4. Likelihood construction for field reliability data

Field reliability data rarely arrive as clean, complete failure times. A repairable pump may still be running when the observation window closes; a batch of bearings may already have failed before the first inspection; a sensor may only reveal that a failure occurred somewhere between two scheduled maintenance visits. Each of these situations carries information about the underlying lifetime distribution, but only if the likelihood function is constructed to reflect what was actually observed. This chapter develops the likelihood contributions for right-censored, left-censored, and interval-censored observations, and shows how to assemble them into a single, correct log-likelihood for maximum likelihood estimation (MLE) or Bayesian inference.

4.1 The Likelihood Principle Under Censoring

For an independent sample of lifetime observations, the likelihood is the joint density of the observed data evaluated at the observed values. When an observation is exact, its contribution is the probability density function (PDF) f(t; θ). When an observation is censored, the exact failure time is unknown, so the contribution becomes the probability of the event that was actually recorded — a survival probability, a cumulative probability, or a difference of cumulative probabilities. The general rule is:

Let T denote the true (unobserved) failure time and θ the parameter vector of the lifetime model. The four observation types and their contributions are:

Observation type	Recorded information	Likelihood contribution
Exact failure	T = t	f(t; θ)
Right-censored	T > t	S(t; θ)
Left-censored	T ≤ t	F(t; θ)
Interval-censored	tL < T ≤ tR	F(tR; θ) − F(tL; θ)

The total log-likelihood for n independent observations is the sum of the log of each contribution. Because the contributions are probabilities (not densities) for censored observations, they remain valid under any monotone transformation of the time axis, which is a key advantage when working with log-time or accelerated life models.

4.2 Right-Censored Observations

Right censoring is the most common pattern in field data: a unit has survived up to a known time t without failing. The contribution is the survival function S(t; θ) = 1 − F(t; θ). For a Weibull model with shape β and scale η,

S(t; β, η) = exp[ −(t / η)^β ]

log L_right = Σ_i log S(t_i; β, η)
           = −Σ_i (t_i / η)^β

Note that right-censored observations contribute no β-dependent density term; they only penalize parameter combinations that predict too few survivors. This is why right censoring alone can still identify both parameters, provided there is a mix of exact failures and censored times.

4.3 Left-Censored Observations

Left censoring occurs when a unit is known to have failed before a certain time, but the exact failure time is unknown. This is typical when the first inspection reveals a failure that occurred sometime earlier, or when a detection limit truncates small failure times. The contribution is the CDF:

F(t; β, η) = 1 − exp[ −(t / η)^β ]

log L_left = Σ_j log F(t_j; β, η)
           = Σ_j log[ 1 − exp( −(t_j / η)^β ) ]

Left censoring is often mishandled by simply discarding the observations. Discarding them biases the estimate of η upward because the earliest failures are systematically removed. Including the F(t) term corrects this bias.

4.4 Interval-Censored Observations

Interval censoring is the general case that subsumes both right and left censoring. The failure time is known only to lie in (tL, tR]. The contribution is the probability mass in that interval:

P(t_L < T ≤ t_R; θ) = F(t_R; θ) − F(t_L; θ)

log L_interval = Σ_k log[ F(t_R,k; θ) − F(t_L,k; θ) ]

Right censoring is the special case tR → ∞ (so F(∞) = 1), and left censoring is the special case tL = 0 (so F(0) = 0). This unification is useful when implementing a single likelihood routine that accepts a lower and upper bound for every observation.

4.5 A Worked Field-Data Example

Consider a fleet of 12 identical industrial valves observed over a 24-month window. Maintenance records yield the following (times in months). These values are illustrative and constructed for teaching purposes, not measured field results.

Unit	Recorded event	Type	(tL, tR]
V01	Failed at 6.2	Exact	(6.2, 6.2]
V02	Failed at 11.8	Exact	(11.8, 11.8]
V03	Failed at 17.5	Exact	(17.5, 17.5]
V04	Failed before first inspection at 3.0	Left	(0, 3.0]
V05	Failed between inspections at 9 and 15	Interval	(9.0, 15.0]
V06	Failed between inspections at 15 and 21	Interval	(15.0, 21.0]
V07–V12	Still running at 24	Right	(24.0, ∞)

Assuming a Weibull lifetime model, the log-likelihood is assembled term by term. For the exact failures, use the density f(t) = (β/η)(t/η)β−1 exp[−(t/η)β]. For V04, use F(3.0). For V05 and V06, use F(tR) − F(tL). For V07–V12, use S(24.0) six times.

log L(β, η) =
  [log f(6.2) + log f(11.8) + log f(17.5)]        # exact
+ log F(3.0)                                       # left
+ log[F(15.0) − F(9.0)] + log[F(21.0) − F(15.0)]   # interval
+ 6 · log S(24.0)                                   # right

Maximizing this expression numerically (e.g., with a general-purpose optimizer or a Bayesian sampler) yields the MLE. A useful sanity check: if the interval-censored terms are mistakenly treated as exact failures at their midpoint, the estimate of β is typically biased downward because the spread of the interval is ignored. Comparing the two fits is a quick diagnostic for whether interval handling matters in a given dataset.

4.6 Numerical Stability and Practical Pitfalls

• Log of differences: For interval-censored terms, compute log(F(tR) − F(tL)) using a numerically stable form. When the interval is narrow, F(tR) − F(tL) can underflow; use log1p or a log-space subtraction such as log F(tR) + log1p(−F(tL)/F(tR)).
• Zero-width intervals: Exact failures should be handled with the density, not with a zero-width interval, to avoid division by zero.
• Infinite upper bounds: Use F(∞) = 1 and S(∞) = 0; never evaluate the CDF at a literal infinity without a guard.
• Mixed censoring: Always verify that the sum of contributions uses the correct function for each observation type; a single mislabeled row can shift the MLE substantially.
• Identifiability: If all observations are right-censored, the likelihood is monotone in η and the MLE may not exist; at least some exact or interval information is needed.

4.7 Visual Summary of Likelihood Contributions

The following diagram maps each observation type to its position on the time axis and to the corresponding likelihood contribution. The shaded regions indicate the probability mass that the contribution captures.

4.8 From Likelihood to Inference

Once the log-likelihood is correctly assembled, the path to inference is standard. For MLE, maximize log L(θ) over the parameter space using a numerical optimizer; the observed Fisher information (negative Hessian) gives asymptotic standard errors. For Bayesian inference, combine the same likelihood with a prior p(θ) to obtain the posterior p(θ | data) ∝ L(θ) p(θ), and sample with MCMC or variational methods. In both cases, the censoring-aware likelihood is the single most important ingredient: an incorrect contribution silently biases every downstream reliability metric, including mean time to failure, B₁₀ life, and warranty predictions.

The practical takeaway is to treat every field observation as an interval (tL, tR] and to write one likelihood routine that dispatches on the interval type. This discipline eliminates the most common source of error in field reliability analysis and makes the model directly comparable across datasets with different inspection schemes.