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Parametric survival and reliability — Examples

1 · Prerequisites

2 · Summary

The examples compare constant exponential and increasing Weibull hazards, work the finite-atom survival product, and calculate an exponential likelihood from event and censored exposure times. A second calculation separates the no-failure, positive-exposure boundary supremum from a zero-exposure data set, where the likelihood is constant. These are model calculations, not evidence that a physical failure process follows either parametric family or censoring assumption.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

Exponential and Weibull hazards with right-censored exposure

Example

Compare an exponential lifetime with rate $λ = 0.2$ and a Weibull lifetime with shape $k = 2$ and scale $η = 3$ . For right-censored exponential data, assume the three event times are iid $E x p (0.2)$ and use fixed censoring times $C_{i} = 5$ with observed records $(U_{i}, δ_{i}) = (2, 1), (5, 0), (3, 1)$ . The resulting subject pairs are iid because the censoring time is deterministic. The two exact event times are density observations under the continuous model, not point events of positive probability.

Facts & Assumptions

Given: Countable Choice, the exponential and Weibull families above, iid event times $T_{1}, T_{2}, T_{3} \sim E x p (0.2)$ for the likelihood calculation, an atomic lifetime law $P (T = 1) = 0.4$ , $P (T = 3) = 0.6$ for the jump calculation, and the fixed censoring design and right-censored records in the Example.

[F1]

Exponential and Weibull densities, survivals, and density hazards are as stated. Exponential and Weibull survival specifications define probability laws

[F2]

At an atom $t$ , $S (t) = S (t^{-}) (1 - q (t))$ ; for finite atomic support the survival values are the corresponding finite product. Survival jumps multiply by one minus the atomic hazard

[F3]

Under the iid, independent, parameter-distinct and common-domination hypotheses of Right-censored exponential likelihood and its boundary maximum, fix a censor parameter $ψ$ . The likelihood is $λ^{d} e^{- λ E} C_{ψ}$ , where $C_{ψ} = \prod_{i : δ_{i} = 1} {\overset{ˉ}{G}}_{ψ} (u_{i}) \prod_{i : δ_{i} = 0} g_{ψ} (u_{i})$ . If $C_{ψ} > 0$ , it is proportional in $λ$ to $λ^{d} e^{- λ E}$ , with unique maximum $d / E$ when $d > 0, E > 0$ . If $C_{ψ} = 0$ , it is zero for every $λ > 0$ .

[F4]

A likelihood is an explicitly chosen density version and is defined only up to reference-measure null sets; it is not the point probability of an exact continuous observation. Likelihood densities

[A1]

The censoring law is the point mass $G = δ_{5}$ , dominated by $ν = δ_{5}$ with density $g (5) = 1$ , and it has no event-rate parameter. Fixed censoring is independent of the event time. Event-time observation, censoring, and truncation schemes, The measure with density $f$ relative to $μ$

[F5]

A single-lifetime survival curve describes a nonrepairable unit; repeated failures of a repaired system require a recurrent-event model. Parametric survival and reliability models

[A2]

Countable Choice is used by the family, jump and likelihood results [F1]–[F3]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Verification

technique · direct

1.1A2F1givenalgebra

For $λ = 0.2$ , [F1] gives constant hazard $0.2$ ; its integral is $Λ (t) = 0.2 t$ , and the family survival is $S (3) = e^{- 0.6}$ . For $k = 2, η = 3$ , the density hazard is $2 t / 9$ , which is increasing on $t > 0$ , and $S (3) = e^{- 1}$ .

1.2A2F2givenalgebra

In the two-point lifetime law, $q (1) = 0.4 / 1 = 0.4$ and $q (3) = 0.6 / 0.6 = 1$ . Thus the jump-product formula gives $S (1) = 1 (1 - 0.4) = 0.6$ and $S (3) = 0.6 (1 - 1) = 0$ , matching the probabilities $P (T > 1)$ and $P (T > 3)$ .

1.3A1A2F3givenalgebra

For the fixed censoring design, $\overset{ˉ}{G} (2) = \overset{ˉ}{G} (3) = 1$ and $g (5) = 1$ , so the full censor-only factor is $C = \overset{ˉ}{G} (2) g (5) \overset{ˉ}{G} (3) = 1 > 0$ . The finite Borel dominator $ν = δ_{5}$ and its specified density satisfy [F3]'s domination hypotheses, while the iid event times and fixed censoring give iid independent subject pairs. Therefore $d = 2$ , $E = 2 + 5 + 3 = 10$ , and [F3] gives $L (λ) = λ^{2} e^{- 10 λ}$ with unique maximum $\hat{λ} = 2 / 10 = 0.2$ .

