Why are rejected lots informative?

Their initial sample counts reveal information hidden by selecting only zero-count samples.

Does nonparametric mean there is no model?

No. Sampling, binomial, and process assumptions remain.

Can the displayed formula handle unequal sample sizes?

Not directly; a suitable extension or different estimator is needed.

AI Glossary / Statistics and probability / Advanced

Hahn estimator

Definition

In quality-control terminology, the Hahn estimator estimates outgoing nonconformance from acceptance-sampling data while allowing underlying lot quality to vary. NIST describes it as a nonparametric estimate of average outgoing quality based on a binomial model with an unobserved distribution of the nonconformance probability. This entry concerns Gerald J. Hahn's acceptance-sampling work, particularly zero-defect plans, rather than unrelated estimators associated with researchers sharing the surname Hahn.

What it means

Under a zero-defect acceptance rule, a lot passes its initial inspection only when the inspected sample contains no nonconforming units. Looking only at those passing samples will therefore always report zero observed nonconformance. That result is built into the selection rule and does not show that the uninspected product is perfect.

Hahn's approach uses information across lots, including lots whose initial samples fail. The pattern of zero and small positive sample counts helps estimate what remains unseen in passing lots.

The exact target matters. Quality among initially accepted lots differs from quality among all lots after rejected lots have been completely inspected and corrected. The worked example explicitly uses the latter rectification setting.

How it works

Consider T equally sized lots, each containing N units, with an initial random sample of m units per lot. Lots with zero sampled nonconformances pass; every other lot undergoes complete inspection and correction before shipment. Assume the inspection and correction process works as specified.

Let T0 count initial samples with zero nonconformances and T1 count those with exactly one. A Hahn proportion estimator reported in the rectification literature is p_hat = T0*T1/[T*(T1 + m*T0)]. A suggested bias-reduction multiplier is (m − 1)/m. Multiplying the adjusted proportion by N*T gives an estimated remaining count across all outgoing lots.

This is a particular equal-lot, equal-sample, zero-acceptance-number formulation. It should not be applied unchanged to variable sampling plans, imperfect inspection, or discretionary shipment of rejected lots. Nonparametric treatment of the unknown quality distribution does not make the procedure assumption-free.

Three practical examples

A supplier estimates residual nonconformance after a documented zero-defect sampling and rectification process.
A quality analyst explains why zero defects in all passing inspection samples is not evidence of zero defects in shipped product.
An AI inspection project reviews whether historical lot labels reflect acceptance selection and correction, rather than treating passing-sample outcomes as unbiased measurements of initial lot quality.

Think of it this way

Think of checking a few pages from each manuscript and approving a manuscript only when those pages contain no errors. The approved pages will look flawless by construction, although unchecked pages may still contain mistakes. Information from manuscripts that failed the initial check helps assess the hidden problem. This analogy explains selection; it does not supply the estimator's assumptions.

Worked example

Suppose an illustrative inspection program processes T = 100 lots of N = 5,000 units each. The initial sample size is m = 50. Eighty samples contain zero nonconforming units, sixteen contain exactly one, and the remaining four contain two or more. All twenty initially failing lots are fully inspected and corrected before shipment.

The unadjusted proportion estimate is (80*16)/[100*(16 + 50*80)] = 1280/401600 = 0.00318725. Applying the suggested multiplier 49/50 gives 0.00312351, or approximately 0.31235% and 3,123.5 parts per million.

There are 500,000 outgoing units, so the adjusted estimated remaining count is 500000*0.00312351 = 1,561.75. It is an estimated count, not an observed integer tally. Independent arithmetic confirms the proportion and total. The estimate refers to all outgoing lots after the stated rectification procedure, not solely to the eighty lots that passed immediately, and it is not a confidence limit.

Illustrative case study

In a fictional electronics supplier, a dashboard reports “zero defects” because it includes only the initial samples from passing lots. A quality specialist notices that this metric is guaranteed by the acceptance rule.

The team reconstructs the initial sample-count distribution for all lots and records which rejected lots were fully corrected. It evaluates an outgoing-quality estimator appropriate to that process and compares the resulting estimates with independent shipment audits. Several months used different sample sizes, so the team does not pool them into the simple formula without further methodological work. The scenario illustrates correct definition and review of evidence, not a claim that the estimator guarantees a customer's quality requirement.

Common misconceptions and mistakes

Zero-defect sampling describes an acceptance number, not a guarantee of defect-free product. Do not confuse the observed fraction in selected passing samples with the fraction in uninspected units. Do not confuse a proportion of nonconforming units with a count of multiple defects per unit.

The simple estimator can be unstable when the relevant frequency counts are sparse. For example, a zero T1 can produce a zero estimate despite evidence of rejected lots; that is not proof of perfection. The bias-reduction multiplier does not establish exact unbiasedness or provide an uncertainty interval. Rectification assumptions are substantive requirements, not bookkeeping details.

Business use

Before using this estimator in reporting, specify the outgoing population, acceptance rule, lot and sample sizes, and treatment of rejected material. Retain the complete initial inspection results and evidence that correction was performed. Review inspector errors, process changes, and deviations from the planned sampling rule.

For management decisions, accompany point estimates with an appropriate uncertainty assessment and independent validation where feasible. An estimator designed for one inspection scheme should not become a universal quality score merely because its arithmetic is simple.

Check your understanding

Why does the example divide by all 100 lots?

Answer: Its target is outgoing quality after rectification across all shipped lots.

Is 1,561.75 an observed defect count?

Answer: No; it is a point estimate derived from the inspection summary.

Related terms

Acceptance sampling
Rectification
Empirical Bayes
Selection bias
average outgoing quality
zero-defect sampling
binomial mixture
nonconformance

Learn next

Next, distinguish incoming quality, initially accepted-lot quality, and post-rectification outgoing quality. Then study interval estimation and methods for unequal lot sizes, varying sample sizes, and imperfect inspection.

Rectification

Sources

Sources: NIST glossary, https://www.itl.nist.gov/div898/handbook/glossary.htm; Greenberg and Stokes (1992), “Estimating Nonconformance Rates After Zero-Defect Sampling With Rectification,” https://www.stat.cmu.edu/technometrics/90-00/vol-34-02/v3402203.pdf, formula and definitions read and scanned equation visually checked. Hahn (1986), https://doi.org/10.1080/00224065.1986.11979007, indexed original abstract consulted; full publisher access was unavailable. The specific rectification formulation and source-access limit are explicit. The worked data are original.

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