Chapter 8: Q13 E (page 473)
Prove that the distribution of\({\hat \sigma _0}^2\)in Examples 8.2.1and 8.2.2 is the gamma distribution with parameters\(\frac{n}{2}\)and\(\frac{n}{{2{\sigma ^2}}}\).
\({\hat \sigma ^2}\) follows Gamma distribution with parameters \(\frac{n}{2}\) and \(\frac{n}{{2{\sigma ^2}}}\).
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By using the table of the t distribution given in the back of this book, determine the value of the integral
\(\int\limits_{ - \infty }^{2.5} {\frac{{dx}}{{{{\left( {12 + {x^2}} \right)}^2}}}} \)
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.)
Suppose that X1,……,Xn form a random sample from a normal distribution for which the mean is known and the variance is unknown. Construct an efficient estimator that is not identically equal to a constant, and determine the expectation and the variance of this estimator.
Suppose that each of two statisticians, A and B, independently takes a random sample of 20 observations from the normal distribution with unknown mean μ and known variance 4. Suppose also that statistician A finds the sample variance in his random sample to be 3.8, and statistician B finds the sample variance in her random sample to be 9.4. For which random sample is the sample mean likely to be closer to the unknown value of μ?
Question:Suppose that a random variable X can take only the five values\({\bf{x = 1,2,3,4,5}}\) with the following probabilities:
\(\begin{aligned}{}{\bf{f}}\left( {{\bf{1}}\left| {\bf{\theta }} \right.} \right){\bf{ = }}{{\bf{\theta }}^{\bf{3}}}{\bf{,}}\,\,\,\,{\bf{f}}\left( {{\bf{2}}\left| {\bf{\theta }} \right.} \right){\bf{ = }}{{\bf{\theta }}^{\bf{2}}}\left( {{\bf{1}} - {\bf{\theta }}} \right){\bf{,}}\\{\bf{f}}\left( {{\bf{3}}\left| {\bf{\theta }} \right.} \right){\bf{ = 2\theta }}\left( {{\bf{1}} - {\bf{\theta }}} \right){\bf{,}}\,\,\,{\bf{f}}\left( {{\bf{4}}\left| {\bf{\theta }} \right.} \right){\bf{ = \theta }}{\left( {{\bf{1}} - {\bf{\theta }}} \right)^{\bf{2}}}{\bf{,}}\\{\bf{f}}\left( {{\bf{5}}\left| {\bf{\theta }} \right.} \right){\bf{ = }}{\left( {{\bf{1}} - {\bf{\theta }}} \right)^{\bf{3}}}{\bf{.}}\end{aligned}\)
Here, the value of the parameter θ is unknown (0 ≤ θ ≤ 1).
a. Verify that the sum of the five given probabilities is 1 for every value of θ.
b. Consider an estimator δc(X) that has the following form:
\(\begin{aligned}{}{{\bf{\delta }}_{\bf{c}}}\left( {\bf{1}} \right){\bf{ = 1,}}\,\,{{\bf{\delta }}_{\bf{c}}}\left( {\bf{2}} \right){\bf{ = 2}} - {\bf{2c,}}\,\,{{\bf{\delta }}_{\bf{c}}}\left( {\bf{3}} \right){\bf{ = c,}}\\{{\bf{\delta }}_{\bf{c}}}\left( {\bf{4}} \right){\bf{ = 1}} - {\bf{2c,}}\,\,{{\bf{\delta }}_{\bf{c}}}\left( {\bf{5}} \right){\bf{ = 0}}{\bf{.}}\end{aligned}\)
Show that for each constant, c\({{\bf{\delta }}_{\bf{c}}}\left( {\bf{X}} \right)\)is an unbiased estimator of θ.
c. Let\({{\bf{\theta }}_{\bf{0}}}\)be a number such that\({\bf{0 < }}{{\bf{\theta }}_{\bf{0}}}{\bf{ < 1}}\). Determine a constant\({{\bf{c}}_{\bf{0}}}\)such that when\({\bf{\theta = }}{{\bf{\theta }}_{\bf{0}}}\)the variance is smaller than the variance \({{\bf{\delta }}_{\bf{c}}}\left( {\bf{X}} \right)\)for every other value of c.
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