righttruncated

Righttruncated refers to a property of data or distributions in which observations are only available up to a fixed upper bound. In a right-truncated setting, any true value that exceeds the truncation point is not observed at all. This is a form of sampling restriction and is distinct from right-censoring, where the value is known to exceed a threshold but its exact value is not.

Mathematically, if X has density f(x; θ) and is truncated above at t, the observed data have a density f_tr(x; θ) = f(x; θ) / F(t; θ) for x ≤ t, and 0 otherwise, where F is the corresponding distribution function. For a sample x1, ..., xn with all xi ≤ t, the likelihood under truncation is L(θ) = ∏ f(xi; θ) / F(t; θ). If the truncation point t is known and fixed, F(t; θ) acts as a normalizing constant. If t is unknown or part of the sampling mechanism, the likelihood becomes more complex and may require joint modeling.

Inference under right-truncation typically uses the truncated likelihood, which adjusts parameter estimates and standard errors to account for the missing tail above the truncation point. Ignoring truncation can bias estimates, particularly for tail-related parameters. Bayesian approaches can incorporate truncation directly via the likelihood or priors, and there are software tools for common truncated distributions such as the truncated normal, truncated exponential, and truncated Pareto.

Applications of right-truncation appear in survey sampling with upper bounds, reliability tests with maximum observable lifetimes, and financial or actuarial contexts where values above a cap are not observed. It is important to distinguish right-truncation from censoring and from left or interval truncation, as each affects inference and interpretation differently.