llK_b {HQM}R Documentation

Local linear kernel

Description

Implements the local linear kernel function.

Usage

llK_b(b,x,y, K)

Arguments

x

A vector of design points where the kernel will be evaluated.

y

A vector of sample data points.

b

The bandwidth to use (a scalar).

K

The kernel function to use.

Details

Implements the local linear kernel

K_{x,b}(u)= \frac{K_b(u)-K_b(u)u^T D^{-1}c_1}{c_0 - c_1^T D^{-1} c_1},

where c_1 = (c_{11}, \dots, c_{1d})^T, D = (d_{ij})_{(d+1) \times (d+1)} with

c_0 = \sum_{i=1}^n \int_0^T K_b(x-X_i(s)) Z_i(s)ds, \\ c_{ij} = \sum_{i=1}^n \int_0^T K_b(x-X_i(s))\{x-X_{ij}(s)\} Z_i(s)ds, \\ d_{jk} = \sum_{i=1}^n \int_0^T K_b(x-X_i(s))\{x-X_{ij}(s)\}\{x-X_{ik}(s)\} Z_i(s)ds,

see also Nielsen (1998).

Value

Matrix output with entries the values of the kernel function at each point.

References

doi:10.1080/03461238.1998.10413997


[Package HQM version 1.0 Index]