For population i , let the lifetime observations follow a three-parameter Weibull distribution with threshold parameter γ i , scale parameter η i , and shape parameter β i . The probability density function is

where η i > 0 and β i > 0 . The corresponding distribution function is

and the survival function is

The threshold parameter γ i determines the lower endpoint of the lifetime distribution. Under the model, failures cannot occur before γ i . The scale parameter η i controls the scale of the lifetimes above the threshold, while the shape parameter β i determines the shape of the distribution and the behavior of its hazard function [10][11].

The hazard function is

Thus, the hazard rate is decreasing when β i < 1 , constant when β i = 1 , and increasing when β i > 1 . This flexibility makes the Weibull distribution useful in fatigue-life and other reliability applications.

The semi-simulated Alloy reliability study includes right-censored observations. Let t i j denote the observed time for unit j in population i , and let

The likelihood contribution for observation j in population i is

Therefore, the likelihood for population i is

The corresponding log-likelihood is

Substituting the Weibull density and survival functions gives

The threshold parameter must satisfy

where

Transformations of constrained parameters are commonly used to improve computational behavior and simplify posterior simulation [34][35]. To enforce the parameter constraints during posterior simulation, the following transformation is used:

where

and ε is a small positive constant. Equivalently,

Because exp ( θ i 3 ) > 0 , this parameterization guarantees that

For every proposed value of θ i 3 . The logarithmic transformations similarly guarantee that β i > 0 and η i > 0 .

The prior distributions assigned to the transformed model parameters and the resulting posterior distribution are presented in the following section.

The specification of priors for the three-parameter Weibull distribution requires consideration of the positivity constraints on the shape and scale parameters, as well as the threshold constraint on the location parameter. Independent prior distributions are assigned to the Weibull parameters for each population. Since β i > 0 and η i > 0 , gamma prior distributions are chosen for the shape and scale parameters. The threshold parameter is handled by introducing the positive distance parameter,

where ε = 10 − 6 is a fixed numerical offset used to keep the threshold parameter away from the boundary t i ( 1 ) . This offset helps prevent numerical instability when evaluating terms involving t i j − γ i , is negligible relative to the scale of the observed fatigue lifetimes, and is not an estimated model parameter. The prior is assigned directly to the transformed parameter

which is defined on the real line. Prior specification for Weibull reliability models and the use of formal prior distributions in Bayesian analysis are discussed in [36]-[39].

In the semi-simulated Alloy fatigue study, the prior distributions are specified as follows:

and

In Equations (3.1) and (3.2), the second parameter represents the scale parameter of the gamma distribution.

The prior for η i is specified on a scale that aligns with the observed fatigue lifetimes, while allowing for substantial uncertainty. The gamma prior for β i respects the positivity constraint on the Weibull shape parameter while remaining sufficiently flexible for the data to inform posterior inference. The prior assigned to θ i 3 is particularly important, as the threshold parameter is related to the smallest observed time through

Larger values of δ i correspond to smaller values of γ i , whereas smaller values of δ i position the threshold nearer to the smallest observed lifetime. Therefore, the prior on θ i 3 determines the proximity of the threshold to the lower bound of the observed data. The Alloy study investigates the sensitivity of posterior conclusions to the prior standard deviation of θ i 3 .

For population i , let

The original model parameters can be recovered from the transformed parameters using the following relationships:

and

Applying the right-censored likelihood from Section 2.2, the posterior density for population i on the transformed scale is proportional to the product of the likelihood, the prior densities, and the Jacobian of the transformation:

Because

the Jacobian term is

No additional Jacobian term is necessary for θ i 3 because the normal prior in Equation (3.3) is assigned directly to θ i 3 = log ( δ i ) .

Therefore, the posterior density targeted in the analysis is

Equivalently, the log-posterior density is given by

where C is a constant independent of the model parameters. The posterior distribution does not have a closed form and is evaluated using the random-walk Metropolis algorithm described in Section 5.

