`kdeem.lse' is used for semiparametric mixture regression based on least squares estimation (Hunter and Young, 2012) using a kernel density-based expectation-maximization (EM)-type algorithm with unspecified homogeneous error distributions.
kdeem.lse(x, y, C = 2, ini = NULL)A list containing the following elements:
posterior probabilities of each observation belonging to each component.
estimated regression coefficients.
estimated precision parameter, the inverse of standard deviation.
estimated mixing proportions.
bandwidth used for the kernel estimation.
an n by p data matrix where n is the number of observations and p is the number of explanatory variables (including the intercept).
an n-dimensional vector of response variable.
number of mixture components. As of version 1.1.0, C must be set to 2.
initial values for the parameters. Default is NULL, which obtains the initial values
using the regmixEM function from the `mixtools' package.
If specified, it can be a list with the form of list(beta, prop, tau, pi, h), where
beta is a p by C matrix for regression coefficients of C components,
prop is an n by C matrix for probabilities of each observation belonging to each component,
caculated based on the initial beta and h, tau is a vector of C precision parameters
(inverse of standard deviation), pi is a vector of C mixing proportions, and
h is the bandwidth for kernel estimation.
As of version 1.1.0, this function can only be used for a two-component mixture-of-regressions model
with independent identically distributed errors. Assuming \(C=2\), the model is defined as follows:
$$f_{Y|\boldsymbol{X}}(y,\boldsymbol{x},\boldsymbol{\theta},g) = \sum_{j=1}^C\pi_jg(y-\boldsymbol{x}^{\top}\boldsymbol{\beta}_j).$$
Here, \(\boldsymbol{\theta}=(\pi_1,...,\pi_{C-1},\boldsymbol{\beta}_1^{\top},\cdots,\boldsymbol{\beta}_C^{\top})\),
and \(g(\cdot)\) represents identical unspecified density functions.
The bandwidth of the kernel density estimation is calculated adaptively using the bw.SJ function from the `stats'
package, which implements the method of Sheather & Jones (1991) for bandwidth selection based on pilot estimation
of derivatives. This function employs weighted least square estimation for \(\beta\) in the M-step (Hunter and Young, 2012),
where the weight is the posterior probability of an observation belonging to each component.
Hunter, D. R., and Young, D. S. (2012). Semiparametric mixtures of regressions. Journal of Nonparametric Statistics, 24(1), 19-38.
Ma, Y., Wang, S., Xu, L., and Yao, W. (2021). Semiparametric mixture regression with unspecified error distributions. Test, 30, 429-444.
kdeem, kdeem.h, bw.SJ
for bandwidth calculation, and regmixEM for initial value calculation.
# See examples for the `kdeem' function.
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