Suppose a type of unit has an exponential life distribution with mean θ,. Suppose that unit i is inspected once at time π, to determine whether it has failed or not, i = 1,. . .,I. (a) Write the sample log likelihood C, distinguishing between failed
Write the sample log likelihood C, distinguishing between failed and unfailed units.
Derive ac/a8, and give the likelihood equation.
Use the wheel data of Section 1, and iteratively calculate the ML estimate θ accurate to two figures. Treat each unit as if it were inspected at the middle of its time interval, and treat the last interval as if all inspections were at 4600 hours
(a) Derive the formula for d2C/dd2, and evaluate it for the wheel
(b) Give the formula for the local estimate of Var(θ).
(f) Evaluate the local estimate of Var( d) for the wheel data, using (d) and (e).
(F) (θ) Calculate positive approximate 95% confidence limits for e,,, using (f).
(h) Express the sample log likelihood in terms of indicator functions, and derive the formula for the true asymptotic variance of θ .
(i) Evaluate the ML estimate of Var(d) for the wheel data, using (c) and (h)
(j) Calculate positive approximate 95% confidence limits for B,, using (i).
(k) On Weibull paper, plot the data, the fitted exponential distribution, and approximate 95% confidence limits for percentiles.