ESL Eqs. (6.23)-(6.24). In one dimension the estimate IS the
empirical distribution convolved with a Gaussian of standard
deviation lambda: F_hat puts mass 1/N at each observation
and is jumpy, and the estimate smooths it by adding independent
Gaussian noise to each point. That is why lambda is a standard
deviation here rather than a generic bandwidth.
Usage
morie_esl_kernel_density(x, data, lambda_ = NULL)
Arguments
- x
evaluation points: a vector, or a matrix with p columns.
- data
sample: a vector, or a matrix with p columns.
- lambda_
kernel standard deviation; Silverman's normal
reference 1.06 sigma N^(-1/5) when NULL, which is a DENSITY
rule and correct here.
Value
list: x, density, lambda, n, p, mass (1-D only),
normaliser, is_convolution, method.
Details
In p dimensions the Gaussian product kernel (6.24) normalises by
N (2 lambda^2 pi)^(p/2) – the exponent is p/2, not p.
References
Hastie, Tibshirani and Friedman (2009), Sec. 6.6,
Eqs. (6.22)-(6.24); Parzen (1962).
Examples
set.seed(1)
morie_esl_kernel_density(c(-1, 0, 1), stats::rnorm(200))$density
#> [1] 0.2309164 0.4073686 0.2288376
