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Chapter 7: Q13E (page 417)

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 胃], where the value of the parameter 胃 is unknown. Suppose also that the prior distribution of 胃 is the Pareto distribution with parameters \({{\bf{x}}_{\bf{0}}}\) and 伪 (\({{\bf{x}}_{\bf{0}}}\)> 0 and 伪 > 0), as defined in Exercise 16 of Sec. 5.7. If the value of 胃 is to be estimated by using the squared error loss function, what is the Bayes estimator of 胃? (See Exercise 18 of Sec. 7.3.)

Short Answer

Expert verified

The Bayes estimator of parameter \(\theta \) is \(\frac{{\alpha + n}}{{\alpha + n - 1}}\max \left\{ {{x_0},{x_1},...,{x_n}} \right\}\)

Step by step solution

01

Given information

\({X_1},{X_2},...,{X_n}\) form a random sample from the uniform distribution on the interval [0, 胃],We need to calculate the Bayes estimator of 胃.

02

Calculation of the Bayes estimator of θ.

If the prior distribution of \(\theta \) has Pareto distribution with parameters \(\alpha \,\,\,{\rm{and}}\,\,\,{x_0}\) ,then the posterior distribution is also Pareto distribution with parameters\(\alpha + n\,\,\,{\rm{and}}\,\,\,\max \left\{ {{x_0},{x_1},...,{x_n}} \right\}\)and it has mean \(\frac{{\alpha + n}}{{\alpha + n - 1}}\max \left\{ {{x_0},{x_1},...,{x_n}} \right\}\).

Here it is given that squared error loss function is used which indicates the Bayes estimator of 胃 is the mean of the posterior distribution.

Hence the required Bayes estimator is \(\frac{{\alpha + n}}{{\alpha + n - 1}}\max \left\{ {{x_0},{x_1},...,{x_n}} \right\}\)

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Most popular questions from this chapter

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\,\,{\bf{and}}\,\,{\bf{\alpha }}\)(see Exercise 16 of Sec. 5.7), where\({{\bf{x}}_{\bf{0}}}\)is unknown and\({\bf{\alpha }}\)is known. Determine the M.L.E. of\({{\bf{x}}_{\bf{0}}}\).

Consider again the problem described in Exercise 7.

a. Which interval 1-inch long had the highest prior probability of containing the value of胃?

b. Which interval 1-inch long has the highest posterior probability of containing the value of胃?

c. Find the values of the probabilities in parts (a) and (b).

Consider again the conditions of Exercise 3. Suppose that after a certain statistician has observed that there were three defective items among the 100 items selected at random, the posterior distribution that she assigns to 胃 is a beta distribution for which the mean is \(\frac{{\bf{2}}}{{{\bf{51}}}}\) and the variance is \(\frac{{{\bf{98}}}}{{\left( {{{\left( {{\bf{51}}} \right)}^{\bf{2}}}\left( {{\bf{103}}} \right)} \right)}}\). What prior distribution had the statistician assigned to 胃?

Assume that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from a distribution that belongs to an exponential family of distributions as defined in Exercise 23 of Sec. 7.3. Prove that\({\bf{T = }}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{\bf{d}}\left( {{{\bf{X}}_{\bf{i}}}} \right)} \) is a sufficient statistic for 胃.

The Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\)and伪\(\left( {{{\bf{x}}_{\bf{0}}}{\bf{ > 0}}\;{\bf{and}}\;{\bf{\alpha > 0}}} \right)\)is defined in Exercise 16 of Sec. 5.7.Show that the family of Pareto distributions is a conjugate family of prior distributions for samples from a uniformdistribution on the interval (0, 胃), where the value of the endpoint胃is unknown.
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