Comparing Exact and Approximate Censored Normal Likelihoods

display

likelihood

normal quantile

sample size, n

25

50

100

theoretical censor rate, r

0.414

random seed

22889

	exact
	approximate

-1.0

-0.5

0.0

0.5

1.0

0.0

0.2

0.4

0.6

0.8

1.0

μ

R

A random sample of size

n

is simulated from a left-censored standard normal distribution with censor point

c=Φ

-1

(r)

, where

r

is the censor rate and

Φ

-1

is the inverse cumulative distribution function of the standard normal distribution. The Demonstration shows the exact relative likelihood function for the mean of the underlying distribution (blue curve) as well as a crude approximation (red curve) that ignores censoring. When

r

is close to zero, the crude approximation is reasonable, but as

r

increases, the exact method becomes more and more accurate. The blue and red lines show corresponding point estimates for the exact and approximate estimates. while the gray line shows the true parameter value. By varying the random seed, you see the stochastic nature of estimates; for example, sometimes the crude estimator may be more accurate, even though overall, on average, it is less accurate when

r>0

.