Ch 9 Interval Estimation
Ch 9. Interval Estimation
Ch 9. Interval Estimation
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<strong>Ch</strong> 9. <strong>Interval</strong> <strong>Estimation</strong><br />
Optimal theory for CI<br />
CI: Length of CI vs Coverage probability<br />
<strong>Ch</strong>apter 9<br />
Intro<br />
Finding <strong>Interval</strong><br />
Estimator<br />
Optimal theory<br />
for CI<br />
Definition<br />
f(x) is a unimodal pdf if f(x) is nondecreasing for x ≤ x ∗ and<br />
nonincreasing for x ≥ x ∗ in which case x ∗ is the mode of the<br />
distribution.<br />
Theorem (Shortest CI based on unimodal distn.)<br />
Let f(x) be a unimodal pdf. If the interval [a, b] satisfies<br />
i. ∫ b<br />
a f(x)dx = 1 − α<br />
ii. f(a) = f(b) > 0<br />
iii. a ≤ x ∗ ≤ b, when x ∗ is a mode of f(x)<br />
Then no other interval estimator satisfying (i) is shorter than<br />
[a, b].
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