semimrGlobal: Semiparametric Mixtures of Nonparametric Regressions with Global EM-type Algorithm

Description

`semimrGlobal' is used to estimate a mixture of regression models, where the mixing proportions and variances remain constant, but the component regression functions are smooth functions ($m(\cdot)$) of a covariate. The model is expressed as follows: $$\sum_{j=1}^C\pi_j\phi(y|m(x_j),\sigma^2_j).$$ This function provides the one-step backfitting estimate using the global EM-type algorithm (GEM) (Xiang and Yao, 2018). As of version 1.1.0, this function supports a two-component model.

Usage

semimrGlobal(x, y, u = NULL, h = NULL, ini = NULL)

Value

A list containing the following elements:

pi: vector of length 2 of estimated mixing proportions.
mu: length(x) by 2 matrix of estimated mean functions at x, $m(x)$.
mu_u: length(u) by 2 matrix of estimated mean functions at grid point u, $m(u)$.
var: vector of length 2 estimated component variances.
lik: final likelihood.

Arguments

x: a vector of covariate values.
y: a vector of response values.
u: a vector of grid points for spline method to estimate the proportions. If NULL (default), 100 equally spaced grid points are automatically generated between the minimum and maximum values of x.
h: bandwidth for the nonparametric regression. If NULL (default), the bandwidth is calculated based on the method of Botev et al. (2010).
ini: initial values for the parameters. Default is NULL, which obtains the initial values using regression spline approximation. If specified, it can be a list with the form of list(pi, mu, var), where pi is a vector of length 2 of mixing proportions, mu is a length(x) by 2 matrix of component means, and var is a vector of length 2 of component variances.

References

Xiang, S. and Yao, W. (2018). Semiparametric mixtures of nonparametric regressions. Annals of the Institute of Statistical Mathematics, 70, 131-154.

Botev, Z. I., Grotowski, J. F., and Kroese, D. P. (2010). Kernel density estimation via diffusion. The Annals of Statistics, 38(5), 2916-2957.

Examples

Run this code

# produce data that matches the description using semimrGen function
# true_mu = (4 - sin(2 * pi * x), 1.5 + cos(3 * pi * x))
n = 100
u = seq(from = 0, to = 1, length = 100)
true_p = c(0.3, 0.7)
true_var = c(0.09, 0.16)
out = semimrGen(n, true_p[1], true_var, u)

x = out$x
y = out$y
true_mu = out$true_mu
true = list(true_p = true_p, true_mu = true_mu, true_var = true_var)

# estimate parameters using semimrGlobal function.
est = semimrGlobal(x, y)