# Question

Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator θ, if the random sample X1, X2, . . . , Xn is taken from each of the distributions having the following pdfs:

(a) f(x; θ) = (1/θ2) x e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(b) f(x; θ) = (1/2θ3) x2 e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(c) f(x; θ) = (1/θ) x(1−θ)/θ, 0 < x < 1, 0 < θ < ∞.

(a) f(x; θ) = (1/θ2) x e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(b) f(x; θ) = (1/2θ3) x2 e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(c) f(x; θ) = (1/θ) x(1−θ)/θ, 0 < x < 1, 0 < θ < ∞.

## Answer to relevant Questions

Find a sufficient statistic for θ, given a random sample, X1, X2, . . . , Xn, from a distribution with pdf Let X1, X2, . . . , Xn be a random sample from N(0, θ), where σ2 = θ > 0 is unknown. Argue that the sufficient statistic are independent. Let X1, X2 be a random sample from the Cauchy distribution with pdf Consider the non-informative prior h(θ1, θ2) ∝ 1 on that support. Obtain the posterior pdf (except for constants) of θ1, θ2 if x1 = 3 and x2 = 7. For ...Assume that the yield per acre for a particular variety of soybeans is N(μ, σ2). For a random sample of n = 5 plots, the yields in bushels per acre were 37.4, 48.8, 46.9, 55.0, and 44.0. (a) Give a point estimate for ...An environmental survey contained a question asking what respondents thought was the major cause of air pollution in this country, giving the choices “automobiles,” “factories,” and “incinerators.” Two versions ...Post your question

0