mixScale function

Continuous Scale Mixture Approach for Normal Scale Mixture Model

`mixScale' is used to estimate a two-component continuous normal scale mixture model, based on a backfitting method (Xiang et al., 2016): [REMOVE_ME]$p(x;\boldsymbol{\theta},f) = \pi f_1(x-\mu_1) + (1-\pi) f_2(x-\mu_2), [REMOVE_ME_2]$

where $\boldsymbol{\theta}=(\pi,\mu_1,\mu_2)$. Here, $f$ is assumed to be a member of $\mathcal{F} = \left\{ f(x) \big| \int\frac{1}{\sigma}\phi(x/\sigma)dQ(\sigma) \right\}$, where $\phi(x)$ is the standard normal density and $Q$ is an unspecified probability measure on positive real numbers.

mixScale(x, ini = NULL, maxiter = 100)

Arguments

• x: a vector of observations.
• ini: initial values for the parameters. Default is NULL, which obtains the initial values using the mixnorm function. If specified, it can be a list with the form of list(pi, mu, sigma), where pi is a vector of 2 mixing proportions, mu is a vector of 2 component means, and sigma is a vector of 2 component (common) standard deviations.
• maxiter: maximum number of iterations for the EM algorithm. Default is 100.

Returns

A list containing the following elements: - mu: estimated component means.

• pi: estimated mixing proportions.

• suppQ: support of Q.

• weightQ: weight of Q corresponding to initial standard deviations.

• loglik: final log-likelihood.

• run: number of iterations after convergence.

Description

`mixScale' is used to estimate a two-component continuous normal scale mixture model, based on a backfitting method (Xiang et al., 2016):

where $\boldsymbol{\theta}=(\pi,\mu_1,\mu_2)$. Here, $f$ is assumed to be a member of $\mathcal{F} = \left\{ f(x) \big| \int\frac{1}{\sigma}\phi(x/\sigma)dQ(\sigma) \right\}$, where $\phi(x)$ is the standard normal density and $Q$ is an unspecified probability measure on positive real numbers.

Examples

require(quadprog)

#-----------------------------------------------------------------------------------------#
# Example 1: simulation
#-----------------------------------------------------------------------------------------#
n = 10
mu = c(-2.5, 0)
sd = c(0.8, 0.6)
pi = c(0.3, 0.7)
set.seed(2023)
n1 = rbinom(n, 1, pi[1])
x = c(rnorm(sum(n1), mu[1], sd[1]), rnorm(n - sum(n1), mu[2], sd[2]))
ini = list(pi = pi, mu = mu, sigma = sd)
out = mixScale(x, ini)

#-----------------------------------------------------------------------------------------#
# Example 2: elbow data
#-----------------------------------------------------------------------------------------#
ini = mixnorm(elbow)
res = mixScale(elbow, ini)

References

Xiang, S., Yao, W., and Seo, B. (2016). Semiparametric mixture: Continuous scale mixture approach. Computational Statistics & Data Analysis, 103, 413-425.

See Also

mixnorm for initial value calculation.