In this section, we are concerned with inferring the characteristics of a full population from a sample drawn from a restricted part of that population.

1Five of the many surveys of these topics are Dhrymes (1984), Maddala (1977b, 1983, 1984), and Amemiya (1984). The last is part of a symposium on censored and truncated regression models. A survey that is oriented toward applications and techniques is Long (1997). Some recent results on non- and semiparametric estimation appear in Lee (1996).

2For example, Lancaster (1990) and Kiefer (1985).

CHAPTER 22 ♦ Limited Dependent Variable and Duration Models 757 22.2.1 TRUNCATED DISTRIBUTIONS

A truncated distribution is the part of an untruncated distribution that is above or below some specified value. For instance, in Example 22.2, we are given a characteristic of the distribution of incomes above $100,000. This subset is a part of the full distribution of incomes which range from zero to (essentially) infinity.

THEOREM 22.1 Density of a Truncated Random Variable

If a continuous random variable x has pdf f (x) and a is a constant, then f (x | x > a) = —-.3

The proof follows from the definition of conditional probability and amounts merely to scaling the density so that it integrates to one over the range above a. Note that the truncated distribution is a conditional distribution.

Most recent applications based on continuous random variables use the truncated normal distribution. If x has a normal distribution with mean / and standard deviation a, then

Prob (x > a) = 1 - $( = 1 -&(a), where a = (a - j)/a and $(.) is the standard normal cdf. The density of the truncated normal distribution is then

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