Optimum exponential quanta1 response. For Problem, suppose that all I units have a common inspection time q, = q, I = I,. . . ,I. Suppose Y of the I units are failed on inspection.
(a) Derive an explicit formula for the ML estimate 4; use part (b) of Problem 9.1.
(b) Derive the formula for the true asymptotic variance of 6; use part (h) of Problem 9.1.
(c) Derive the optimum inspection time q* that minimizes the true asymptotic variance. It is a multiple c of the true unknown do, q* = c0,. Numerically find c
(d) Evaluate the minimum variance for = q*, and compare it with the variance of θ for a complete observed sample.
Inpractice, one must guess a value
(F)Explain how to use "exact" confidence limits for a binomial proportion to get exact limits for θ,.
Problem
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.