Methods of choosing histogram width and the smoothing parameter of kernel density estimators by use of data are studied. They are based on estimators of risk functions corresponding to mean integrated squared error and the Kullback-Leibler information measure. Two closely related risk function estimators are given, one of Which is derived from cross-validation. In examples with simulated and real data the methods are applied to estimation of probability densities and the rate function of a time-depend- ent Poisson process.

The laws of large numbers

Empirical Choice of Histograms and Kernel Density Estimators

Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator

The time histogram method is the most basic tool for capturing a time dependent rate of neuronal spikes. Generally in the neurophysiological literature, the bin size that critically determines the goodness of the fit of the time histogram to the underlying spike rate has been subjectively selected by individual researchers. Here, we propose a method for objectively selecting the bin size from the spike count statistics alone, so that the resulting bar or line graph time histogram best represents the unknown underlying spike rate. For a small number of spike sequences generated from a modestly fluctuating rate, the optimal bin size may diverge, indicating that any time histogram is likely to capture a spurious rate. Given a paucity of data, the method presented here can nevertheless suggest how many experimental trials should be added in order to obtain a meaningful time-dependent histogram with the required accuracy.

A Method for Selecting the Bin Size of a Time Histogram

Let (X k) k≥1 be a sequence of random variables with common distribution function F(x) = P(X 1 ≤ x). Define the empirical distribution function 

$$ {{F}_{n}}(x) = \frac{1}{n}\# \{ 1 \leqslant i \leqslant n:{{X}_{1}} \leqslant x\} , $$

and the empirical process by $ \sqrt {n} ({{F}_{n}}(x) - F(x)) $ In this chapter we provide a survey of classical as well as modern techniques in the study of empirical processes of dependent data. We begin with a sketch of the early roots of the field in the theory of uniform distribution mod 1, of sequences defined by X k = {n k ω}, ω ∈ [0, 1], dating back to Weyl’s celebrated 1916 paper. In the second section we provide the essential tools of empirical process theory, and we prove Donsker’s classical empirical process invariance principle for i.i.d. processes. The third section provides an introduction to the subject of weakly dependent random variables. We introduce a variety of mixing concepts, provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums. The empirical process of weakly dependent data is investigated in the fourth section, where we put special emphasis on almost sure approximation techniques. The fifth section is devoted to the empirical distribution of U-statistics, defined as 

$$ Un(x) = {{\left( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right)}^{{ - 1}}}\# \{ 1 \leqslant i < j \leqslant n:h({{X}_{i}},{{X}_{j}}) \leqslant x\} $$

for some symmetric kernel h. We give some applications, e.g., to dimension estimation in the analysis of time series, and prove weak convergence of the corresponding empirical process. Empirical processes of long-range dependent data are the topic of the sixth section. We give an introduction to the area of long-range dependent processes, provide important technical tools for the study of their partial sums and investigate the limit behavior of the empirical process. It turns out that the limit process is of a completely different type as in the case of independent or weakly dependent data, and that this has important consequences for various functionals of the empirical process. The final section is devoted to pair correlations, i.e., U-statistics empirical processes over short intervals associated with the kernel h(x, y) = |x − y|

Empirical Process Techniques for Dependent Data

(1972) Nonparametric Probability Density Estimation: I A Summary of Available Methods Technometrics: Vol 14, No 3, pp 533-546

The laws of large numbers

Citations

Empirical Choice of Histograms and Kernel Density Estimators

Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator

A Method for Selecting the Bin Size of a Time Histogram

Empirical Process Techniques for Dependent Data

Nonparametric Probability Density Estimation: I. A Summary of Available Methods

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