Lecture

Nonparametric Statistics: Estimation and Optimization

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Description

This lecture covers nonparametric statistics, focusing on estimating the distribution without assuming a parametric form. Topics include empirical distribution functions, consistency without assumptions, kernel density estimators, bandwidth selection, and the bias-variance tradeoff. The instructor discusses the plug-in principle, optimal bandwidth selection, and the asymptotic risk of kernel density estimators.

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Kernel density estimation

In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form.

Multivariate kernel density estimation

Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. Apart from histograms, other types of density estimators include parametric, spline, wavelet and Fourier series. Kernel density estimators were first introduced in the scientific literature for univariate data in the 1950s and 1960s and subsequently have been widely adopted.

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to . Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) satisfying and .

Empirical distribution function

In statistics, an empirical distribution function (commonly also called an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

Nonparametric statistics

Nonparametric statistics is the type of statistics that is not restricted by assumptions concerning the nature of the population from which a sample is drawn. This is opposed to parametric statistics, for which a problem is restricted a priori by assumptions concerning the specific distribution of the population (such as the normal distribution) and parameters (such the mean or variance).

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