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Chapter 8: Q 19SE (page 529)

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the exponential distribution with unknown parameter 尾. Show that if n is large, the distribution of the M.L.E. of 尾 will be approximately a normal distribution with mean 尾 and variance\(\frac{{{{\bf{\beta }}^{\bf{2}}}}}{{\bf{n}}}\).

Short Answer

Expert verified

Proved.

Step by step solution

01

Given information

Suppose that\({X_1},...,{X_n}\) from a random sample from the exponential distribution with an unknown parameter \(\beta \).

02

Showing part

The p.d.f. of an exponential distribution is,

\(f\left( {x\left| \beta \right.} \right) = \beta \exp \left( { - \beta x} \right)\)

Then,

Taking log,

\(\begin{align}\lambda \left( {x\left| \beta \right.} \right) &= \log f\left( {x\left| \beta \right.} \right)\\ &= \log \left( \beta \right) - \beta x\end{align}\)

\(\begin{align}\lambda '\left( {x\left| \beta \right.} \right) &= \frac{1}{\beta } - x\\\lambda ''\left( {x\left| \beta \right.} \right) &= - \frac{1}{{{\beta ^2}}}\end{align}\)

The Fisher information is,

\(\begin{align}I\left( \beta \right) &= - {E_\theta }\left( { - \frac{1}{{{\beta ^2}}}} \right)\\ &= \frac{1}{{{\beta ^2}}}\end{align}\)

The distribution of the M.L.E of \(\beta \) will be approximately the normal distribution with the mean is \(\beta \) and the variance is \(\frac{1}{{{\beta ^2}}}\)

Hence, (Proved)

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Most popular questions from this chapter

Suppose that the five random variables \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{5}}}\) are i.i.d. and that each has the standard normal distribution. Determine a constantcsuch that the random variable

\(\frac{{{\bf{c}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}} \right)}}{{{{\left( {{\bf{X}}_{\bf{3}}^{\bf{2}}{\bf{ + X}}_{\bf{4}}^{\bf{2}}{\bf{ + X}}_{\bf{5}}^{\bf{2}}} \right)}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}\)

will have atdistribution.

Suppose that \({X_1},...,{X_n}\) form a random sample from the exponential distribution with unknown mean 渭. Describe a method for constructing a confidence interval for 渭 with a specified confidence coefficient \(\gamma \left( {0 < \gamma < 1} \right)\)
  1. Construct a 2脳2 orthogonal matrix for which the first row is as follows: \(\left( {\begin{align}{}{\frac{{\bf{1}}}{{\sqrt {\bf{2}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{2}} }}}\end{align}} \right)\)
  2. Construct a 3脳3 orthogonal matrix for which the first row is as follows: \(\left( {\begin{align}{}{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}\end{align}} \right)\)

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,\,\,{\bf{,E}}\left( {\bf{\tau }} \right){\bf{ = 2,E}}\left( {{{\bf{\tau }}^{\bf{2}}}} \right){\bf{ = 5}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {\left| {\bf{\mu }} \right|{\bf{ < 1}}{\bf{.412}}} \right){\bf{ = 0}}{\bf{.5}}\)Determine the prior hyperparameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)

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 胃 is unknown (0 鈮� 胃 鈮� 1).

a. Verify that the sum of the five given probabilities is 1 for every value of 胃.

b. Consider an estimator 未c(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 胃.

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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