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Chapter 7: Q10E (page 425)

Question : Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from a distribution for which the p.d.f. f (x|胃 ) is as follows:

\({\bf{f}}\left( {{\bf{x|\theta }}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}{{\bf{e}}^{{\bf{ - }}\left| {{\bf{x - \theta }}} \right|}}\,\,\,\,{\bf{, - }}\infty {\bf{ < x < }}\infty \)

Also, suppose that the value of 胃 is unknown (鈭掆垶 < 胃 < 鈭�). Find the M.L.E. of 胃.

Short Answer

Expert verified

The median of the random sample \({X_1},...,{X_n}\)

Step by step solution

01

Given information

\({X_1},...,{X_n}\)form a random sample from a distribution for which the pdf is . f (x|胃 ) is as follows:

\(f\left( {x|\theta } \right) = \frac{1}{2}{e^{ - \left| {x - \theta } \right|}}\,\,\,\,, - \infty < x < \infty \). We need to calculate the M.L.E of 胃

02

Calculation of M.L.E of θ

The likelihood function ,

\(\begin{array}{c}L\left( \theta \right) = \log \left[ {\frac{1}{{{2^n}}}\exp \left( { - \sum\limits_{i = 1}^n {\left| {{x_i} - \theta } \right|} } \right)} \right]\\ = \log \left( {{2^{ - n}}} \right) - \sum\limits_{i = 1}^n {\left| {{x_i} - \theta } \right|} \\ = - \left( {n\log 2 + \sum\limits_{i = 1}^n {\left| {{x_i} - \theta } \right|} } \right)\end{array}\)

The M.L.E. can be obtained by minimizing \(\sum\limits_{i = 1}^n {\left| {{x_i} - \theta } \right|} \).We know that \(\sum\limits_{i = 1}^n {\left| {{x_i} - \theta } \right|} \)is minimum when \({X_1},...,{X_n}\)are measured about median. So, the M.L.E is the median of the sample \({X_1},...,{X_n}\).

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Most popular questions from this chapter

Consider a distribution for which the pdf. or the p.f. is \(f\left( {x|\theta } \right)\) , where the parameter 胃 is a k-dimensional vector belonging to some parameter space\(\Omega \) . It is said that the family of distributions indexed by the values of 胃 in\(\Omega \) is a k-parameter exponential family, or a k-parameter Koopman-Darmois family, if \(f\left( {x|\theta } \right)\)can be written as follows for \(\theta \in \Omega \)and all values of x:

\(f\left( {x|\theta } \right) = a\left( \theta \right)b\left( x \right)\exp \left( {\sum\limits_{i = 1}^k {{c_i}\left( \theta \right){d_i}\left( x \right)} } \right)\)

Here, a and \({c_1},...,{c_k}\) are arbitrary functions of 胃, and b and \({d_1},...,{d_k}\) are arbitrary functions of x. Suppose now that \({X_1},...,{X_n}\) form a random sample from a distribution which belongs to a k-parameter exponential family of this type, and define the k statistics \({T_1},...,{T_k}\) as follows:

\({T_i} = \sum\limits_{j = 1}^n {{d_i}\left( {{X_j}} \right)} \)

Show that the statistics \({T_1},...,{T_k}\)are jointly sufficient statistics for 胃.

Suppose that a single observation X is to be taken from the uniform distribution on the interval \(\left[ {{\bf{\theta - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{,\theta + }}\frac{{\bf{1}}}{{\bf{2}}}} \right]\), the value of 胃 is unknown, and the prior distribution of 胃 is the uniform distribution on the interval [10, 20]. If the observed value of X is 12, what is the posterior distribution of 胃?

Question: In a clinical trial, let the probability of successful outcome 胃 have a prior distribution that is the uniform distribution on the interval\(\left[ {0,1} \right]\), which is also the beta distribution with parameters 1 and 1. Suppose that the first patient has a successful outcome. Find the Bayes estimates of 胃 that would be obtained for both the squared error and absolute error loss functions.

Question: Prove that the method of moments estimator for the parameter of a Bernoulli distribution is the M.L.E.

Suppose that a random sample \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) is drawn from the Pareto distribution with parameters \({{\bf{x}}_{\bf{0}}}\,\,\,\,{\bf{and}}\,\,\,{\bf{\alpha }}\).

a. If \({{\bf{x}}_{\bf{0}}}\) is known and \({\bf{\alpha > 0}}\) unknown, find a sufficient statistic

b. If \({\bf{\alpha }}\) is known and \({{\bf{x}}_{\bf{0}}}\) unknown, find a sufficient statistic.
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