CHAPTER 1: An introduction to time series and forecasting
CHAPTER 1: An introduction to time series and forecasting
CHAPTER 1: An introduction to time series and forecasting
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The variance of ˆµ is<br />
we have V ar(ˆµ) → 0 as n → ∞.<br />
V ar(ˆµ) = 1 n 2 (V ar(X 1) + V ar(X 2 ) + ...V ar(X n )) = σ2<br />
n<br />
Imagine the situation where all the X i ’s are “perfectly” correlated, i.e.<br />
Cov(X i , X j ) = σ 2<br />
Corr(X i , X j ) = 1<br />
We still estimate µ by<br />
ˆµ = (X 1 + X 2 + · · · + X n )/n<br />
the variance is then<br />
V ar(ˆµ) = 1 n 2 V ar(X 1 + X 2 + · · · + X n )<br />
= 1 n∑<br />
n { V ar(X 2 i ) + 2 ∑ Cov(X i , X j )}<br />
i=1<br />
i
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