1.4F5given

The single-lifetime hazards above do not summarize repeated failures after repair.

2.1F1F4step 1.3

The event observations at $2$ and $3$ are evaluated through their Lebesgue density values. Accordingly, [F4] makes the likelihood a density evaluation, not the point probability of observing either exact time.

3.1step 1.1step 1.2step 1.3step 2.1step 1.4∎

Earlier steps 1.1, 1.2, 1.3, 1.4, and 2.1 verify the continuous hazard, atomic jump, censored exposure calculation, recurrent-event qualification, and density interpretation under their stated models.

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

No observed failures with positive versus zero exposure

Example

Assume Countable Choice.

Compare two right-censored exponential samples with no observed failures. In each scenario, assume the two event times are iid $E x p (λ)$ . In the first, two subjects are followed until fixed censoring time $5$ and both are censored there. In the second, both subjects are censored immediately at time $0$ . The deterministic censoring times make the subject pairs iid in each sample. Here $λ > 0$ is the common event-rate parameter in both samples.

Facts & Assumptions

Given: Right-censored records use $U_{i} = \min (T_{i}, C_{i})$ and $δ_{i} = 1 {T_{i} \leq C_{i}}$ ; in each scenario the two event times are iid $E x p (λ)$ and the censoring time is fixed and independent of the event times.

[F1]

For $d = \sum_{i} δ_{i}$ and $E = \sum_{i} U_{i}$ , the event likelihood factor is $λ^{d} e^{- λ E}$ . If $d = 0, E > 0$ , its supremum occurs as $λ ↓ 0$ and is not attained for $λ > 0$ ; if $d = 0, E = 0$ , it is constant. The boundary $λ = 0$ is not a proper exponential law on finite event times. Right-censored exponential likelihood and its boundary maximum

[F2]

An exponential event time has $P (T > 0) = 1$ . Exponential and Weibull survival specifications define probability laws

[A1]

The two censoring designs form the family $G_{c} = δ_{c}$ for $c \in {0, 5}$ , dominated by the fixed finite measure $ν = δ_{0} + δ_{5}$ with specified density $g_{c} (u) = 1_{{c}} (u)$ ; in particular $g_{c} (c) = 1$ . In scenario one $C_{i} = 5$ , and in scenario two $C_{i} = 0$ . Event-time observation, censoring, and truncation schemes, The measure with density $f$ relative to $μ$

[A2]

Countable Choice is inherited from the exponential-law supplier [F2] and the likelihood supplier [F1]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Verification

technique · direct

1.1A1A2F1givenalgebra

In scenario one the records are $(U_{i}, δ_{i}) = (5, 0)$ for both subjects, so $d = 0$ , $E = 10$ , and $C_{5} = g_{5} (5)^{2} = 1$ . By [F1], the event likelihood factor is $e^{- 10 λ}$ , whose supremum $1$ is approached only as $λ ↓ 0$ .

1.2A1A2F1F2givenalgebra

In scenario two, [F2] gives $T_{i} > 0$ almost surely, while $C_{i} = 0$ . Thus $U_{i} = 0$ and $δ_{i} = 0$ almost surely for both subjects, so $d = 0$ , $E = 0$ , and $C_{0} = g_{0} (0)^{2} = 1$ . By [F1], the event likelihood factor is $1$ for every $λ > 0$ .

1.3A2F1given

The value $λ = 0$ is only a formal likelihood boundary for the finite-time exponential model; including it as an actual law requires an extended event-time model that admits $T = \infty$ .

2.1F1step 1.1step 1.2algebra

These samples have the same failure count but different likelihood maximizers: positive exposure gives only a boundary supremum, whereas zero exposure gives every positive rate as a maximizer. Thus the no-failure boundary claim requires $E > 0$ , in agreement with the full case distinction in [F1].

3.1step 1.1step 1.2step 1.3∎

Earlier steps 1.1, 1.2, and 1.3 establish the two boundary cases and the required positive exposure condition.

Parametric survival and reliability — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Exponential and Weibull hazards with right-censored exposure

Example

Facts & Assumptions

Verification

No observed failures with positive versus zero exposure

Example

Facts & Assumptions

Verification

Sources