MixSemiRob (version 1.1.0)

kdeem: Kernel Density-based EM-type algorithm for Semiparametric Mixture Regression with Unspecified Error Distributions

Description

`kdeem' is used for semiparametric mixture regression using a kernel density-based expectation-maximization (EM)-type algorithm with unspecified homogeneous or heterogenous error distributions (Ma et al., 2012).

Usage

kdeem(x, y, C = 2, ini = NULL, maxiter = 200)

Value

A list containing the following elements:

posterior

posterior probabilities of each observation belonging to each component.

beta

estimated regression coefficients.

tau

estimated precision parameters, the inverse of standard deviation.

pi

estimated mixing proportions.

h

bandwidth used for the kernel estimation.

Arguments

x

an n by p data matrix where n is the number of observations and p is the number of explanatory variables (including the intercept).

y

an n-dimensional vector of response variable.

C

number of mixture components. Default is 2.

ini

initial values for the parameters. Default is NULL, which obtains the initial values using the kdeem.lse function. 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.

maxiter

maximum number of iterations for the algorithm. Default is 200.

Details

It can be used for a semiparametric mixture of linear regression models with unspecified component error distributions. The errors can be either homogeneous or heterogenous. The model is as follows: $$f_{Y|\boldsymbol{X}}(y,\boldsymbol{x},\boldsymbol{\theta},g) = \sum_{j=1}^C\pi_j\tau_jg\{(y-\boldsymbol{x}^{\top}\boldsymbol{\beta}_j)\tau_j\}.$$ Here, \(\boldsymbol{\theta}=(\pi_1,...,\pi_{C-1},\boldsymbol{\beta}_1^{\top},..,\boldsymbol{\beta}_C^{\top},\tau_1,...,\tau_C)^{\top}\), \(g(\cdot)\) is an unspecified density function with mean 0 and variance 1, and \(\tau_j\) is a precision parameter. For the calculation of \(\beta\) in the M-step, this function employs the universal optimizer function ucminf from the `ucminf' package.

References

Ma, Y., Wang, S., Xu, L., & Yao, W. (2021). Semiparametric mixture regression with unspecified error distributions. Test, 30, 429-444.

See Also

kdeem.h, kdeem.lse, and ucminf for beta calculation.

Examples

n = 300
C = 2
Dimen = 2
Beta.true.matrix = matrix(c(-3, 3, 3, -3), Dimen, C)
PI.true = c(0.5, 0.5)
x = runif(n)
X = cbind(1, x)
Group.ID = Rlab::rbern(n, prob = 0.5)
Error = rnorm(n, 0, 1)
n1 = sum(Group.ID)
n2 = n - n1
y = rep(0, n)
err = rep(0, n)

for(i in 1:n){
  if(Group.ID[i] == 1){
    err[i] = Error[i]
    y[i] = X[i, ] %*% Beta.true.matrix[, 1] + err[i]
  } else {
    err[i] = 0.5 * Error[i]
    y[i] = X[i, ] %*% Beta.true.matrix[, 2] + err[i]
  }
}
Result.kdeem.lse = kdeem.lse(x, y)
Result.kdeem.h = kdeem.h(x, y, 2, Result.kdeem.lse, maxiter = 200)
Result.kdeem = kdeem(x, y, 2, Result.kdeem.lse, maxiter = 200)