semimrLocal {MixSemiRob}

Semiparametric Mixtures of Nonparametric Regressions with Local EM-type Algorithm

Description

‘semimrLocal’ 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 local EM-type algorithm (LEM) (Xiang and Yao, 2018). As of version 1.1.0, this function supports a two-component model.

Usage

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

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 with length(x) rows, and var is a vector of length 2 of variances.

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.

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.

See Also

semimrGlobal, semimrGen

Examples

# 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 semimrLocal function.
est = semimrLocal(x, y)