Unidentified competing failure modes. A certain product contains many components which can fail, including a motor. The motor manufacturer reimburses the product manufacturer for each motor failure on warranty. In a particular production period, the method of motor manufacture was changed, resulting in a defective motor with a high failure rate. The following model and analyses were used to predict the motor manufacturer's liability. The cause of product failure was not identified. (a) Prior to any motor problem, experience with all other failure modes indicated that (1) the product had a small proportion n that were found failed when installed and (2) the remaining proportion (1 - n) followed a Weibull distribution for time t to failure with scale parameter a and shape parameter p near 1. Write this cumulative distribution function for the product, assuming no motor failures.

(b) The defective motor was assumed to have a Weibull life distribution with scale parameter a' and shape parameter p'. Assuming these motors are not failed on installation (not in the proportion a), write the combined cumulative distribution function for time t to failure due to both motors and all other causes, assuming motor failures are independent of all others.

(c) Ordinarily the Weibull distribution is written in terms of the scale parameter. Rewrite the distributions from (a) and (b) in terms of small percentiles f_p = α - [ ln(1 - P)]"B and frP, = α' - [In( 1 - P')]"B'; eliminating a and a'. Ths reparametrization makes the ML calculations converge more easily, when P and P' are chosen near the corresponding sample fractions failed.

(d) The accompanying figure shows maximum likelihood estimates and the local estimate of their covariance matrix for the coefficients C1, ..., C5 where

Here to, and f0, are expressed in months and C1 in radians. Calculate two-sided (approximate) 95% confidence limits for π,ß p and ß'. Is p near 1 as expected? Is ß' significantly different from 1, and what does it indicate regarding future numbers of motor failures in service? Is πi significantly different from zero, suggesting π in the model improves the fit?

(d)   Write the algebraic formula for the fraction failing from all causes on a warranty of t* months. Calculate the numerical ML estimate of this fraction for t* =60 months.

(f) Write the algebraic formula for the fraction failing from all other causes (except motor) on a warranty of t* months. Calculate the numerical ML estimate of this fraction for z* =60 months.

(g) The fraction from (e) minus the fraction from (f) is the increase in failures due to motors and is paid by the motor manufacturer. Write the algebraic formula for this difference, and calculate its numerical ML estimate for t* =60 months.

(h) Numerically calculate two-sided approximate 95% confidence limits for the difference in (G).

 (i) Explain why one-sample prediction limits are preferable to the confidence limits in (h), and discuss how much the two types of limits differ in this application.

(j) Criticize the model in (b).