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 θ is unknown (0 ≤ θ ≤ 1).
a. Verify that the sum of the five given probabilities is 1 for every value of θ.
b. Consider an estimator δc(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 θ.
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.