For population i , let

denote the vector of three-parameter Weibull parameters, where γ i is the threshold parameter, η i is the scale parameter, and β i is the shape parameter.

The best population is defined as the population with the largest threshold parameter. Thus, population i is best if

This definition is motivated by reliability applications, where a larger threshold parameter indicates a greater lower endpoint of the lifetime distribution and, consequently, a later earliest possible failure time.

Accordingly, this criterion is intended for engineering decisions in which protection against early failure, or equivalently a later model-implied lower lifetime bound, is the primary objective. It is not intended as a universal ordering of Weibull populations. When the engineering objective concerns median lifetime, an upper lifetime percentile, expected lifetime, reliability at a specified time, or another performance measure, the corresponding reliability functional should determine the ranking criterion. The percentile comparisons reported in this paper are therefore supplementary analyses that illustrate how the population rankings may change under alternative reliability objectives; they are not part of the formal threshold-based subset-selection rule.

Let i * denote the index of the best population:

Because the posterior distributions considered here are continuous, exact ties among the threshold parameters occur with probability zero.

After observing the right-censored lifetime data, uncertainty about the Weibull parameters is represented by their posterior distribution. The posterior probability that population i is best is defined as

Here, t and d represent the observed times and censoring indicators for all populations. The values p 1 , ⋯ , p k summarize the posterior evidence supporting each population as the best. Because one population is identified as best in each posterior draw,

In practice, these probabilities are estimated from the retained posterior draws. After burn-in and thinning, suppose M draws of the threshold parameters are available:

For draw m , the population having the largest sampled threshold parameter is recorded as the best population. The posterior probability that population i is best is estimated by

Here, I ( ⋅ ) denotes the indicator function. Therefore, p ^ i represents the proportion of retained posterior draws in which population i has the largest threshold parameter.

Let S denote a selected subset of populations. Correct selection occurs when the selected subset contains the best population:

Under the Bayesian model, the posterior probability of correct selection is

Since p i is the posterior probability that population i is best, this probability can be written as

This posterior probability differs from the frequentist probability of correct selection. The frequentist probability is a pre-data measure evaluated over repeated samples with fixed parameter values. In contrast, Equation (4.9) is conditional on the observed Alloy data and the specified prior model.

Let P * denote the desired posterior probability level. To construct the selected subset, the estimated posterior probabilities of being best are ordered from largest to smallest:

where p ^ ( r ) denotes the rth largest estimated posterior probability and i ( r ) denotes the population index associated with that probability.

The Bayesian selected subset is

where m is the smallest integer satisfying

Thus, S B is the smallest subset, determined by the ordering of the estimated posterior probabilities, whose estimated posterior probability of containing the best population meets or exceeds P * .

If a single population has a dominant posterior probability of being best, the selected subset may consist solely of that population. When posterior uncertainty is higher, additional populations are included until the desired posterior probability threshold is achieved. In the Alloy study, the primary analysis sets P * = 0.95 .

Although the formal definition of the best population and the Bayesian selected subset are based on the threshold parameter γ i , the same posterior comparison can be applied to Weibull lifetime percentiles. For a three-parameter Weibull distribution, the pth percentile for population i is

The 10th, 50th, and 90th percentiles are examined:

The quantity t p , i indicates the time by which a proportion p of units from population i is expected to have failed according to the Weibull model.

For each retained posterior draw, the largest value of t p , i among the populations is identified. The posterior probability that population i has the largest pth percentile is estimated by

This calculation is performed separately for p = 0.10 , 0.50 , and 0.90.

These percentile-based comparisons do not define the formal Bayesian selected subset, which remains based on the threshold parameter γ i . Rather, they serve as supplementary reliability summaries to evaluate whether populations with greater posterior support for the largest threshold parameter also tend to exhibit larger lifetime percentiles.

Let

denote the log-posterior density for population i on the transformed scale. At iteration m , a candidate value is generated from

where c is a tuning constant and Σ ^ pilot , i is a covariance matrix estimated from a preliminary random-walk run.

