In Exercise 9.17, suppose that the populations are normally distributed with σ 1 2 = σ 2 2 = σ 2 . Show that ∑ i = 1 n ( X i − X ¯ ) 2 + ∑ i = 1 n ( Y i − Y ¯ ) 2 2 n − 2 is a consistent estimator of σ 2 . 9.17 Suppose that X 1 , X 2 ,…, X n and Y 1 , Y 2 ,…, Y n are independent random samples from populations with means µ 1 and µ 2 and variances σ 1 2 and σ 2 2 , respectively. Show that X ¯ − Y ¯ is a consistent estimator of µ 1 – µ 2.
In Exercise 9.17, suppose that the populations are normally distributed with σ 1 2 = σ 2 2 = σ 2 . Show that ∑ i = 1 n ( X i − X ¯ ) 2 + ∑ i = 1 n ( Y i − Y ¯ ) 2 2 n − 2 is a consistent estimator of σ 2 . 9.17 Suppose that X 1 , X 2 ,…, X n and Y 1 , Y 2 ,…, Y n are independent random samples from populations with means µ 1 and µ 2 and variances σ 1 2 and σ 2 2 , respectively. Show that X ¯ − Y ¯ is a consistent estimator of µ 1 – µ 2.
In Exercise 9.17, suppose that the populations are normally distributed with
σ
1
2
=
σ
2
2
=
σ
2
. Show that
∑
i
=
1
n
(
X
i
−
X
¯
)
2
+
∑
i
=
1
n
(
Y
i
−
Y
¯
)
2
2
n
−
2
is a consistent estimator of σ2.
9.17 Suppose that X1, X2,…, Xn and Y1, Y2,…,Yn are independent random samples from populations with means µ1 and µ2 and variances
σ
1
2
and
σ
2
2
, respectively. Show that
X
¯
−
Y
¯
is a consistent estimator of µ1 –µ2.
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