Question

2. A certain type of electronic component has a lifetime X (in hours) with probability density function given by otherwise. where θ 0. Let X1, . . . , Xn denote a simple random sample of n such electrical components. . Find an expression for the MLE of θ as a function of X1 Denote this MLE by θ ·Determine the expected value and variance of θ. » What is the MLE for the variance of X? Show that θ satisfies the Cramr-Rao Inequality, and hence that θ is a uniformly minimum-variance unbiased estimator (UMVUE) » Now suppose that you plan to independently test the lifetime of n-3 such electrical components. If the true value of θ is actually θ0-130. find a bound such that the probability that the esti- mation error (î.е.Ιθ-θο!) is less than this bound is at least 95% Explain your approach . If the three such components, tested independently, had lifetimes hours, respectively, what is the maximum of 120, 130 and 128 hours, likelihood estimate of θ ? If indeed the true value of θ actually is Ho-130, what can you say, if anything, about the actual error between your estimate of θ and its true value? Discuss

Answer 1