mixpf {MixSemiRob}

Profile Likelihood Method for Normal Mixture with Unequal Variance

Description

‘mixpf’ is used to estimate the following C-component univariate normal mixture model, using the profile likelihood method (Yao, 2010), with the assumption that the ratio of the smallest variance to the largest variance is k:

f(x;\boldsymbol{\theta}) = \sum_{j=1}^C\pi_j\phi(x;\mu_j,\sigma_j^2),

where \boldsymbol{\theta}=(\pi_1,\mu_1,\sigma_1,..,\pi_{C},\mu_C,\sigma_C)^{\top} is the parameter to estimate, \phi(\cdot;\mu,\sigma^2) is the normal density with a mean of \mu and a standard deviation of \sigma, and \pi's are mixing proportions that sum up to 1. Once the results are obtained, one can also find the maximum likelihood estimate (MLE) of k by plotting the likelihood vs. k for different k values and finding the maximum interior mode in the likelihood. See examples below.

Usage

mixpf(x, k = 0.5, C = 2, nstart = 20)

Arguments

x

a vector of observations.

k

ratio of the smallest variance to the largest variance. Default is 0.5.

C

number of mixture components. Default is 2.

nstart

number of initializations to try. Default is 20.

Value

A list containing the following elements:

mu

vector of estimated component means.

sigma

vector of estimated component standard deviations.

pi

vector of estimated mixing proportions.

lik

final likelihood.

References

Yao, W. (2010). A profile likelihood method for normal mixture with unequal variance. Journal of Statistical Planning and Inference, 140(7), 2089-2098.

Examples

set.seed(4)
n = 100
u = runif(n, 0, 1)
x2 = (u <= 0.3) * rnorm(n, 0, 0.5) + (u > 0.3) * rnorm(n, 1.5, 1)

# please set ngrid to 200 to get a smooth likelihood curve
ngrid = 5
grid = seq(from = 0.01, to = 1, length = ngrid)
likelihood = numeric()
for(i in 1:ngrid){
  k = grid[i]
  est = mixpf(x2, k)
  lh = est$lik
  likelihood[i] = lh
}

# visualize likelihood to find the best k
plot(grid, likelihood, type = "l", lty = 2, xlab = "k", ylab = "profile log-likelihood")