Censored Distribution#

This is not a distribution per se, but a modifier of univariate distributions.

A censored distribution arises when the observed data is limited to a certain range, and values outside this range are not recorded. For instance, in a study aiming to measure the impact of a drug on mortality rates it may be known that an individual’s age at death is at least 75 years (but may be more). Such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75. Censoring can also happen when a value falls outside the range of a measuring instrument. For example, if a bathroom scale only measures up to 140 kg, and a 160-kg person is weighed, the observer would only know that the individual’s weight is at least 140 kg.

Key properties and parameters#

Parameters:

dist : (PreliZ distribution) Univariate distribution to be censored.
lower : (float, int, or np.inf) Lower (left) censoring point, np.inf indicates no lower censoring.
upper : (float, int, or np.inf) Upper (right) censoring point, np.inf indicates no upper censoring.

Probability Density Function (PDF)#

Given a base distribution with cumulative distribution function (CDF) and probability density mass/function (PDF). The pdf of a Censored distribution is:

\[\begin{split} \begin{cases} 0 & \text{for } x < \text{lower}, \\ \text{CDF}(\text{lower}) & \text{for } x = \text{lower}, \\ \text{PDF}(x) & \text{for } \text{lower} < x < \text{upper}, \\ 1-\text{CDF}(\text{upper}) & \text {for } x = \text{upper}, \\ 0 & \text{for } x > \text{upper}, \end{cases} \end{split}\]

where lower and upper are the lower and upper bounds of the censored distribution, respectively.

Cumulative Distribution Function (CDF)#

The given expression can be written mathematically as:

\[\begin{split} \begin{cases} 0 & \text{for } x < \text{lower}, \\ \text{CDF}(x) & \text{for } \text{lower} < x < \text{upper}, \\ 1 & \text{for } x > \text{upper}, \end{cases} \end{split}\]