mixpf function

Profile Likelihood Method for Normal Mixture with Unequal Variance

`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 $k$: [REMOVE_ME]$f(x;\boldsymbol{\theta}) = \sum_{j=1}^C\pi_j\phi(x;\mu_j,\sigma_j^2), [REMOVE_ME_2]$

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.

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.

Returns

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.

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 $k$:

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.

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")

References

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