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Chapter 7: Q14E (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 the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean \(\lambda \)is either 1.0 or 1.5, and the prior p.f. of \(\lambda \) is as follows:

\(\xi \left( {1.0} \right) = 0.4\)and\(\xi \left( {1.5} \right) = 0.6\)

If a roll of tape selected at random is found to have three defects, what is the posterior p.f. of \(\lambda \) ?

Identify the components of the statistical model (as defined in Definition 7.1.1) in Example 7.1.3.

Question: Suppose that the lifetime of a certain type of lamp has an exponential distribution for which the value of the parameter \({\bf{\beta }}\) is unknown. A random sample of n lamps of this type are tested for a period of T hours and the number X of lamps that fail during this period is observed, but the times at which the failures occurred are not noted. Determine the M.L.E. of \({\bf{\beta }}\) based on the observed value of X.

Suppose that the proportion \(\theta \) of defective items in a large manufactured lot is unknown, and the prior distribution of \(\theta \) is the uniform distribution on the interval \(\left[ {0,1} \right]\). When eight items are selected at random from the lot, it is found that exactly three of them are defective. Determine the posterior distribution of \(\theta \).

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.)
See all solutions

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