ⓘ Truncated distribution. In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability ..

Truncated distribution

ⓘ Truncated distribution

In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the schools cutoff dates if only a direct approach to the school were used to obtain information.

Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as censoring, as opposed to the truncation here.


1. Definition

The following discussion is in terms of a random variable having a continuous distribution although the same ideas apply to discrete distributions. Similarly, the discussion assumes that truncation is to a semi-open interval y ∈ {\displaystyle fx}, with cumulative distribution function F x {\displaystyle Fx} both of which have infinite support. Suppose we wish to know the probability density of the random variable after restricting the support to be between two constants so that the support, y = a, b" {\displaystyle y=a,b"}. That is to say, suppose we wish to know how X {\displaystyle X} is distributed given a < X ≤ b {\displaystyle a y={\frac {\int _{y}^{\infty }xgxdx}{1-Fy}}}

where again g x {\displaystyle gx} is g x = f x {\displaystyle gx=fx} for all x > y {\displaystyle x> y} and g x = 0 {\displaystyle gx=0} everywhere else.

Letting a {\displaystyle a} and b {\displaystyle b} be the lower and upper limits respectively of support for the original density function f {\displaystyle f} which we assume is continuous, properties of E u X | X > y) {\displaystyle EuX|X> y)}, where u {\displaystyle u} is some continuous function with a continuous derivative, include:

i lim y → a E u X | X > y) = E u X) {\displaystyle \lim _{y\to a}EuX|X> y)=EuX)}

ii lim y → b E u X | X > y) = u b {\displaystyle \lim _{y\to b}EuX|X> y)=ub}

iii ∂ ∂ y ={\frac {1}{2}}ub}

Provided that the limits exist, that is: lim y → c u ′ y = u ′ c {\displaystyle \lim _{y\to c}uy=uc}, lim y → c u y = u c {\displaystyle \lim _{y\to c}uy=uc} and lim y → c f y = f c {\displaystyle \lim _{y\to c}fy=fc} where c {\displaystyle c} represents either a {\displaystyle a} or b {\displaystyle b}.


2. Examples

The truncated normal distribution is an important example.

The Tobit model employs truncated distributions. Other examples include truncated binomial at x=0 and truncated poisson at x=0.


3. Random truncation

Suppose we have the following set up: a truncation value, t {\displaystyle t}, is selected at random from a density, g t {\displaystyle gt}, but this value is not observed. Then a value, x {\displaystyle x}, is selected at random from the truncated distribution, f x | t = T r x {\displaystyle fx|t=Trx}. Suppose we observe x {\displaystyle x} and wish to update our belief about the density of t {\displaystyle t} given the observation.

First, by definition:

f x = ∫ x ∞ f x | t g t d t {\displaystyle fx=\int _{x}^{\infty }fx|tgtdt}, and F a = ∫ x a. Let g t and f x | t be the densities that describe t and x respectively. Suppose we observe a value of x and wish to know the distribution of t given that value of x. g t | x = f x | t g t f x = 1 t ln ⁡ T − ln ⁡ x) for all t > x. {\displaystyle gt|x={\frac {fx|tgt}{fx}}={\frac {1}{t\lnT-\lnx)}}\quad {\text{for all }}t> x.}
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