Snapshot 1: This shows the quantile plot of the data used in the thumbnail. You see from the plot that

and that there were five censored values, so the empirical censor rate was 20%.

Snapshot 2: The initial point is obtained as the mean of

random variables generated from the right-truncated normal distribution

with mean parameter

, where

is the sample mean of original data, including the censored values. Some crude data augmentation algorithms used in practice stop at this point. The plot shows this does not usually provide an accurate result.

Snapshot 3: Increasing the number of iterations from 0 to 5 greatly improves the accuracy. The average of these iterations is very close to the deterministic EM result.

Snapshot 4: Increasing both the number of iterations and the number of simulations shows convergence to the deterministic EM result.

Snapshot 5: Using only

iterations and

simulations but increasing the data sample size from

to

results in estimates that more accurately estimate the true value,

. Also note the decrease in variability as the iterations increase.

The Monte Carlo EM algorithm for the censored normal distribution is discussed in [1, p. 533].

A new discovery illustrated in this Demonstration is that a more accurate estimate may be obtained by using the average of the Monte Carlo EM iterates, as shown by the fact that the horizontal blue line segment has nearly the same ordinate as the red one. This is what is usually done in Markov chain Monte Carlo (MCMC) applications [2], and the Monte Carlo EM algorithm may be viewed as a special case of MCMC [3].

[1] C. R. Robert and G. Casella,

*Monte Carlo Statistical Methods*, New York: Springer, 2004.

[2] C. J. Geyer, "Introduction to Markov Chain Monte Carlo" in

*Handbook of Markov Chain Monte Carlo* (S. Brooks, A. Gelman, G. L. Jones, and X.-L. Meng, eds.), CRC Press: Boca Raton, 2011.

[3] D. A. van Dyk and X.-L. Meng, "The Art of Data Augmentation,"

*Journal of Computational and Graphical Statistics*,

**10**(1), 2001 pp. 1–50.

doi:10.1198/10618600152418584.