Chapter 8: Q 4E (page 469)
For the conditions of Exercise 2, how large a random sample must be taken in order that\({\bf{P}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - \theta |}} \le {\bf{0}}{\bf{.1}}} \right) \ge {\bf{0}}{\bf{.95}}\)for every possible value ofθ?
The needed sample size is \( \ge 1537\)
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Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with meanμand variance\({{\bf{\sigma }}^{\bf{2}}}\), and let\({{\bf{\hat \sigma }}^{\bf{2}}}\)denote the sample variance. Determine the smallest values ofnfor which the following relations are satisfied:
Suppose that\({{\bf{X}}_{\bf{1}}},{\bf{ \ldots }},{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown meanμand known variance\({{\bf{\sigma }}^{\bf{2}}}\). Let\({\bf{\Phi }}\)stand for the c.d.f. of the standard normal distribution, and let\({{\bf{\Phi }}^{{\bf{ - 1}}}}\)be its inverse. Show that
the following interval is a coefficient\(\gamma \)confidence interval forμif\({{\bf{\bar X}}_{\bf{n}}}\)is the observed average of the data values:
\(\left( {{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}{\bf{,}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ + }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}} \right)\)
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 \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)
Assume thatX1, . . . , Xnfrom a random sample from the normal distribution with meanμand variance \({\sigma ^2}\). Show that \({\hat \sigma ^2}\)has the gamma distribution with parameters \(\frac{{\left( {n - 1} \right)}}{2}\)and\(\frac{n}{{\left( {2{\sigma ^2}} \right)}}\).
Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{{\bf{X}}_{\bf{n}}}\)form a random sample from a distribution for which the p.d.f. is as follows:
\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\theta }} \right.} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{\theta }}{{\bf{x}}^{{\bf{\theta - 1}}}}}&{{\bf{for}}\,\,{\bf{0 < x < 1,}}}\\{\bf{0}}&{{\bf{otherwise,}}}\end{align}} \right.\)
where the value of θ is unknown (θ > 0). Determine the asymptotic distribution of the M.L.E. of θ. (Note: The M.L.E. was found in Exercise 9 of Sec. 7.5.)
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