Because the proposal distribution is symmetric, the proposal-density terms cancel in the Metropolis-Hastings ratio. The candidate is accepted with probability

Equivalently, computation is performed on the log scale using

If the proposed value is not accepted, the current value is retained for the next iteration.

For each population, starting values are obtained by fitting shifted Weibull models across a grid of possible values for δ i . These values are then used to initialize

A preliminary random-walk run of 8000 iterations is conducted for each chain. The empirical covariance matrix from this pilot run is used as Σ ^ pilot , i , which accounts for the posterior scales and dependencies among the three transformed parameters. In the main random-walk proposal, the tuning constant is set to c = 1.2 .

Four independent chains are run for each population. Each chain contains 200,000 iterations, of which the first 20,000 are discarded as burn-in. Every third post-burn-in draw is retained, yielding 60,000 retained draws per chain. The retained draws from the four chains are pooled for posterior estimation and subset selection.

Because the five populations are modeled independently and independent priors are assigned across populations, the joint posterior distribution factors into the product of the five population-specific posterior distributions. Joint posterior draws used in the subset-selection calculations are therefore formed by combining the corresponding retained draws from the separately fitted population models.

Convergence and sampling performance for all five populations are assessed using traceplots, autocorrelation functions, acceptance rates, rank-normalized split and folded-split R ^ diagnostics, and bulk and tail effective sample sizes. Traceplots and autocorrelation functions are examined separately within each chain for the transformed parameters

The R ^ , bulk ESS, and tail ESS diagnostics are calculated jointly across the four chains while preserving chain identity. These diagnostics are evaluated for the Weibull parameters γ i , η i , β i , and δ i , and for the selected lifetime percentiles t 0.10 , i , t 0.50 , i , and t 0.90 , i . After convergence has been assessed, the retained draws from the four chains are combined for posterior summaries and subset-selection probabilities. The resulting numerical summaries and representative graphical displays are presented alongside the Alloy study results in Section 6.

Before constructing the five-population study, a three-parameter Weibull distribution was fit to the original Alloy data using the right-censored likelihood. The maximum likelihood estimates were

The estimated shape parameter is greater than one, corresponding to an increasing fitted hazard rate. This behavior is consistent with a fatigue process in which the instantaneous risk of failure increases as the number of cycles accumulates.

Figure 1 provides two complementary graphical assessments of the fitted model. The fitted Weibull survival curve follows the Kaplan-Meier estimate closely over most of the observed range and remains positive near 300 thousand cycles, as expected from the censored specimens. On the Weibull probability scale, the transformed Kaplan-Meier points are approximately linear, with most points lying close to the fitted line or within its approximate pointwise confidence band. These results support the three-parameter Weibull distribution as a reasonable working model for the Alloy fatigue-life data.

(a) Kaplan-Meier estimate and fitted Weibull survival curve

(b) Weibull probability plot with fitted line and approximate confidence band

Figure 1.Graphical assessment of the three-parameter Weibull fit to the original Alloy T7987 fatigue-life data.

A Bayesian three-parameter Weibull model was first fitted to the original Alloy T7987 fatigue-life data. The posterior mean estimates of γ , η , and β from this fit were adopted as the baseline generating values for Population P1. The remainder of the semi-simulated study was constructed relative to this Bayesian baseline. Populations P2-P5 were generated with progressively larger threshold parameters, while their scale and shape parameters were kept close to the posterior estimates obtained from the original Alloy data. Thus, the five populations retain broadly similar fatigue-life behavior, while their intended ordering is determined primarily by the threshold parameter γ .

Table 1 gives the parameter values used in the construction and the resulting censoring summaries. Each population contains 72 observations, with censoring percentages ranging from 6.94% to 12.50%. Since P5 has the largest generating threshold parameter, γ 5 = 100.993 , it is the true best population under the formal threshold-based definition.

Table 1.Construction and censoring summary for the semi-simulated Alloy fatigue-life populations.

