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Estimating the discrete time, hazard rate subject to left-truncation

estimate_hazard.Rd

This function calculates the estimated hazard rate, \(\hat{\lambda}\), for the entire vector of observable support. It is the discrete time, hazard rate estimator adjusted for random left-truncation, right-censoring, and competing risks (Lautier et al., 2026a).

Usage

estimate_hazard(
  lifetime,
  trunc_time = NULL,
  censoring_indicator = NULL,
  event_type = NULL,
  support_lifetime_rv = NULL,
  carry_hazard = FALSE,
  ci_level = 0.95
)

Arguments

lifetime

A numeric vector representing the minimum of the observed termination time or right-censoring time.

trunc_time

A numeric vector representing the observed left-truncation time. If NULL (the default), this becomes a zero vector (i.e., there is no left-truncation).

censoring_indicator

An indicator to identify if an observation is right-censored (1) or complete (0). Defaults to a zero vector if NULL (i.e., there are no right-censored observations). An observation is only treated as an event (of a given type) if its right-censoring indicator is 0.

event_type

An optional vector of event identifiers (e.g., "Default" and "Prepayment") for the case of competing risks. When this field has more than one unique value, cause-specific hazards are estimated for each unique event type.

support_lifetime_rv

A vector of time points at which to evaluate the hazard. If NULL (the default), it is calculated by calc_osup() as a sequence from \(\Delta + 1\) to \(\xi\) (that is, max(lifetime)).

carry_hazard

A logical indicator to determine whether or not zero hazard rate estimates should be replaced by the last non-zero hazard rate estimate (i.e., “carried-forward”). Defaults to FALSE.

ci_level

A number between 0 and 1 indicating the desired confidence interval width. Defaults to 0.95.

Value

An object of class alife (or alife_multi, when event_type has more than one unique value). These objects are each a data.frame with the time point, hazard estimate, standard error, and asymptotic confidence interval for all time points in the support. We emphasize that the column called se_log_hazard represents the standard error of the hazard rate on the log scale.

Details

Returns the vector of hazard rate estimates from Lautier et al. (2026a, Theorem 2.1) with asymptotic confidence intervals from Lautier et al. (2026a, Lemma 1). When there are no competing risks, this function returns the equivalent estimates and asymptotic confidence intervals from Lautier et al. (2023). When there are no competing risks and no right-censoring, the estimates and asymptotic confidence intervals follow from Lautier et al. (2026b).

References

Lautier, J. P., Pozdnyakov, V., Yan, J. (2023). Pricing time-to-event contingent cash-flows: A discrete-time survival analysis approach. Insurance: Mathematics and Economics, 110, 53-71. https://doi.org/10.1016/j.insmatheco.2023.02.003

Lautier, J. P., Pozdnyakov, V., Yan, J. (2026a). On the convergence of credit risk in current consumer automobile loans. Journal of the Royal Statistical Society Series A: Statistics in Society, 189 (1), 196-221. https://doi.org/10.1093/jrsssa/qnae137

Lautier, J. P., Pozdnyakov, V., Yan, J. (2026b). Estimating a discrete distribution subject to random left-truncation with an application to structured finance. Econometrics and Statistics, 37, 174-198. https://doi.org/10.1016/j.ecosta.2023.05.005