Reference no: EM131056467
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.
|
Simple macroeconomic model of an economy
: A simple macroeconomic model of an economy is estimated at end of 2009 giving the following information:
|
|
Write a time-stamped string to a second file
: The timestamp should be recorded in text form as the number of milliseconds since the start of the script session. Recording the timestamp in text form has the advantage that the resulting file is human-readable. Within string, real newline charac..
|
|
Chain of workout facilities
: HealthWorks Inc, a chain of workout facilities, is looking at a new type of exercise equipment for all of its stores. The total amount of investment (an immediate outflow) will be $550 today and the equipment is expected to last for 5 years with no s..
|
|
What is the discounted payback period for this investment
: The Seattle Corporation has been presented with an investment opportunity which will yield cash flows of $30,000 per year in Years 1 through 4, $35,000 per year in Years 5 through 9, and $40,000 in Year 10. What is the discounted payback period for t..
|
|
Optimum exponential quanta1 response
: Optimum exponential quanta1 response. For Problem 9.1, 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.
|
|
Considering the purchase of new parking system
: A small town in Ohio is considering the purchase of a new parking system that would enhance the collection of parking fees while providing additional convenience to shoppers. The hardware requires an immediate outflow (an investment today) of $1350.
|
|
Problems of inflation or unemployment
: Businesses spend $1,000 in new investment spending. And, foreigners spend $500 on exports. In order to avoid any problems of inflation or unemployment, the government should have a budget deficit or surplus of:
|
|
Where does the data appear in the file and why
: Where does the data appear in the file
|
|
Describe the variables and their scale of measurement
: Identify a research question from your professional life or career specialization that can be addressed by a correlation. Indicate why a correlation would be the appropriate analysis for this research question.
|