VGAM (version 0.7-1)

mix2normal1: Mixture of Two Univariate Normal Distributions

Description

Estimates the five parameters of a mixture of two univariate normal distributions by maximum likelihood estimation.

Usage

mix2normal1(lphi="logit", lmu="identity", lsd="loge",
            iphi=0.5, imu1=NULL, imu2=NULL, isd1=NULL, isd2=NULL,
            qmu=c(0.2, 0.8), esd=FALSE, zero=1)

Arguments

lphi

Link function for the parameter $\phi$. See below for more details. See Links for more choices.

lmu

Link function applied to each $\mu$ parameter. See Links for more choices.

lsd

Link function applied to each $\sigma$ parameter. See Links for more choices.

iphi

Initial value for $\phi$, whose value must lie between 0 and 1.

imu1, imu2

Optional initial value for $\mu_1$ and $\mu_2$. The default is to compute initial values internally using the argument qmu.

isd1, isd2

Optional initial value for $\sigma_1$ and $\sigma_2$. The default is to compute initial values internally based on the argument qmu.

qmu

Vector with two values giving the probabilities relating to the sample quantiles for obtaining initial values for $\mu_1$ and $\mu_2$. The two values are fed in as the probs argument into

esd

Logical indicating whether the two standard deviations should be constrained to be equal. If set TRUE, the appropriate constraint matrices will be used.

zero

An integer specifying which linear/additive predictor is modelled as intercepts only. If given, the value or values must be from the set ${1,2,\ldots,5}$. The default is the first one only, meaning $\phi$ is a single parameter even when there ar

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Numerical problems can occur. Half-stepping is not uncommon. If failure to converge occurs, try obtaining better initial values, e.g., by using iphi and qmu etc.

This function uses a quasi-Newton update for the working weight matrices (BFGS variant). It builds up approximations to the weight matrices, and currently the code is not fully tested. In particular, results based on the weight matrices (e.g., from vcov and summary) may be quite incorrect, especially when the arguments weights is used to input prior weights.

Details

The probability function can be loosely written as $$f(y) = \phi \, N(\mu_1,\sigma_1^2) + (1-\phi) \, N(\mu_2, \sigma_2^2)$$ where $\phi$ is the probability an observation belongs to the first group. The parameters $\mu_1$ and $\mu_2$ are the means, and $\sigma_1$ and $\sigma_2$ are the standard deviations. The parameter $\phi$ satisfies $0 < \phi < 1$. The mean of $Y$ is $\phi \mu_1 + (1-\phi) \mu_2$ and this is returned as the fitted values. By default, the five linear/additive predictors are $(logit(\phi), \mu_1, \log(\sigma_1), \mu_2, \log(\sigma_2))^T$. If esd=TRUE then $\sigma_1 = \sigma_2$ is enforced.

References

Everitt, B. S. and Hand, D. J. (1981) Finite Mixture Distributions. London: Chapman & Hall.

See Also

normal1, Normal, mix2poisson.

Examples

n = 1000
mu1 =  99   # Mean IQ of geography professors
mu2 = 150   # Mean IQ of mathematics professors
sd1 = sd2 = 16
phi = 0.3
y = ifelse(runif(n) < phi, rnorm(n, mu1, sd1), rnorm(n, mu2, sd2))

# Good idea to have trace=TRUE:
fit = vglm(y ~ 1, mix2normal1(esd=TRUE), maxit=200)
coef(fit, matrix=TRUE) # the estimates
c(phi, mu1, sd1, mu2, sd2) # the truth

# Plot the results
xx = seq(min(y), max(y), len=200)
plot(xx, (1-phi)*dnorm(xx, mu2, sd2), type="l", xlab="IQ",
     main="Red=estimate, blue=truth", col="blue")
phi.est = logit(coef(fit)[1], inverse=TRUE)
sd.est = exp(coef(fit)[3])
lines(xx, phi*dnorm(xx, mu1, sd1), col="blue")
lines(xx, phi.est * dnorm(xx, Coef(fit)[2], sd.est), col="red")
lines(xx, (1-phi.est) * dnorm(xx, Coef(fit)[4], sd.est), col="red")
abline(v=Coef(fit)[c(2,4)], lty=2, col="red")
abline(v=c(mu1, mu2), lty=2, col="blue")