancer studies.

In this paper, we give a strong motivation, based on new statistical problems mostly concerned with high frequency data, for the construction of second generation wavelets. These new wavelets basically differ from the classical ones in the fact that, instead of being constructed on the Fourier basis, they are associated with different orthonormal bases such as bases of polynomials. We give in the introduction three statistical problems where these new wavelets are clearly helpful. These examples are revisited in the core of the paper, where the use of the wavelets are enlightened. The construction of these new wavelets is given as well as their important concentration properties in spectral and space domains. Spaces of regularity associated with these new wavelets are studied, as well as minimax rates of convergence for nonparametric estimation over these spaces.

Journal of Nonparametric Statistics, 2012

2011

We consider the estimation of a two dimensional continuous-discrete density function with applications to competing risks. We construct two new wavelet estimators (non-adaptive and adaptive) for the joint density function taking into account this special continuous-discrete structure. The rates of convergence of the proposed estimators are established under the L2 risk over Besov balls. Our main result proves that our adaptive wavelet estimator (based on hard thresholding) attains a sharp rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.

2008

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.

Journal of Multivariate Analysis, 2018

Wavelet estimators for a probability density f enjoy many good properties, however they are not 'shapepreserving' in the sense that the final estimate may not be non-negative or integrate to unity. A solution to negativity issues may be to estimate first the square-root of f and then square this estimate up. This paper proposes and investigates such an estimation scheme, generalising to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on nearestneighbour-balls. The theoretical properties of the proposed estimator are obtained, and it is shown to reach the optimal rate of convergence uniformly over large classes of densities under mild conditions. Simulations show that the new estimator performs as well in general as the classical wavelet estimator, while automatically producing estimates which are bona fide densities.

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 [email protected]. 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

1998

In this contribution, the statistical properties of the wavelet estimator of the long-range dependence parameter introduced in Abry et al. (1995) are discussed for a stationary Gaussian process. This contribution complements the heuristical discussion presented in Abry et al. (1999), by taking into account the correlation between the wavelet coecients (which is discarded in the mentioned reference) and the bias

Statistics, 2014

In this paper, a mixture model under multiplicative censoring is considered. We investigate the estimation of a component of the mixture (a density) from the observations. A new adaptive estimator based on wavelets and a hard thresholding rule is constructed for this problem. Under mild assumptions on the model, we study its asymptotic properties by determining an upper bound of the mean integrated squared error over a wide range of Besov balls. We prove that the obtained upper bound is sharp.

Random Operators and Stochastic Equations, 2006

In the paper we present conditions for uniform convergence with probability one of wavelet expansions of ϕ-sub-Gaussian (in particular, Gaussian) random processes defined on the space R. It is shown that upon certain conditions for the bases of wavelets the wavelet expansions of stationary almost sure continuous Gaussian processes and wavelet expansions of fractional Brownian motion converge uniformly with probability one on any finite interval.

The Annals of Statistics, 1996

Density estimation is a commonly used test case for non-parametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coecients. Minimax rates of convergence are studied over a large range of Besov function classes B s;p;q and for a range of global L 0 p error measures, 1 p 0 < 1. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p 0 > p , some form of non-linearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error (p 0 = 2).

Journal of Nonparametric Statistics, 1997

Adaptive sampling schemes with multiple sampling rates have the potential to significantly improve the efficiency and effectiveness of methods for signal analysis. For example, in the case of equipment which transmits data continuously, multi-rate methods can reduce the cost of transmission. For equipment which transmits data only periodically they can reduce the costs of both storage and transmission. When multiple sampling rates are used in connection with wavelet estimators, the most natural algorithms for rate-switching are arguably those based on threshold-crossings by wavelet coefficients. In this paper we study the performance of such algorithms, and show that even simple threshold-crossing rules can achieve near-optimal convergence rates. A new mathematical model is suggested for assessing performance, combining the simplicity and familiarity of global approaches with an account of the local variation to which multi-rate sampling responds.

2011

In this paper, we consider estimating copulas for time series, under mixing conditions, using wavelet expansions. The proposed estimators are based on estimators of densities and distribution functions. Some statistical properties of the estimators are derived and their performance assessed via simulations. Empirical applications to real data are also given.