The censored three-parameter Weibull model was fit separately to each population using the random-walk Metropolis algorithm described in Section 5. Table 2 reports the posterior means and 95% credible intervals for the principal Weibull parameters. The posterior mean of γ i is largest for P5, followed by P4 and P3. However, the credible intervals for P3-P5 overlap substantially, reflecting the similarity of the populations and the uncertainty introduced by right-censoring. The posterior intervals for η i and β i also overlap, as expected from the construction of the study.

Table 2.Random-walk posterior means and 95% credible intervals for the Weibull parameters.

Posterior means are shown in black; 95% credible intervals are shown in brown.

Figure 2 displays the random-walk posterior densities of the five threshold parameters on a common horizontal scale. The densities shift toward larger values from P1 to the upper-threshold populations, but substantial overlap remains among P3, P4, and P5. This overlap explains why the posterior evidence is distributed across several possible best populations rather than being concentrated entirely on P5.

Figure 2.Random-walk posterior densities of the threshold parameters for the five semi-simulated Alloy populations.

The ranked posterior probabilities of being best are shown in Table 3. P5 has the largest posterior probability, 0.6702, followed by P3 with probability 0.2178 and P4 with probability 0.1062. P2 has only a small probability of being best, and the estimated probability for P1 is zero to four decimal places.

Table 3.Random-walk posterior probabilities of being best, cumulative probabilities, and inclusion in the selected subset for P * = 0.95 .

The Monte Carlo uncertainty in the estimated posterior probabilities was assessed using a binomial approximation. Let X i denote the number of retained draws within a chain for which population i has the largest threshold parameter. Then

where p i is the posterior probability that population i has the largest threshold. Therefore,

With n = 60000 retained draws per chain, the upper bound is approximately 0.00204. Thus, under the binomial approximation, the Monte Carlo uncertainty in the estimated posterior probabilities of being best is small. Serial dependence in the MCMC draws is assessed separately through the chain-aware effective sample size diagnostics.

P5 alone does not attain the target posterior probability P * = 0.95 . Adding P3 raises the cumulative probability to 0.8880, which remains below the target. After P4 is added, the cumulative probability becomes 0.9942. Therefore, the Bayesian selected subset is

The selected subset contains the true best population while acknowledging that the observed censored data do not separate P5 sufficiently from P3 and P4 to justify selecting P5 alone.

The posterior Weibull percentiles provide a complementary comparison because they depend jointly on γ i , η i , and β i . Table 4 reports the posterior mean of each percentile and the posterior probability that the corresponding population has the largest percentile.

Table 4.Random-walk posterior means and probabilities of having the largest Weibull percentile.

For the early-life percentile t 0.10 , P5 has the largest posterior mean and the largest probability of being best, although that probability is only 0.4670. The posterior means of the median t 0.50 are nearly equal for P2-P5, and the probability of having the largest median is distributed across these populations. For t 0.90 , P4 and P2 receive the largest posterior probabilities. Thus, the upper-tail ranking differs from the threshold-based ranking because upper percentiles are particularly sensitive to uncertainty in the scale and shape parameters.

The posterior probabilities that the threshold-based and percentile-based criteria identify the same population are 0.3804, 0.2095, and 0.1281 for t 0.10 , t 0.50 , and t 0.90 , respectively. The decline in agreement illustrates that the formal threshold criterion and the broader percentile comparisons answer related but distinct reliability questions. The selected subset in this paper remains defined by the largest threshold parameter.

The convergence and sampling diagnostics described in Section 5.3 were evaluated for all five populations. Figure 3 shows the random-walk traceplots for P5 as a representative case. The four chains remain stable and explore common posterior regions without visible long-term drift. The log ( δ 5 ) chains exhibit greater variability than the log ( β 5 ) and log ( η 5 ) chains because the threshold-gap parameter is directly related to the lower-endpoint constraint.

Figure 3.Random Walk traceplots for Population P5. The columns correspond to log ( β 5 ) , log ( η 5 ) , and log ( δ 5 ) , and the rows correspond to the four chains. The horizontal axis indexes the 60,000 retained post-burn-in draws from each chain after retaining every third draw.

