TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 24

CHAPTER 47 Density Estimation. [Sim96] is a very encompassing text. A more elementary introduction with good explanations is [WJ95]. This also has some plots with datasets relevant to economics, see pp. 1, 11, and there is a R and Splus-package called KernSmooth associated with it | CHAPTER 47 Density Estimation Sim96 is a very encompassing text. A more elementary introduction with good explanations is WJ95 . This also has some plots with datasets relevant to economics see pp. 1 11 and there is a R and Splus-package called KernSmooth associated with it but this package does not contain the datasets . A more applied book is BA97 which goes together with the R and Splus-package sm. . How to Measure the Precision of a Density Estimator Let f be the estimated density and f the true density. Then for every fixed value u the estimation error at u is f u f u . This is a random variable which depends on u as a nonrandom parameter. Its expected value is the bias at u E f u f u 1031 1032 47. DENSITY ESTIMATION and the expected value of its square is the MSE at u E u f u 2 . This is a measure of the precision of the density estimate at point u only. The overall deviation of the estimated density from the true density can be measured by the integrated squared error ISE f f u f u du This is a random variable for each observation vector it gives a different number. The mean integrated squared error MISE is the expected value of the ISE and at the same time as long as integration and formation of the expected value can be interchanged it is the integral of the MSE at u over all us MISE J ZTO u f u 2 du JuJ to E u f u 2 du. The asymptotic value of this is called AMISE. The MSE is the variance plus the squared bias. In density estimation bias arises if one overmoothes and variance increases if one undersmoothes. . The Histogram Histograms are density estimates. They are easy to understand easy to construct and do not require advanced graphical tools. Here the number of bins is important. Too few bins lead to oversmoothing too many to undersmoothing. Sim96 p. 16 has some math how to compute the MISE of a histogram and which bin size is optimal. If the underlying distribution . THE HISTOGRAM 1033 is Normal the optimal bin width is h rn-1 3 This

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