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eduKateSG · Why Mathematics? · Rényi-Type Tests for Crossing Reliability Curves: Cumulative Scores, Maximum Excursions, Censoring and Interpretation
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes practical when early and late reliability differences point in opposite directions. A final summed log-rank score can cancel to nearly zero even though the curves separated substantially along the way. Rényi-type procedures follow the cumulative weighted observed-minus-expected score over event time and use its largest absolute excursion, standardised by its total variance, as the test statistic.
This guide develops the score process, supremum, Brownian-motion calibration, crossing curves, censoring, truncation ranges and interpretation. A large maximum identifies evidence of some departure from equal curves, not a durable advantage for one design. The time of the maximum is descriptive after selection and must be paired with curve plots and effect estimates.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
Positive early score contributions can be offset by negative late contributions in a final sum. Design A may fail more during burn-in but less during wear-out. This is the central mathematical mechanism in why cancellation hides crossings. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A near-zero terminal score does not mean curves stayed close. Plot the cumulative score path. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind event-time score increments. At each pooled failure time, compute observed minus expected failures for one group, optionally multiplied by a planned weight. In the illustrative case, One extra-than-expected A failure moves the path upward under the chosen sign. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Group coding controls direction. State the sign convention.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat cumulative score process as a model-checking problem rather than a vocabulary exercise. Z(t) sums score increments only up to time t, creating a path rather than one terminal number. The path can rise early and fall later. Recompute the example from the raw values and label every unit. Successive values are strongly correlated. Do not treat time points as independent tests.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of supremum statistic becomes useful only when its mechanism is explicit. A Rényi-type statistic uses sup_t |Z(t)| divided by a variance scale, capturing the largest excursion. Consider the worked illustration: If the path reaches 3.2 before ending at .2, the maximum records 3.2. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The selected maximum time is data-dependent. Use correct null calibration.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.2307/2556114. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use one-sided and two-sided versions to practise sensitivity analysis. sup Z(t) targets one directional excursion, while sup |Z(t)| protects both directions. A two-sided design detects either group becoming worse at some time. Before accepting the result, identify the quantity that would change the conclusion most. Choosing sidedness after inspection is invalid. Pre-specify the scientific direction.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for variance standardisation. The cumulative score variance aggregates weighted hypergeometric event-time variances. The initial example is: Divide the maximum by the square root of planned terminal variance. Then create a nearby case where the usual shortcut becomes unreliable. A wrong variance range changes scale. Declare the terminal analysis time.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect brownian-motion approximation to a real decision while protecting the boundary between calculation and judgement. Under the null, the standardised score process approaches a Gaussian process related to Brownian motion after information-time transformation. Critical values concern a path maximum, not a single normal variable. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The stochastic process is an inferential tool, not a physical failure model. Separate process limit from mechanism.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Cumulative scores 0,.8,1.4,.5,-.3 have absolute maximum 1.4 even though the final value is -.3. The test preserves the earlier departure. This is the central mathematical mechanism in a simple path example. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Five points are too few for asymptotic confidence. Sketch the path before calculating. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind crossing reliability curves. A curve crossing often generates a score path that changes direction as risk-set allocations reverse. In the illustrative case, Early weakness and late strength leave a characteristic excursion. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Not every score reversal implies literal curve crossing. Inspect survival and hazard plots.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat weighted rényi family as a model-checking problem rather than a vocabulary exercise. Planned event weights can emphasise early, middle or late information before taking the supremum. Fleming–Harrington weights can define different score paths. Recompute the example from the raw values and label every unit. Trying many weights adds multiplicity. Use one plan or calibrated combination.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of ordinary log-rank comparison becomes useful only when its mechanism is explicit. Log-rank standardises the terminal score, while Rényi-type tests examine the maximum along the path. Consider the worked illustration: They agree more closely under sustained same-direction effects. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Rényi need not be more powerful for proportional hazards. Compare alternatives by simulation.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use kolmogorov–smirnov connection to practise sensitivity analysis. Both ideas use a maximum discrepancy, but censored survival data require risk-set score processes and censoring-aware limits. Directly applying an uncensored KS table is wrong. Before accepting the result, identify the quantity that would change the conclusion most. The null process depends on the survival design. Use the censored-data procedure.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.1093/biomet/68.1.99. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for right censoring. Censored systems contribute to risk sets until their last observed time under independent censoring. The initial example is: Late withdrawals reduce the information range. Then create a nearby case where the usual shortcut becomes unreliable. Differential informative censoring can mimic crossings. Compare censoring by group.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect choosing the time range to a real decision while protecting the boundary between calculation and judgement. Restrict the supremum to a pre-specified interval with adequate risk sets in both groups. Analyse 0–1000 hours if both designs retain support. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Extending into a two-unit tail can create unstable maxima. Set a minimum risk threshold.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Ties require joint expected counts and variance at each shared time. Three failures recorded at day 10 form one increment block. This is the central mathematical mechanism in tied failures. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Arbitrary tie ordering can change the path. Use a declared tie formula. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind p-value computation. Critical values may use asymptotic process distributions, permutation under exchangeability or resampling. In the illustrative case, Bootstrap whole systems to preserve event histories. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A naive normal p-value ignores maximisation. Record the calibration method.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat time of maximum excursion as a model-checking problem rather than a vocabulary exercise. The argmax can guide descriptive investigation of when separation was strongest. The largest excursion may occur near 700 hours. Recompute the example from the raw values and label every unit. Post-selection uncertainty makes it unsuitable as a precise change-point estimate. Label it exploratory.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of multiple groups becomes useful only when its mechanism is explicit. Extensions use vector score processes and quadratic or maximum norms. Consider the worked illustration: Three materials create multidimensional departures. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Interpretation becomes harder and pairwise follow-ups need control. Plan contrasts in advance.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use effect sizes after a path test to practise sensitivity analysis. Report survival differences, restricted means or time-specific hazards to quantify what the test detected. Show early loss and late gain separately. Before accepting the result, identify the quantity that would change the conclusion most. The supremum statistic has no direct hour or probability unit. Keep evidence and magnitude distinct.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for simulation and power. Simulate proportional, delayed, crossing and null curves with realistic censoring. The initial example is: Rényi-type tests may gain power for crossing alternatives. Then create a nearby case where the usual shortcut becomes unreliable. One favourable scenario does not establish general superiority. Publish the full scenario grid.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect decision interpretation to a real decision while protecting the boundary between calculation and judgement. A detected crossing can warn that one design is not uniformly better across mission ages. Choose design by the intended operating window and consequences. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A global departure does not prescribe switching time. Combine with mission-specific effects.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Create ten score increments that cancel at the end, plot cumulative sums and identify the absolute maximum. Reorder two tied increments and explain why ties need joint handling. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Toy paths are illustrative. Check the risk-set table. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind parent and educator guidance. Rényi tests connect cumulative sums, maxima, graphs, variance and cancellation. In the illustrative case, Ask why a journey can wander far yet finish near the start. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Use a number-line walk first. Then connect steps to failures.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat common misconceptions as a model-checking problem rather than a vocabulary exercise. The supremum is not a largest survival-probability gap, its time is not a confirmed change point and a significant result does not show one group is always better. Path calibration differs from a z test. Recompute the example from the raw values and label every unit. Censoring can affect late excursions. Report curves and support.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub, the eduKate Mathematics Learning System, the approach to learning mathematics, the Singapore Mathematics curriculum overview, the cross-subject learning approach and the previous reliability article. Each destination has its own role: subject navigation, learning method, school-to-advanced progression, or continuity with the latest reliability mechanism.
The mathematics of research and next reading becomes useful only when its mechanism is explicit. Fleming and colleagues developed modified Kolmogorov–Smirnov procedures for censored data, while Nair studied related maximum weighted discrepancies and bands. Consider the worked illustration: Reproduce a cancelling score path before software. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The procedure detects departure, not engineering cause. Read process calibration beside examples.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why cancellation hides crossings
Setup: Use this chapter’s illustrative case: Design A may fail more during burn-in but less during wear-out.
Task: Plot the cumulative score path.
Checking target: A near-zero terminal score does not mean curves stayed close.
Practice case 2: Event-time score increments
Setup: Use this chapter’s illustrative case: One extra-than-expected A failure moves the path upward under the chosen sign.
Task: State the sign convention.
Checking target: Group coding controls direction.
Practice case 3: Cumulative score process
Setup: Use this chapter’s illustrative case: The path can rise early and fall later.
Task: Do not treat time points as independent tests.
Checking target: Successive values are strongly correlated.
Practice case 4: Supremum statistic
Setup: Use this chapter’s illustrative case: If the path reaches 3.2 before ending at .2, the maximum records 3.2.
Task: Use correct null calibration.
Checking target: The selected maximum time is data-dependent.
Practice case 5: One-sided and two-sided versions
Setup: Use this chapter’s illustrative case: A two-sided design detects either group becoming worse at some time.
Task: Pre-specify the scientific direction.
Checking target: Choosing sidedness after inspection is invalid.
Practice case 6: Variance standardisation
Setup: Use this chapter’s illustrative case: Divide the maximum by the square root of planned terminal variance.
Task: Declare the terminal analysis time.
Checking target: A wrong variance range changes scale.
Practice case 7: Brownian-motion approximation
Setup: Use this chapter’s illustrative case: Critical values concern a path maximum, not a single normal variable.
Task: Separate process limit from mechanism.
Checking target: The stochastic process is an inferential tool, not a physical failure model.
Practice case 8: A simple path example
Setup: Use this chapter’s illustrative case: The test preserves the earlier departure.
Task: Sketch the path before calculating.
Checking target: Five points are too few for asymptotic confidence.
Practice case 9: Crossing reliability curves
Setup: Use this chapter’s illustrative case: Early weakness and late strength leave a characteristic excursion.
Task: Inspect survival and hazard plots.
Checking target: Not every score reversal implies literal curve crossing.
Practice case 10: Weighted Rényi family
Setup: Use this chapter’s illustrative case: Fleming–Harrington weights can define different score paths.
Task: Use one plan or calibrated combination.
Checking target: Trying many weights adds multiplicity.