Linear regression with censored data is a regression model, the dependent variable of which is censored, that is, the variable is transformed if it is less or (greater) than a certain threshold.
Unlike the censored data model, in the truncated data model, observation is entirely excluded if the response is less (or greater) of a certain threshold.
Let the time series be given according to the linear regression model:
The εt errors are a sequence of independent identically distributed random variables. Errors can have the normal and logistic probability distribution or the probability distribution of the first-type extremum (Gumbel distribution). The main characteristics of these distributions are given in the table:
| Characteristic\Distribution | Normal | Logistic | Gumbel |
| Density |  |
 |
 |
| Distribution function |  |
 |
 |
| Mean: | 0 | 0 | -0.57721566… |
| Variance | 1 |  |
 |
The censored regression is defined as follows:
The truncated regression is defined as follows: yt = zt is present only if lt < zt < rt.
On estimating censored or truncated regression, it is necessary to use the observations y1,…,yT and l1,…,lT, r1,…,rt to estimate the parameters β1,…+βn,σ.
See also:
Library of Methods and Models | Linear Regression | ISmCensoredTruncatedRegression