Table 5 provides a compact summary of the random-walk diagnostics. The rank-normalized split/folded-split R ^ , bulk effective sample size, and tail effective sample size were calculated jointly across the four chains while preserving chain identity (Vehtari et al. [40]). Bulk ESS assesses sampling efficiency in the central region of the posterior distribution, whereas tail ESS assesses sampling efficiency in the posterior tails.

The mean random-walk acceptance rates range from 0.356 to 0.367. All maximum R ^ values are at most 1.0001. Across the model parameters γ , η , β , and δ , the minimum bulk and tail ESS values are 57515.1 and 74180.7, respectively. Across the Weibull percentiles t 0.10 , t 0.50 , and t 0.90 , the corresponding minimum values are 60267.6 and 79309.0. Autocorrelation functions were also examined separately within each chain for every population and decreased with increasing lag; they are not displayed to avoid repetitive graphical material.

Table 5.Summary of chain-aware random-walk MCMC diagnostics.

Taken together, the trace plots, autocorrelation functions, acceptance rates, rank-normalized R ^ values, and bulk and tail effective sample sizes indicate satisfactory convergence and sampling efficiency for the retained random-walk posterior draws.

Sensitivity to the prior on the threshold-gap parameter was examined by varying the standard deviation of the Normal prior for θ i 3 = log ( δ i ) over 1, 2, and 3. Figure 4 shows that the posterior means of γ i remain relatively stable across the three specifications. The credible intervals widen, particularly through their lower limits, as the prior becomes more diffuse. P5 retains the largest posterior mean threshold under all three prior settings, indicating that the main threshold-based conclusion is not driven by the selected prior standard deviation.

A similar sensitivity analysis was done by varying the location parameter of the Normal prior distribution for θ i 3 , log ( δ i ) , over log ( 0.5 ) , log ( 1 ) , and log ( 2 ) , while the prior standard deviation was held fixed at 2. As before the posterior means of γ i remained relatively stable. The corresponding 95% credible intervals widened as the prior median of δ i increased, primarily through decreases in their lower limits. The upper credible limits remained similar, and the intervals overlapped substantially across the prior specifications. Although the posterior probabilities of being best also varied numerically, the population ordering remained P5 > P3 > P4 > P2 > P1, and the selected subset at P * = 0.95 remained {P5, P3, P4}. Thus, the main subset-selection conclusion was not sensitive to the prior location values considered.

Figure 4.Prior sensitivity of the posterior threshold estimates. Each panel shows the posterior mean and 95% credible interval for γ i as the prior standard deviation of log ( δ i ) varies.

Posterior predictive checks were performed for all five fitted populations. Replicated lifetime datasets were generated from the posterior predictive distribution and subjected to the same right-censoring mechanism as the observed datasets. The checks included the minimum, mean, standard deviation, numbers of failures and censored observations, median, and lower- and upper-tail quantiles.

Figure 5 shows representative checks for the minimum, mean, and standard deviation of P5. In each panel, the vertical line marks the observed statistic. The observed values lie within the main body of the corresponding posterior predictive distributions. Across all five populations and all examined statistics, the posterior predictive p-values ranged from 0.267 to 0.797, with no values near zero or one. The fitted censored Weibull models therefore reproduce the principal features of the observed datasets, including their lower endpoints, centers, spreads, tails, and censoring patterns.

Overall, the semi-simulated Alloy study demonstrates the role of Bayesian subset selection in a censored reliability setting. P5 receives the largest posterior probability of being best, but the evidence is not sufficiently concentrated to select P5 alone at P * = 0.95 . The resulting subset {P5, P3, P4} preserves the target posterior probability while representing the uncertainty among the leading populations. The calculations in this section were carried out using the R-code in Appendix A.

Figure 5.Representative posterior predictive checks for Population P5. The histograms show replicated values of the minimum, mean, and standard deviation; the vertical lines mark the corresponding observed values.