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Chapter 7: Q9E (page 448)
The uniform distribution on the interval [a, b], where the value of a is known and the value of b is unknown\(\left( {b > a} \right)\):\(T = \max \left\{ {{X_1},...{X_n}} \right\}\)
Statistic T is sufficient statistic.
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Question: Let \({{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}\) be distinct numbers. Let Y be a discrete random variable with the following p.f.:
\(\begin{array}{c}{\bf{f}}\left( {\bf{y}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\,{\bf{if}}\,{\bf{y}} \in \left\{ {{{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}} \right\}\\ = 0\,otherwise\end{array}\)
Prove that Var(Y ) is given by Eq. (7.5.5).
Question: Consider the conditions of Exercise 2 again. Suppose that the prior distribution of 胃 is as given in Exercise 2, and suppose again that 20 items are selected at random from the shipment.
a. For what number of defective items in the sample will the mean squared error of the Bayes estimate be a maximum?
b. For what number the mean squared error of the Bayes estimate will be a minimum?
Suppose that the vectors \(\left( {{X_1},{Y_1}} \right),\left( {{X_2},{Y_2}} \right),...,\left( {{X_n},{Y_n}} \right)\) form a random sample of two-dimensional vectors from a bivariate normal distribution for which the means, the variances, and the correlation are unknown. Show that the following five statistics are jointly sufficient:
\(\sum\limits_{i = 1}^n {{X_i}} ,\sum\limits_{i = 1}^n {{Y_i}} ,\sum\limits_{i = 1}^n {X_i^2} ,\sum\limits_{i = 1}^n {Y_i^2} \,\,\,\,{\rm{and}}\,\,\,\,\sum\limits_{i = 1}^n {{X_i}{Y_i}} \)
Question : Suppose that the two-dimensional vectors \(\left( {{{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{Y}}_{\bf{1}}}} \right){\bf{,}}\left( {{{\bf{X}}_{\bf{2}}}{\bf{,}}{{\bf{Y}}_{\bf{2}}}} \right){\bf{,}}...{\bf{,}}\left( {{{\bf{X}}_{\bf{n}}}{\bf{,}}{{\bf{Y}}_{\bf{n}}}} \right)\) form a random sample from a bivariate normal distribution for which the means of X and Y are unknown but the variances of X and Y and the correlation between X and Y are known. Find the M.L.E.鈥檚 of the means.
Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval \(\left( {{{\bf{\theta }}_{\bf{1}}}{\bf{,}}{{\bf{\theta }}_{\bf{2}}}} \right)\) , where both \({{\bf{\theta }}_{\bf{1}}}\)and \({{\bf{\theta }}_{\bf{2}}}\) are unknown . Find the M.L.E.鈥檚 \({{\bf{\theta }}_{\bf{1}}}\) and \({{\bf{\theta }}_{\bf{2}}}\).
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