A. Wood - Softare Reliability. Growth Models (798489), страница 9
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Then, siice the confidence interval for the parameter a is also derivedfrom the parameter b, the confidence interval for parameter a does not scale linearly.For the classical least squares technique, the following equation must be solved for the 0-0model (see Section 2.3.2):(A9) L jW =1(1n«fj-fj.l)/(lj-tj.l))-ln(b) -In(a-fj))2If we replace the number of failures f j by k/j and replace the test time lj by k 2lj in Equation(A9), the result isW(AlO) Lj =1 (In((fj-fj.l)l(lj-tj_l)) + In(k 1) -In(kz) -In(b) -In(a-k/j))zIf we replace a with k a and b with b1k , we get the Equation (A9) with the same21minimum. Thus, a scales linearly. The confidence intervals for a are:a ± t w - Z,<XI2 (var(a))0.52Since var(kl a) = k 1 var(a), the confidence intervals scale linearly.For the alternative least squares technique, the equation to minimize is:W(All) L j =1 (fj - ~(tj))ZWbtj= Lj =1 (fj - a(l_e- ))2 for the 0-0 modelIf we replace the number of failures f j by k f j and replace the test time lj by k lj in Equation12(All), the result is(A12) L.
w (k f. _ a(l_e- k2btj ))2I =11IIf we replace a with k a and b with b/k in Equation (A12), we get an equation with the12same minimum as Equation (All). Thus, a scales linearly. The first method of findingconfidence intervals for the alternative least squares technique is the same as the classicalleast squares technique.
As shown above, these confidence intervals scale linearly. Thesecond technique defines the confidence intervals for a as a ± ZI-<XIZ (a)0.5. If we replace awith k 1a, the confidence intervals are k a ± ZI-al2 k 0.5a0.5. Therefore, the confidence11interval for parameter a does not scale linearly.30Distributed by~TANDEMCorporate Information Center10400 N. Tantau Ave., LOC 248-07Cupertino, CA 95014-0726.
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