Downloaded from https://www.everyspec.com AMCP 7G.MM Parzan (Ref. I, p. 143) has shown that: The Maximum Likelihood solution is obtained by setting Eq. B-4 to zero and solving for &, I. The conditional Cdf of the failure time A; where denotes the Maximum Likelihood is solution. Cdf{t Is;T}- m(f)/m(T), for O0s t s Ti - n/tj In (T/ti) (B-5a the condition is that the number of failures " nIT 5 (B-5b) n is known. The uncertainties are evaluated by the 2. The failure times behave Pi if they Fischer Information matrix. For this pur- were s-independent, pose, the s-expected values of the second derivatives of L are needed.
The Pr( n;T I Is the same for a nonhomo- geneous process as it Is for a homogeneous Additional notation follows: one, I.e.,, .Pr{ix$T}- ex p- m(T)I[,(T)JN/nl (B-2) c = PInt C = P1n T The Likelihood is D - EL, IEL22 -EL212 L;* - Enjpdf{tj nt.T11]Pr(n; }. (8-3) ELjj - s.expected value of the second It is tedious but straightforward to show derivative of L with respect to the that (the random variables, tt and n, are the variables i and/*; #I corresponds constants; the 2 parameters a and 0 are the to a, #2 corresponds to variables): The second partial derivatives of L are:
L - -In L* - - In (no) Lj a2L - T m() B 6a - (W - )Z 4 In,tj + a TO + constant " aTa m(T) (B-)a) (B-4) 2 L L 81ho &InP Since, in the Likelihood methods, monotonic transformations. of the variables or Likelihood (P In T)cv TO Cm (T) (B-6b) don't change the basic results, all the partial derivatives are with respect to In a and 2_L In 0, because the results are more convenient. L - 'L - - 0 7n t It is again straightforward to show that: (n In T)oTO + (P In TNT)'as 2; n,["'0]4-CL-(T) 8i in tj - n + Cm(T)J (B-6o) +P(1l T)aTA (B-4b) The random variables t 1 and n appear only i '! ... .