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Prairie & Zimmer

Rules of Concordance

The basis for concordance is the determination of the 50% Iower confidence bound based on the cumulative data.

The observed failure ratÄ™ is defined to be concordant with the reliability requirement if the Iower 50% confidence bound, based on x failures in n trials, includes the reliability requirement, expressed in terms of unreliability. Generally, the cumulative count begins at the time of the last initial sampling period.

The Iower one-sided 100(1 - y)% confidence limit on the parameter Q, for a binomial yariable X, is the value of Q which satisfies

[1]

where x is the observed number of failures. For the present application, we want to determine n for a fixed x and Q = Ql, where Ql is the requirement. Since n is an integer, EÄ…uation [1] is written as

[2]

to obtain at least 100(1 - g)% confidence. Then EÄ…uation (2] can be written

[3]

Thus the reÄ…uired value of n will be the smallest n that satisfies [3].

With the n and Ql values most often considered for the reliability reÄ…uirements, the Poisson approximation to the binomial is satisfactory. The usual cnteria for the use of the Poisson approximation to the binomial are that QL be smali and n be large, often interpreted as the criterion that nQL ^ 5. For the rangÄ™ of values considered with the reliability reÄ…uirements of interest for the techniÄ…ues of this paper, that criterion is easily satisfied.

In this case, [3] becomes: