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Chapter 8: Q 15SE (page 529)

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

\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\theta }} \right.} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{\theta }}{{\bf{x}}^{{\bf{\theta - 1}}}}}&{{\bf{for}}\,\,{\bf{0 < x < 1,}}}\\{\bf{0}}&{{\bf{otherwise,}}}\end{align}} \right.\)

where the value of 胃 is unknown (胃 > 0). Determine the asymptotic distribution of the M.L.E. of 胃. (Note: The M.L.E. was found in Exercise 9 of Sec. 7.5.)

Short Answer

Expert verified

The asymptotic distribution of \(\theta \) is follows a standard normal distribution.

Step by step solution

01

Given information

Suppose that \({X_1},...{X_n}\) from a random sample from a distribution for which p.d.f. is,

\(f\left( {x\left| \theta \right.} \right) = \left\{ {\begin{align}{}{\theta {x^{\theta - 1}}}&{for\,\,0 < x < 1,}\\0&{otherwise,}\end{align}} \right.\)

02

Finding the asymptotic distribution of \({\bf{\theta }}\) 

The p.d.f. is given by,

\(f\left( {x\left| \theta \right.} \right) = \left\{ {\begin{align}{{}{}}{\theta {x^{\theta - 1}}}&{for\,\,0 < x < 1,}\\0&{otherwise,}\end{align}} \right.\)

Then,

Taking log in left side.

\(\log f\left( {x\left| \theta \right.} \right) = \lambda \left( {x\left| \theta \right.} \right)\)

So,

\(\lambda \left( {x\left| \theta \right.} \right) = \log \theta + \left( {\theta - 1} \right)\log x\)

Then,

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

The Fisher information is,

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

It follows that the asymptotic distribution of \(\frac{{{n^{\frac{1}{2}}}}}{\theta }\left( {{{\hat \theta }_n} - \theta } \right)\) is standard normal distribution.

Therefore, the asymptotic distribution of \(\theta \) is follows a standard normal distribution.

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Most popular questions from this chapter

Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with known mean 渭 and unknown precision \(\tau \left( {\tau > 0} \right)\). Suppose also that the prior distribution of \(\tau \) is the gamma distribution with parameters\({\alpha _0}\,\,\,{\rm{and}}\,\,\,\,{\beta _0}\left( {{\alpha _0} > 0\,\,\,{\rm{and}}\,\,\,{\beta _0} > 0} \right)\) . Show that the posterior distribution of \(\tau \) given that \({X_i} = {x_i}\) (i = 1, . . . , n) is the gamma distribution with parameters \({\alpha _0} + \frac{n}{2}\,\,\,\,{\rm{and}}\,\,\,\,\,{\beta _0} + \frac{1}{2}\sum\limits_{i = 1}^N {{{\left( {{x_i} - \mu } \right)}^2}} \).

Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with mean 渭 and variance \({\sigma ^2}\) . Assuming that the sample size n is 16, determine the values of the following probabilities:

\(\begin{align}a.\,\,\,\,P\left( {\frac{1}{2}{\sigma ^2} \le \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \mu } \right)}^2} \le 2{\sigma ^2}} } \right)\\b.\,\,\,\,P\left( {\frac{1}{2}{\sigma ^2} \le \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - {{\bar X}_n}} \right)}^2} \le 2{\sigma ^2}} } \right)\end{align}\)

Consider again the conditions of Exercise 19, and let\({{\bf{\hat \beta }}_{\bf{n}}}\)n denote the M.L.E. of 尾.

a. Use the delta method to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

b. Show that\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}{\bf{ = }}{{\bf{\bar X}}_{\bf{n}}}\), and use the central limit theorem to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

Question:Suppose that a specific population of individuals is composed of k different strata (k 鈮� 2), and that for i = 1,...,k, the proportion of individuals in the total population who belong to stratum i is pi, where pi > 0 and\(\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{\bf{ = 1}}} \). We are interested in estimating the mean value 渭 of a particular characteristic among the total population. Among the individuals in stratum i, this characteristic has mean\({{\bf{\mu }}_{\bf{i}}}\)and variance\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\), where the value of\({{\bf{\mu }}_{\bf{i}}}\)is unknown and the value of\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\)is known. Suppose that a stratified sample is taken from the population as follows: From each stratum i, a random sample of ni individuals is taken, and the characteristic is measured for each individual. The samples from the k strata are taken independently of each other. Let\({{\bf{\bar X}}_{\bf{i}}}\)denote the average of the\({{\bf{n}}_{\bf{i}}}\)measurements in the sample from stratum i.

a. Show that\({\bf{\mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{\bf{\mu }}_{\bf{i}}}} \), and show also that\({\bf{\hat \mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{{\bf{\bar X}}}_{\bf{i}}}} \)is an unbiased estimator of 渭.

b. Let\({\bf{n = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{n}}_{\bf{i}}}} \)denote the total number of observations in the k samples. For a fixed value of n, find the values for which the variance \({\bf{\hat \mu }}\)will be a minimum.

Question:For the conditions of Exercise 2, find an unbiased estimator of \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}\). Hint: \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}{\bf{ = E}}\left( {{{\bf{X}}^{\bf{2}}}} \right){\bf{ - Var}}\left( {\bf{X}} \right)\)
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