Chapter 8: Q2 E (page 468)
Find the mode of theχ2 distribution withmdegrees of
freedom(m=1,2, . . .).
-
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Consider again the conditions of Exercise 19, and let\({{\bf{\hat \beta }}_{\bf{n}}}\)n denote the M.L.E. of β.
a. Use the delta method to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).
b. Show that\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}{\bf{ = }}{{\bf{\bar X}}_{\bf{n}}}\), and use the central limit theorem to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).
Question:Suppose that a random variable X has the geometric distribution with an unknown parameter p (0<p<1). Show that the only unbiased estimator of p is the estimator \({\bf{\delta }}\left( {\bf{X}} \right)\) such that \({\bf{\delta }}\left( {\bf{0}} \right){\bf{ = 1}}\) and \({\bf{\delta }}\left( {\bf{X}} \right){\bf{ = 0}}\) forX>0.
Suppose that \({X_1},...,{X_n}\) are i.i.d. having the normal distribution with mean μ and precision \(\tau \) given (μ, \(\tau \) ). Let (μ, \(\tau \) ) have the usual improper prior. Let \({\sigma '^{2}} = \frac{{s_n^2}}{{n - 1}}\) . Prove that the posterior distribution of \(V = \left( {n - 1} \right){\sigma '^{2}}\tau \) is the χ2 distribution with n − 1 degrees of freedom.
Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,\,\,{\bf{,E}}\left( {\bf{\tau }} \right){\bf{ = 2,E}}\left( {{{\bf{\tau }}^{\bf{2}}}} \right){\bf{ = 5}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {\left| {\bf{\mu }} \right|{\bf{ < 1}}{\bf{.412}}} \right){\bf{ = 0}}{\bf{.5}}\)Determine the prior hyperparameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)
Show that two random variables \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)cannot have the joint normal-gamma distribution such that
\({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,{\bf{,Var}}\left( {\bf{\mu }} \right){\bf{ = 1,E}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}\,\,{\bf{and}}\,\,{\bf{Var}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
