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Chapter 8: Q8E (page 513)

Question:Suppose that a random variable X has the geometric distribution with an unknown parameter p (0<p<1). Show that the only unbiased estimator of p is the estimator \({\bf{\delta }}\left( {\bf{X}} \right)\) such that \({\bf{\delta }}\left( {\bf{0}} \right){\bf{ = 1}}\) and \({\bf{\delta }}\left( {\bf{X}} \right){\bf{ = 0}}\) forX>0.

Short Answer

Expert verified

the only unbiased estimator of p is the estimator \(\delta \left( X \right)\) such that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\).

Step by step solution

01

Given information

Therefore, the pdf of X is \(P\left( {X = x} \right) = p{\left( {1 - p} \right)^x},x \ge 0\)

02

Define the unbiased estimator

An estimator \(\delta \left( X \right)\,\,of\,\,g\left( \theta \right)\) is unbiased if \(E\left( {\delta \left( X \right)} \right) = g\left( \theta \right)\) for all possible values of \(\theta \) .

Therefore,

\(\begin{aligned}{}E\left( {\delta \left( X \right)} \right) &= p\\ \Rightarrow \sum\limits_{x = 0}^n {p{{\left( {1 - p} \right)}^{x - 1}}} \delta \left( X \right) &= p\end{aligned}\)

03

Do the required calculation

It is given that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\), therefore, the above equation becomes,

\(\begin{aligned}{}p{\left( {1 - p} \right)^0}\delta \left( 0 \right) + \ldots + p{\left( {1 - p} \right)^n}\delta \left( n \right)\\ &= p \times 1 + 0 + \ldots 0\\ &= p\end{aligned}\)

Therefore, the only unbiased estimator of p is the estimator \(\delta \left( X \right)\) such that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\).

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Most popular questions from this chapter

The study on acid concentration in cheese included a total of 30 lactic acid measurements, the 10 given in Example 8.5.4 on page 487 and the following additional 20:

1.68, 1.9, 1.06, 1.3, 1.52, 1.74, 1.16, 1.49, 1.63, 1.99, 1.15, 1.33, 1.44, 2.01, 1.31, 1.46, 1.72, 1.25, 1.08, 1.25.

a. Using the same prior as in Example 8.6.2 on page 498, compute the posterior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) based on all 30 observations.

b. Use the posterior distribution found in Example 8.6.2 on page 498 as if it were the prior distribution before observing the 20 observations listed in this problem. Use these 20 new observations to find the posterior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)and compare the result to the answer to part (a).

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.)

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\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

Suppose that a random sample is to be taken from the Bernoulli distribution with an unknown parameter,p. Suppose also that it is believed that the value ofpis in the neighborhood of 0.2. How large must a random sample be taken so that\(P\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - p|}}} \right) \ge 0.75\)when p=0.2?

For the conditions of Exercise 5, use the central limit theorem in Sec. 6.3 to find approximately the size of a random sample that must be taken so that \(P\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}} - {\bf{p|}}} \right) \ge 0.95\) whenp=0.2.
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