Nonparametric Statistics with Applications to Science and Engineering with R, 2nd Edition

11Density Estimation

George McFly: Lorraine, my density has brought me to you.

Lorraine Baines: What?

George McFly: Oh, what I meant to say was

Lorraine Baines: Wait a minute, don't I know you from somewhere?

George McFly: Yes. Yes. I'm George, George McFly.

I'm your density. I mean your destiny.

From the movie Back to the Future, 1985

Probability density estimation goes hand in hand with nonparametric estimation of the cumulative distribution function discussed in Chapter 10. There, we noted that the density function provides a better visual summary of how the random variable is distributed across its support. Symmetry, skewness, disperseness, and unimodality are just a few of the properties that are ascertained when we visually scrutinize a probability density plot.

Recall, for continuous i.i.d. data, the empirical density function places probability mass on each of the observations. While the plot of the empirical distribution function (EDF) emulates the underlying distribution function, for continuous distributions, the empirical density function takes no shape beside the changing frequency of discrete jumps of across the domain of the underlying distribution – see Figure ...