March 19, 2013

Density Estimation (Advanced Data Analysis from an Elementary Point of View)

The desirability of estimating not just conditional means, variances, etc., but whole distribution functions. Parametric maximum likelihood is a solution, if the parametric model is right. Histograms and empirical cumulative distribution functions are non-parametric ways of estimating the distribution: do they work? The Glivenko-Cantelli law on the convergence of empirical distribution functions, a.k.a. "the fundamental theorem of statistics". More on histograms: they converge on the right density, if bins keep shrinking but the number of samples per bin keeps growing. Kernel density estimation and its properties: convergence on the true density if the bandwidth shrinks at the right rate; superior performance to histograms; the curse of dimensionality again. An example with cross-country economic data. Kernels for discrete variables. Estimating conditional densities; another example with the OECD data. Some issues with likelihood, maximum likelihood, and non-parametric estimation. Simulating from kernel density estimates and from histograms.

Reading: Notes, chapter 15

Advanced Data Analysis from an Elementary Point of View

Posted at March 19, 2013 10:30 | permanent link

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