# Monte Carlo Expectation-Maximization (EM) Algorithm

Monte Carlo Expectation-Maximization (EM) Algorithm

The Monte Carlo expectation-maximization (EM) algorithm is used to estimate the mean in a random sample of size from a left-censored standard normal distribution with censor point, , where is the censor rate and is the inverse cumulative distribution function of the standard normal distribution. The random sample consists of non-censored observations and censored observations. You can see these observations in the quantile plot along with the empirical censor rate, =(n-m)/n.

n

c=(r)

-1

Φ

r

-1

Φ

m

n-m

c

Each Monte Carlo iteration uses simulations of latent censored values in the right-truncated normal distribution on to provide an estimate of the mean. Monte Carlo iterations are done and these are shown as the blue points. The blue line segment shows the corresponding average for the blue points. The short red line segment on the right shows the deterministic EM estimate.

M

(-∞,c)

N

You can see that using the average improves the accuracy, since the blue line showing the average is closer to the red line than the individual points. Also, you see that as the number of iterations increases, the points lie closer, illustrating the well-known result that the iterations converge to the deterministic value. The gray line shows the true parameter value, , and this Demonstration shows that as the sample size increases, both the Monte Carlo and deterministic EM algorithm converge to the true value. See Details for further discussion.

μ=0

n