64 Censored and truncated distributions
Say experimenters A and B have a behavioral assay in which they time how long it takes for a mouse to descend a pole. Experimenter A records the descent times as follows.
- If the mouse descends the pole in less than \(y_\mathrm{max}\) seconds (typically 60 seconds), he records the descent time.
- If the mouse does not descend the pole in less than \(y_\mathrm{max}\) seconds, he does not record anything. He simply ignores that trial.
Experimenter B takes a different approach. She records the descent times as follows.
- If the mouse descends the pole in less than \(y_\mathrm{max}\) seconds, she records the descent time.
- If the mouse does not descend the pole in less than \(y_\mathrm{max}\) seconds, she records the descent time as \(y_\mathrm{max}\) seconds.
Let \(y\) be the time to descend the pole for patient experimenters. That is, \(y\) can be greater than \(y_\mathrm{max}\) seconds. Let \(f(y)\) be the probability density function of descent time and \(F(y)\) be the corresponding CDF.
a) Write down an expression for \(f_\mathrm{trunc}(y)\), the probability density function for experimenter A’s observations in terms of \(f(y)\) and \(F(y)\). This is called a truncated distribution.
b) Write down an expression for \(f_\mathrm{cens}(y)\), the probability density function for experimenter B’s observations that are not \(y_\mathrm{max}\). Also write an expression for \(\pi(y_\mathrm{max})\), which is the probability that experimenter B records \(y_\mathrm{max}\). This is called a censored distribution.