Confidence Limits Determination

This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiply censored) data. The methodology used is the so-called Fisher Matrix Bounds, described in Nelson [30] and Lloyd & Lipow [25].

Approximate Estimates of the Mean and Variance of a Function

Single Parameter Case

For simplicity, consider a one-parameter distribution represented by a general function G, which is a function of one parameter estimator, say (G). Then, in general, the expected value of G() can be found by:

(7)

where G() is some function of , such as the reliability function, and is the population moment or parameter such that E() = as n . The term O () is a function of n, the sample size, and tends to zero, as fast as as n . For example, in the case of = and G(x) = , then E(G()) = + O() where O () = , thus as n , E(G()) = (μ and σ are the mean and standard deviation, respectively). Using the same one parameter distribution, the variance of the function G() can then be estimated by:

(8)

Two Parameter Case

Repeating the previous method for the case of a two parameter distribution, it is generally true that for a function G, which is a function of two parameter estimators, say G(, ), that,

(9)

and:

(10)

Note that the derivatives of Eqn. (10) are evaluated at = and = , where E() and E() .

Variance and Covariance Determination of the Parameters

The determination of the variance and covariance of the parameters is accomplished via the use of the Fisher information matrix. For a two-parameter distribution, and using maximum likelihood estimates, the log likelihood function for censored data (without the constant coefficient) is given by:

Then the Fisher information matrix is given by:

where = and = .

So for a sample of N units where R units have failed, S have been suspended and P have failed within a time interval, and N = R + M + P, one could obtain the sample local information matrix by:

(11)

By substituting in the values of the estimated parameters, in this case and , and inverting the matrix, one can then obtain the local estimate of the covariance matrix or:

(12)

Then the variance of a function (Var(G)) can be estimated using Eqn. (10). Values for the variance and covariance of the parameters are obtained from Eqn. (12).

Once they are obtained, the approximate confidence bounds on the function are given as:

(13)

Approximate Confidence Intervals on the Parameters

In general, MLE estimates of the parameters are asymptotically normal, thus if is the MLE estimator for , in the case of a single parameter distribution, estimated from a sample of n units, and if:

then:

(14)

for large n. If one now wishes to place confidence bounds on , at some confidence level , bounded by the two end points C1 and C2, and where:

then from Eqn. (14):

where Kα is defined by:

Now by simplifying Eqn. (15), one can obtain the approximate confidence bounds on the parameter at a confidence level or:

If must be positive, then ln is treated as normally distributed. The two-sided approximate confidence bounds on the parameter , at confidence level , then become:

(two-sided upper) (16)

(two-sided lower) (17)

The one-sided approximate confidence bounds on the parameter , at confidence level can be found from:

(one-sided upper)

(one-sided lower)

The same procedure can be repeated for the case of a two or more parameter distribution. Lloyd and Lipow [24] elaborate on this procedure.

Percentile Confidence Bounds (Type 1 in ALTA)

Percentile confidence bounds are confidence bounds around time. For example, when using the 1-parameter exponential distribution, the corresponding time for a given exponential percentile (i.e. y-ordinate or unreliability, Q = 1 - R) is determined by solving the unreliability function for the time, T, or:

(18)

Percentile bounds (Type 1) return the confidence bounds by determining the confidence intervals around and substituting into Eqn. (18). The bounds on were determined using Eqns. (16) and (17), with its variance obtained from Eqn. (12).

Reliability Confidence Bounds (Type 2 in ALTA)

Type 2 bounds in ALTA are confidence bounds around reliability. For example, when using the 1-parameter exponential distribution, the reliability function is:

(19)

Reliability bounds (Type 2) return the confidence bounds by determining the confidence intervals around and substituting into Eqn. (19). The bounds on were determined using Eqns. (16) and (17), with its variance obtained from Eqn. (12).

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