Lifetime Distributions in Statistics

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What are lifetime distributions?

Lifetime distribution models describe component or unit lifetimes. These models encompass all possible lifetimes for units manufactured with a specific design, materials, and manufacturing process [2]. Lifetime distributions are extensively used in reliability analysis, which deals with failure times such as lifetimes or time-to-event data. An “event” here can mean death, disease, component failure or some other measurable event.

Many lifetime distributions are linked to the measurement of extreme values. For example, in series connection, the system halts when the first component breaks. Similarly, in parallel connection, the system ceases to function when the last component breaks [3].

List of basic lifetime distributions

Seven lifetime distributions are commonly used for non-repairable populations such as fuses or lightbulbs, where individual components that fail are removed permanently from the population. Over time, the system can be fixed by replacing failed units from comparable or different populations. However, the original population gradually diminishes until all units eventually fail [2].

1. Birnbaum-Saunders distribution: originally proposed as a model for lifetimes of materials with cyclic patterns of strain and stress.
2. Exponential distribution: often used to model the longevity of electrical or mechanical devices.
3. Extreme value distribution: used to model lifetimes of events with extreme stresses, such as earthquakes and hurricanes.
4. Gamma distribution: models the lifetime of items that are subject to a “wear-out” process.
5. Proportional hazards model (PHM): while not a true lifetime distribution, the PHM is used to model the relationship between covariates and the hazard rate. It can also model lifetime distributions such as the exponential distribution, the Weibull distribution, and the lognormal distribution.
6. Lognormal distribution: often used to model the lifetime of products, such as airplanes and appliances.
7. Weibull distribution: can model the lifetime of mechanical components, such as bearings or gears.

The statistical literature contains other, very sophisticated distributions to analyze lifetime data. However, these distributions multiple parameters cause difficulties with estimations, due to their complexity [4].

Representation of lifetimes

There are five main ways to represent the distribution of a continuous nonnegative random variable T in reliability. They have been widely used since the 1950s [5]:

1. Probability density function (PDF): gives the probability that an item’s lifetime will be a certain value.
2. Survival function: gives the probability an item will survive beyond a certain time.
3. Hazard rate: gives the instantaneous rate of failure of an item: can be interpreted as the probability an item that has survived up to a certain point in time will fail in the next instant.
4. Cumulative hazard function: gives the total amount of failure that has occurred up to a certain point in time.
5. Mean residual life function: gives the expected remaining lifetime of an item at a certain time.

These representations are mathematically equivalent: which one you use depends on whether the representation has a tractable (i.e., easily solvable) form or if a plot of the representation gives you valuable information compared to an equation [5].

The cumulative distribution function (CDF) is another valuable tool for assessing failure probabilities because it gives us the probability a randomly selected unit will fail by time t. Other less common ways to represent distributions in survival analysis are moment generating functions (MGF), characteristic functions, Mellin transformation, density quantile function and the total time on test transform.

References

[1] EvgSkv, CC.0, via Wikimedia Commons

[2] NIST. 8.1.2.1. Repairable systems, non-repairable populations and lifetime distribution models. Retrieved September 4, 2023 from: https://www.itl.nist.gov/div898/handbook/apr/section1/apr121.htm

[3] Wolfram. Distributions in Reliability Analysis. Retrieved September 4, 2023 from: https://reference.wolfram.com/language/guide/DistributionsUsedInReliabilityAnalysis.html

[4] Eliwa MS, Altun E, Alhussain ZA, Ahmed EA, Salah MM, Ahmed HH, El-Morshedy M. A new one-parameter lifetime distribution and its regression model with applications. PLoS One. 2021 Feb 19;16(2):e0246969. doi: 10.1371/journal.pone.0246969. PMID: 33606720; PMCID: PMC7894911.

[5] Leemis, L. Lifetime Distribution Identities. IEEE TRANSACTIONS ON RELIABILITY, VOL. R-35, NO. 2, 1986 JUNE