Theory of Statistics - George Mason University

184 2 Distribution Theory and Statistical Models
The support of a distribution may also be changed by censoring. “Censoring”
refers what is done to a random variable, so strictly speaking, we do
not study a “censored distribution”, but rather the distribution of a random
variable that has been censored.
There are various types of censoring. One type is similar to the truncation
of a distribution, except that if a realization of the random variable occurs
outside of the truncated support, that fact is observed, but the actual value
of the realized is not known. This type of censoring is called “type I” fixed
censoring, and in the case that the support is an interval, the censoring is
called “right” or “left” if the truncated region of the support is on the right
(that is, large values) or on the left (small values). A common situation in
which type I fixed censoring occurs is when the random variable is a survival
time, and several observational units are available to generate data. Various
probability distributions such as exponential, gamma, Weibull, or lognormal
may be used to model the survival time. If an observational unit survives
beyond some fixed time, say tc, only that fact is recorded and observation of
the unit ceases.
In another kind of fixed censoring, also illustrated by observation of failure
times of a given set of say n units, the realized failure times are recorded
until say r units have failed. This type of censoring is called “type II” fixed
censoring.
If an observational unit is removed prior to its realized value being observed
for no particular reason relating to that unobserved value, the censoring is
called “random censoring”.
Censoring in general refers to a failure to observe the realized value of
a random variable but rather to observe only some characteristic of that
value. As another example, again one that may occur in studies of survival
times, suppose we have independent random variables T1 and T2 with some
assumed distributions. Instead of observing T1 and T2, however, we observe
X = min(T1, T2) and G, an indicator of whether X = T1 or X = T2. In this
case, T1 and T2 are censored, but the joint distribution of X and G may be
relevant, and it may be determined based on the distributions of T1 and T2.
2.8 Mixture Families
In applications it is often the case that a single distribution models the observed
data adequately. Sometimes two or more distributions from a single
family of distributions provide a good fit of the observations, but in other
cases, more than one distributional family is required to provide an adequate
fit. In some cases most of the data seem to come from one population but
a small number seem to be extreme outliers. Some distributions, such as a
Cauchy, are said to be “outlier-generating”, but often such distributions are
difficult to work with (because they have infinite moments, for example). Mix-
Theory of Statistics c○2000–2013 James E. Gentle