Expectation and variance of combinations of estimators

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Question statement

Suppose $W,\;\;X,\;\;Y,\;\;$ and $Z$ are i.i.d. variables with mean $\mu=\var{m}$, standard deviation $\sigma=\var{sd}$

Find the expectation and variance of each of the following estimators of $\mu$.

Give any introductory information the student needs.

			Name	Type	Generated Value

cy3	integer	6
cy2	integer	9
cy1	integer	-5
t3	integer	1
correct2	string	C
correct1	string	A
t1	integer	1
unb3	integer	1
unb2	integer	0
unb1	integer	1
s3	integer	1
tw	integer	1
s1	integer	1
s7	integer	-1
tol	number	0.01
cx1	integer	-2
cx2	integer	9
cx3	integer	0
sqsum1	number	93
sqsum3	number	733
sqsum2	number	235
cz2	integer	3
cz3	integer	-21
cz1	integer	0
v1	number	837
v2	number	1.94
v3	number	6597
e1	integer	-10
e3	integer	-10
e2	decimal	dec("-8.79")
a	string	A
c	string	C
b	string	B
tx	integer	0
cw1	integer	8
cw3	integer	16
cw2	integer	8
m	integer	-10
wrong	string	B
sx	integer	1
q	string	X
p	string	Z
su2	integer	29
s2	integer	33
sd	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..9)

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

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cz3
s3
sqsum3

Advice
"Part c)" - prompt

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$\simplify[std]{A = {cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}$

$ \operatorname{E}[A]=\;\;$Gap 0.$\;\;\;\;\operatorname{Var}(A)=\;\;\;$Gap 1.

Input both as integers.

Advice

We use the result throughout these solutions that for independent random variables $R$ and $S$ then $\operatorname{E}[aR+bS]=a \operatorname{E}[R]+b\operatorname{E}[S]$ and $\operatorname{Var}(aR+bS)=a^2\operatorname{Var}(R)+b^2\operatorname{Var}(S)$.

a)
\[\begin{eqnarray*} \operatorname{E}[A] &=& \operatorname{E}[\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\ &=& \simplify[std]{{cw1} * {m} + {cx1} * {m} + {cy1} * {m} + {cz1} * {m}}\\ &=& \var{e1}\\ \\ \\ \operatorname{Var}(A) &=& \operatorname{Var}[\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\ &=& \simplify[std]{ {cw1 ^ 2} * Var(W) + {cx1 ^ 2} * Var(X) + {cy1 ^ 2} * Var(Y) + {cz1 ^ 2} * Var(Z)}\\ &=& \simplify[std]{{cw1 ^ 2} * {sd ^ 2} + {cx1 ^ 2} * {sd ^ 2} + {cy1 ^ 2} * {sd ^ 2} + {cz1 ^ 2} * {sd ^ 2} }\\ &=& \var{v1} \end{eqnarray*} \]

b)
\[\begin{eqnarray*} \operatorname{E}[B] &=& \frac{1}{\var{S2}}\left(\operatorname{E}[\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\right)\\ &=& \frac{1}{\var{S2}}\left(\simplify[std]{{cw2} * {m} + {cx2} * {m} + {cy2} * {m} + {cz2} * {m}}\right)\\ &=& \var{e2}\\ \\ \\ \operatorname{Var}(B) &=& \frac{1}{\var{S2^2}}\left(\operatorname{Var}[\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\right)\\ &=& \frac{1}{\var{S2^2}}\left(\simplify[std]{ {cw2 ^ 2} * Var(W) + {cx2 ^ 2} * Var(X) + {cy2 ^ 2} * Var(Y) + {cz2 ^ 2} * Var(Z)}\right)\\ &=& \frac{1}{\var{S2^2}}\left(\simplify[std]{{cw2 ^ 2} * {sd ^ 2} + {cx2 ^ 2} * {sd ^ 2} + {cy2 ^ 2} * {sd ^ 2} + {cz2 ^ 2} * {sd ^ 2} }\right)\\ &=& \var{v2} \end{eqnarray*} \]
c)
\[\begin{eqnarray*} \operatorname{E}[C] &=& \operatorname{E}[\simplify[std,collectNumbers]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\ &=& \simplify[std,collectNumbers]{{cw3} * {m} + {cx3} * {m} + {cy3} * {m} + {cz3} * {m}}\\ &=& \var{e3}\\ \\ \\ \operatorname{Var}(C) &=& \operatorname{Var}[\simplify[std]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\ &=& \simplify[std]{ {cw3 ^ 2} * Var(W) + {cx3 ^ 2} * Var(X) + {cy3 ^ 2} * Var(Y) + {cz3 ^ 2} * Var(Z)}\\ &=& \simplify[std]{{cw3 ^ 2} * {sd ^ 2} + {cx3 ^ 2} * {sd ^ 2} + {cy3 ^ 2} * {sd ^ 2} + {cz3 ^ 2} * {sd ^ 2} }\\ &=& \var{v3} \end{eqnarray*} \]

We see that $\var{Correct1},\;\;\var{Correct2}\;\;$ are unbiased estimators for $\mu=\var{m}$ as their expectations are $\var{m}$.

The most efficient estimator is $B$ as it has the smallest variance.

f)
Since $\var{p}$ and $\var{q}$ are independent we have:

$\operatorname{E}[\var{p}\var{q}]=\operatorname{E}[\var{p}]\operatorname{E}[\var{q}] = \var{m}\times \var{m} = \var{m^2}$

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Expectation and variance of combinations of estimators

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Expectation and variance of combinations of estimators

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Taxonomy: mathcentre

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Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Generated value: `integer`