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Chapter 7: Q9E (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:

\(\begin{array}{c}{\bf{f}}\left( {{\bf{x|\theta }}} \right){\bf{ = \theta }}{{\bf{x}}^{{\bf{\theta - 1}}\,\,\,}}{\bf{,0 < \theta < 1}}\\{\bf{ = 0}}\,\,\,\,{\bf{otherwise}}\end{array}\)

Also, suppose that the value of 胃 is unknown (胃 > 0). Find the M.L.E. of 胃

Short Answer

Expert verified

\(\frac{{ - n}}{{\sum\limits_{i = 1}^n {\log {x_i}} }}\)

Step by step solution

01

Given information

\({X_1},...,{X_n}\)form a random sample and the pdf is

\(\begin{array}{c}f\left( {x|\theta } \right) = \theta {x^{\theta - 1\,\,\,}},0 < \theta < 1\\ = 0\,\,\,\,{\rm{otherwise}}\end{array}\)

We need to calculate the M.L.E. of 胃

02

Step-2: Calculation of the M.L.E. of θ

Since \(0 < \theta < 1\) the likelihood function is as follows

\(f\left( {x|\theta } \right) = {\theta ^n}{\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)^{\theta - 1}}\)

Taking logarithm and differentiating with respect to \(\theta \) we get

\(\frac{{\partial L\left( \theta \right)}}{{\partial \theta }} = \frac{n}{\theta } + \sum\limits_{i = 1}^n {\log {x_i}} \)

Now for maximum value we have \(\frac{{\partial L\left( \theta \right)}}{{\partial \theta }} = 0\)

\( \Rightarrow \hat \theta = \frac{{ - n}}{{\sum\limits_{i = 1}^n {\log {x_i}} }}\)

Hence the M.L.E. is \(\hat \theta = \frac{{ - n}}{{\sum\limits_{i = 1}^n {\log {x_i}} }}\)

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Most popular questions from this chapter

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.

Question: Consider again the conditions in Exercise 2, but suppose also that it is known that\(\)\(\frac{{\bf{1}}}{{\bf{2}}} \le {\bf{p}} \le \frac{{\bf{2}}}{{\bf{3}}}\). If the observations in the random sample of 70 purchases are as given in Exercise 2, what is the M.L.E. of p?

Question: Let \({{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}\) be distinct numbers. Let Y be a discrete random variable with the following p.f.:

\(\begin{array}{c}{\bf{f}}\left( {\bf{y}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\,{\bf{if}}\,{\bf{y}} \in \left\{ {{{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}} \right\}\\ = 0\,otherwise\end{array}\)

Prove that Var(Y ) is given by Eq. (7.5.5).

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from an exponential distribution for which the value of the parameter 尾 is unknown (尾 > 0). Find the M.L.E. of 尾.

Suppose that the number of defects in a 1200-foot roll of magnetic recording tape has a Poisson distribution for which the value of the mean 胃 is unknown and that the prior distribution of 胃 is the gamma distribution with parameters 伪 = 3 and 尾 = 1. When five rolls of this tape are selected at random and inspected, the numbers of defects found on the rolls are 2, 2, 6, 0, and 3. Determine the posterior distribution of 胃.
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