righttruncation

Righttruncation, in statistics and data analysis, refers to a sampling or observation mechanism in which values above a certain threshold are not recorded or included in the sample. In a right-truncated dataset, only observations X that satisfy X ≤ c are observed, while values greater than c are missing by design. This contrasts with right censoring, where the value is known to exceed a threshold but the exact value is not observed.

Mathematically, if the underlying variable X has density f and cumulative distribution F, and observations are retained only if X ≤ c, then the observed data follow the conditional distribution of X given X ≤ c. Consequently, the observed density is f(x)/F(c) for x ≤ c, and there is no data beyond c. For a parametric model with parameter θ, the likelihood of a right-truncated sample {x1,…,xn} is proportional to the product ∏ fθ(xi) / Fθ(c), or equivalently the log-likelihood is ∑ log fθ(xi) − n log Fθ(c). Estimation must account for this truncation; ignoring it can produce biased results.

Inference for right-truncated data can be performed with maximum likelihood under the truncation constraint, or with nonparametric approaches such as Lynden-Bell’s estimator for the distribution function under truncation. Applications appear in astronomy (flux-limited surveys), econometrics, reliability, and other fields where measurement or sampling limits exclude large observations. Important considerations include the degree of truncation, identifiability of parameters when c is unknown, and the potential for selection bias if truncation is not properly modeled.