Finding unbiased estimator constraints

6. A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\).

1. Show that \(\frac { 2 } { 3 } X _ { 1 } - \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(\mu\). An unbiased estimator for \(\mu\) is given by \(\hat { \mu } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
2. Show that \(\operatorname { Var } ( \hat { \mu } ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\).
3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { \mu }\) has minimum variance.

7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),

1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
4. Find the variance of \(p _ { 2 }\).
5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.

6. A continuous uniform distribution on the interval \([ 0 , k ]\) has mean \(\frac { k } { 2 }\) and variance \(\frac { k ^ { 2 } } { 12 }\). A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from this distribution.

1. Show that \(\frac { 2 } { 3 } X _ { 1 } + \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(k\). An unbiased estimator for \(k\) is given by \(\hat { k } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
2. Show that \(\operatorname { Var } ( \hat { k } ) = \left( a ^ { 2 } - 2 a + 2 \right) \frac { k ^ { 2 } } { 6 }\)
3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { k }\) has minimum variance, and calculate this minimum variance.

6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]

1. Show that \(a + b = 1\)
2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
4. 1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
   2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)