3 a Note on Density Estimation for Binary Sequences

2006

We consider quantization of signals in probabilistic framework. In practice, signals (or random processes) are observed at sampling points. We study probabilistic models for run-length encoding (RLE) method. This method is characterized by the compression efficiency coefficient (or quantization rate) and is widely used, for example, in digital signal and image compression. Some properties of RLE quantization rate are investigated. Statistical inference for mean RLE quantization rate is considered. In particular, the asymptotical normality of mean RLE quantization rate estimates is studied. Numerical experiments demonstrating the rate of convergence in the obtained asymptotical results are presented.

Annals of The Institute of Statistical Mathematics, 1995

A b s t r a c t . Let {X~,n _> 1} be a strictly stationary sequence of associated random variables defined on a probability space (f~, B, P) with probability density function f(x) and failure rate function r(x) for X1. Let f,~(x) be a kerneltype estimator of f(x) based on X1,..., X~. Properties of f,~(x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimator rn(x) of r(x) based on fn(X) and P,~(x), the empirical survival function, is proposed. The estimator rn(x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.

2018

Recalling recent results on the characterization of threshold-based sampling as quasi-isometric mapping, mathematical implications on the metric and topological structure of the space of event sequences are derived. In this context, the space of event sequences is extended to a normed space equipped with Hermann Weyl's discrepancy measure. Sequences of finite discrepancy norm are characterized by a Jordan decomposition property. Its dual norm turns out to be the norm of total variation. As a by-product a measure for the lack of monotonicity of sequences is obtained. A further result refers to an inequality between the discrepancy norm and total variation which resembles Heisenberg's uncertainty relation.

IEEE Transactions on Information Theory, 1992

The problem of the nonparametric estimation of a probability distribution is considered from three viewpoints: the consistency in total variation, the consistency in information divergence, and consistency in reversed order information divergence. These types of consistencies are relatively strong criteria of convergence, and a probability distribution cannot he consistently estimated in either type of convergence without any restrictions on the class of probability distributions allowed. Histogram-based estimators of distribution are presented which, under certain conditions, converge in total variation, in information divergence, and in reversed order information divergence to the unknown probability distribution. Some a priori information about the true probability distribution is assumed in each case. As the concept of consistency in information divergence is stronger than that of convergence in total variation, additional assumptions are imposed in the cases of informational divergences. Index Tens-Consistent distribution estimation, total variation, information divergence, reversed order information divergence, histogram-based estimate. E CONSIDER the problem of estimating an un-W known probability distribution p, defined on an arbitrary measurable space ( X , 231, based on independent, identically distributed (i.i.d.) observations XI ,.*e, X,, from p. Here, X could be the real line [w or the Euclidean space Rd, d 2 1, in which case % is the collection of Bore1 sets. As generic notation for the distribution estimate of a set A we use = P . , * ( A X l , X * , . . . > X , , ) > (1 .1> where p.,* is a measurable function of its arguments. In this paper, we examine density estimation problems and related distribution estimation problems that are mo-Manuscript received March 6, 1990. A. R.

Applied Algebra and Number Theory

Large families of binary sequences of the same length are considered and a new measure, the cross-correlation measure of order k is introduced to study the connection between the sequences belonging to the family. It is shown that this new measure is related to certain other important properties of families of binary sequences. Then the size of the cross-correlation measure is studied. Finally, the cross-correlation measures of two important families of pseudorandom binary sequences are estimated.

Statistics & Probability Letters, 1990

Rate of convergence for density estimators based on Haar series are derived under very mild condition: the unknown density has to be of bounded variation. These estimators are histograms on dyadic intervals.

arXiv: Statistics Theory, 2020

In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate--optimal estimators in the case of integer $p\geq 2$. We demonstrate that, depending on parameters of Nikolskii's class and the norm index $p$, the risk asymptotics ranges from inconsistency to $\sqrt{n}$--estimation. The results in this paper complement the minimax lower bounds derived in the companion paper \cite{gl20}.

1999

In this paper, we present some general results determining minimax bounds on statistical risk for density estimation based on certain information-theoretic considerations. These bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence. This work was supported by NSF Grants ECS-9410760 and DMS-9505168. AMS 1991 subject classi cations. Primary 62G07; secondary 62B10, 62C20, 94A29. Key words and phrases. minimax risk, density estimation, metric entropy, Kullback-Leibler distance. of -packing sets is called the packing -entropy or Kolmogorov capacity of S with distance function d and is denoted M d ( ).

Applied Mathematical Sciences, 2013

In this paper, the estimation of a multivariate probability density f of mixing sequences, using wavelet method is considered. We investigate the rate of the L 2-almost sure convergence of wavelet estimators. Optimal rate, up to a logarithm, of convergence of estimators when f belongs to the Sobolev space H s 2 (R d) with s > 0 is established.

Journal of Multivariate Analysis, 2014

Many real phenomena may be modelled as random closed sets in R d , of different Hausdorff dimensions. Of particular interest are cases in which their Hausdorff dimension, say n, is strictly less than d, such as fiber processes, boundaries of germ-grain models, and n-facets of random tessellations. A crucial problem is the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous random sets, as defined by the authors in a series of recent papers. While the case n = 0 (random vectors, point processes, etc.) has been, and still is, the subject of extensive literature, in this paper we face the general case of any n < d; pointwise density estimators which extend the notion of kernel density estimators for random vectors are analyzed, together with a previously proposed estimator based on the notion of Minkowski content. In a series of papers, the authors have established the mathematical framework for obtaining suitable approximations of such mean densities. Here we study the unbiasedness and consistency properties, and identify optimal bandwidths for all proposed estimators, under sufficient regularity conditions. We show how some known results in literature follow as particular cases. A series of examples throughout the paper, both non-stationary, and stationary, are provided to illustrate various relevant situations.