What if there’s no lifetime data? Periodic product ships or sales data and returns (complaints, failures, repairs, or spares sales) contain reliability information. Neutrosophic statistics provide a way to make nonparametric estimates of products’ or parts’ reliability without lifetime data.

Distinguish Lifetime Data from Without…

Table 1. “Nevada” table of cohort sizes “Ships” and their failure counts (lifetimes) from https://mtbfreport.com/nevada-format/

	Ships	Jan	Feb	Mar	Apr	May	June
Jan	5357	3	6	5	4	1	8
Feb	4222		7	9	12	23	16
Mar	2345			2	13	4	23
Apr	6132				4	8	12
May	2456					2	10
June	???						???
Returns	20512	3	13	16	33	38	69

Table 2. Without lifetimes, the cohort “Ships” column and the bottom row “Returns” (failures, repairs, complaints, spare parts’ sales, etc.) from bottom row in Nevada table 1.

	Ships	Returns
Jan	5357	3
Feb	4222	13
Mar	2345	16
Apr	6132	33
May	2456	38
June	???	69

Periodic ships and returns counts, without lifetimes, are statistically sufficient to make nonparametric reliability function estimates. Ships and returns data are available, with a little work, from revenue and service cost data required by GAAP. Parts’ ships and returns counts are available from product data, BoMs, and gozinto theory [George, 2023].

Neutrosophic statistics provides an alternative, to simulate feasible Nevada tables of cohort failure counts that add up to the period returns and allow the use of the Kaplan-Meier reliability estimator, without lifetime data.

Nonparametric Reliability Estimators from Ships and Returns

The nonparametric maximum likelihood reliability function estimator (npmle) maximizes likelihood, ΠPoisson[Returns(t), λG(t)], t=1,2,…, as a function of nondecreasing service time distribution function G(t) = 1-R(t) (R(t)=reliability function) and λ, an assumed or estimated Poisson ships rate [George, July 2022]. This is because the output of M/G/∞ self-service system is Poisson [Mirasol]. The nonstationary likelihood ΠPoisson[Returns(t), λ(t)G(t)] works for nonstationary M(t)/G/∞ too.

The nonparametric least squares actuarial failure rate function estimator (nplse) a(s) minimizes difference between observed returns and actuarial hindcasts ∑n(t-s)a(s), s=1,2,…t with installed base n(t-s). The actuarial failure rate function a(s) = (R(s+1)-R(s))/R(s).

The nonparametric Kaplan-Meier reliability estimator requires lifetime data (or a facsimile). The Greenwood variance of the Kaplan-Meier estimator errs, because it ignores cohort size randomness [George, May 2025]. So estimate cohort reliability variance-covariance matrix from Nevada table cohort lifetimes for confidence bounds on Kaplan-Meier npmle [George “Uncertainty”], if you have lifetime data. This could work for neutrosophic simulated lifetime data in Nevada tables too.

Cohort Lifetimes are Unknown Without Lifetime Data! So Simulate Them?

Table 2 “Returns” or failures in any period could have come from any previously or concurrently shipped cohort! How many combinations of a specified number k of nonnegative integers (periodic cohort returns) could add up to a specified sum S (monthly returns or failures)?

(S+k-1)!/(S!*(k-1)!) = Excel COMBIN(S+k-1,k-1)

Table 3. How many combinations??? S denotes period returns Jan.-May from table 2.

k	1	2	3	4	5
S	3	13	16	33	38
COMBIN(S+k-1,k-1)	1	14	153	7140	111930

It is impractical to enumerate all combinations, so simulate some. This is not a “Bootstrap” method [https://en.wikipedia.org/wiki/Bradley_Efron/]. Bootstrap methods resample from known or estimated distribution(s). No, it’s a simple “Neutrosophic” method [Smarandache, Alhasan et al, and Dasan et al].

What is Neutrosophistry?

“Of or relating to a general form of logic in which each proposition has separate values for truth, falsehood, and indeterminacy” [https://en.wiktionary.org/wiki/neutrosophic/]. Google says, “Neutrosophic statistics is a generalization of classical and interval statistics that analyzes data containing indeterminacy—such as vague, incomplete, or contradictory information.” Monthly ships and returns counts obviously (logic) contain reliability information, without lifetime data by cohort. Why not construct (proposition) Nevada tables with cohort returns counts by month that add up to observed monthly returns sums? Then compute Kaplan-Meier reliability estimate for each simulation, repeat, and average the Kaplan-Meier estimates? Why not compute Kaplan-Meier reliability estimates from each simulated cohort and estimate their variance-covariance matrix? Could even weight (truth, falsehood, and indeterminacy) each simulated Nevada table cohort with observed cohort sizes, ships, population proportions, failure count proportions?

Table 4. Neutrosophic Nevada table Excel formulas; bottom row “Returns” come from table 2.

	A	B	C	D	E	F
1		Jan	Feb	March	April	May
2	Jan	3	=RANDBETWEEN(0,13)	=RANDBETWEEN(0,16)	Etc.	Etc.
3	Feb		=13-C2	=RANDBETWEEN(D2,16)	Etc.	Etc.
4	Mar			=16-SUM(D2:D3)	Etc.	Etc.
5	Apr				Etc.	Etc.
6	May					Etc.
7	Returns	3	13	16	33	38

Notice that the January cohort’s RANDBETWEEN(0,13), RANDBETWEEN(0,16),… could be larger than subsequent 13-C2, RANDBETWEEN(D2,16),… simulated returns? In general, simulated monthly returns may be decreasing even though reliability is changing by cohort. One solution to this is to scramble the formulas in each column randomly, although the combinations in table 3 grow drastically.

Neutrosophic Statistics

Simulate possible cohort lifetimes that add up to total returns (bottom row in a Nevada table) and compute the Kaplan-Meier reliability estimates repeatedly. Excel’s RANDBETWEEN() function returns integers. An Excel Macro or VBA code could repeat this, generate larger samples, and scramble the column entries.

This is not the same as “Neutrosophic Reliability” which computes system reliability functions from component parametric reliability functions with confidence intervals on reliability function parameters [Alhasan et al].

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