Optimal estimator construction

6 The continuous random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\), and the independent continuous random variable \(Y\) has mean \(2 \mu\) and variance \(3 \sigma ^ { 2 }\). Two observations of \(X\) and three observations of \(Y\) are taken and are denoted by \(X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) respectively.

1. Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, \(T _ { 1 }\), of \(\mu\).
2. The estimator \(T _ { 2 }\), where \(T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is an unbiased estimator of \(\mu\). Find the value of the constant \(c\).
3. Determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient.
4. Find the values of the constants \(a\) and \(b\) for which $$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$ is an unbiased estimator of \(\sigma ^ { 2 }\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a parameter.

1. Find \(\mathrm { E } ( X )\). Hence show that \(\frac { 4 } { 3 } X\) is an unbiased estimator of \(k\). Three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), and the largest value of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is denoted by \(M\).
2. Write down an expression for \(\mathrm { P } ( M \leqslant x )\), and hence show that the probability density function of \(M\) is $$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
3. Find \(\mathrm { E } ( M )\) and use your answer to construct an unbiased estimator of \(k\) based on \(M\).

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 \(\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3\) is an unbiased estimator for \(\mu\). [3]

An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. [(b)] Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]

\includegraphics{figure_6} Figure 1 shows a square of side 1 and area \(l^2\) which lies in the first quadrant with one vertex at the origin. A point \(P\) with coordinates \((X, Y)\) is selected at random inside the square and the coordinates are used to estimate \(l^2\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([0, l]\). [You may assume that E\((X^n Y^m) = \) E\((X^n)\)E\((Y^m)\), where \(n\) is a positive integer.]

1. Use integration to show that E\((X^n) = \frac{l^{n+1}}{n+1}\). [3]

The random variable \(S = kXY\), where \(k\) is a constant, is an unbiased estimator for \(l^2\).

1. [(b)] Find the value of \(k\). [3]
2. Show that Var \(S = \frac{7l^4}{9}\). [3]

The random variable \(U = q(X^2 + Y^2)\), where \(q\) is a constant, is also an unbiased estimator for \(l^2\).

1. [(d)] Show that the value of \(q = \frac{3}{2}\). [3]
2. Find Var \(U\). [3]
3. State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(l^2\). [1]

The point (2, 3) is selected from inside the square.

1. [(g)] Use the estimator chosen in part (f) to find an estimate for the area of the square. [1]

TOTAL FOR PAPER: 75 MARKS