Study Guide – STAT 543, Final Exam
The final is comprehensive but below is a rough outline of the new material covered on the final exam.
The new material (i.e., material since exam II) mostly corresponds to material in the textbook from
Chapter 9. You should also look over the study guides for exams I & II as well.
Topics under Interval Estimation
1. General definitions
(a) To estimate a real-valued θ ∈ Θ ⊂ R: Interval estimator IX = [L(X ), U (X )] where
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L(X ) ≤ U (X ); upper interval estimator (−∞, U (X )] (or 1-sided upper intervals); lower
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interval estimators [L(X ), ∞) (or 1-sided lower intervals);, coverage probability of a interval
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estimator Pθ (θ ∈ IX ), confidence coefficient (C.C.) minθ∈Θ Pθ (θ ∈ IX );confidence intervals
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(CIs) (i.e., an interval estimator with a C.C. level)
(b) If θ ∈ Θ ⊂ Rp is a vector of parameter values (i.e., θ = (θ1 , . . . , θp )) instead of a single realvalue, then we have “confidence sets/regions” C(X ) for θ (where C(X ) is not an interval
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subset of R if θ is a vector, instead C(X ) ⊂ Rp ). Also, note that, even if θ is real-valued
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(not a vector), C(X ) can denote a “confidence set” for θ (i.e., a set of plausible values of θ
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determined by the data) which could be an interval C(X ) = IX but doesn’t always have to
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be (i.e., could be just some subset C(X ) ⊂ R when θ is real-valued).
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2. Methods of finding interval/set estimators
(a) Inverting a test
i. For each θ0 , let ϕθ0 (X ) be a simple (i.e, either 0,1) test of size α for testing H0 : θ = θ0
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vs H1 : θ 6= θ0 ; get the acceptance region A(θ0 ) = {X : ϕθ0 (X ) = 0} for ϕθ0 (X ); then
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get a confidence set for θ with C.C 1 − α as C(X ) = {θ0 ∈ Θ : X ∈ A(θ0 )}.
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ii. If p = 1 (or θ is real-valued), inverting a test of H0 : θ = θ0 vs H1 : θ > θ0 (i.e., one-sided
alternative) often gives a one-sided lower interval [L(X ), ∞); if θ is real-valued, inverting
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a one-sided test of H0 : θ = θ0 vs H1 : θ < θ0 often gives a one-sided upper interval
(−∞, U (X )].
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(b) Pivotal Quantities
i. Pivot or pivotal quantity Q(X , θ) (i.e., a function of data and parameter) has same
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distribution for any θ ∈ Θ ⊂ Rp (see examples from class)
ii. Confidence region C(X ) for θ ∈ Θ ⊂ Rp based on Q(X , θ): pick −∞ ≤ a < b ≤ ∞ where
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P (a ≤ Q(X , θ) ≤ b) = 1 − α and get C(X ) = {θ ∈ Θ : a ≤ Q(X , θ) ≤ b} as confidence
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set with C.C. 1 − α
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iii. Asymptotically pivotal quantity Qn ≡ Qn (X1 , . . . , Xn , θ), where Qn −→ Q as n → ∞
for any θ ∈ Θ ⊂ Rp ; distribution of Q doesn’t depend on θ
iv. Large sample confidence sets based on asymptotically pivotal quantities (see examples
from class based on CLT or likelihood-ratio statistics)
v. Variance stabilizing transformation g(·) for a sequence of real-valued estimators {θ̂n } of
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θ ∈ R, where n(θ̂n − θ) −→ N {0, σ 2 (θ)} for θ ∈ Θ ⊂ R (limiting variance σ 2 (θ) > 0
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may depend on θ), satisfies g 0 (θ)σ(θ) = 1 for any θ. Then, by delta method, n(g(θ̂n ) −
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g(θ)) −→ N {0, 1} for any θ.
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vi. Mood, Graybill, Boes’ Method (Tail area method). This involves finding a statistic T
with cdf FT (t|θ) = P (T ≤ t|θ) (here θ ∈ Θ ⊂ R is real-valued). The cdf FT (t|θ) should
be an increasing or decreasing function of θ (for any fixed possible value t of T ). Let
0 < α1 , α2 < 1 such that α1 + α2 = α ∈ (0, 1) (e.g., could take α1 = α2 = α). For any
possible value t of T , fix t and define a confidence interval [θL (t), θU (t)] (i.e, endpoints
are possible values for θ as a function of t) by FT (t|θU (t)) = P (T ≤ t|θU (t)) = α1 ,
P (T ≥ t|θL (t)) = α2 if F (t|θ) is decreasing in θ (or as FT (t|θL (t)) = P (T ≤ t|θL (t)) =
α1 , P (T ≥ t|θU (t)) = α2 if F (t|θ) is increasing in θ). Then for each value t of T , a CI
[θL (t), θU (t)] for θ is defined. It holds that C.C.≡ minθ∈Θ P (θ ∈ [θL (T ), θU (T )]) ≥ 1 − α
if T is a discrete r.v. for any θ and that C.C.≡ minθ∈Θ P (θ ∈ [θL (T ), θU (T )]) = 1 − α if
T is continuous r.v. for any θ.
(c) Bayes intervals/sets
i. Suppose θ ∈ Θ ⊂ R has a prior pdf. Get the posterior pdf fθ|x given the data x and,
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with respect to x, define a 1 − α credible set C(x) where C(x) fθ|x (θ)dθ = 1 − α. For
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any possible data X , C(X ) represents a Bayesian interval/set estimator of θ.
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ii. C(x) is a highest posterior density (HPD) credible 1 − α credible set if, for each possible
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x, C(x) = {θ ∈ Θ : fθ|x (θ) ≥ c} for some c > 0 chosen so that C(x) fθ|x (θ)dθ = 1 − α.
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(d) (Non-parametric) Bootstrap CIs.
i. Let Tn = t(X1 , . . . , Xn , θ) be a function of the data and a population parameter θ;
bootstrap version Tn∗ = t(X1∗ , . . . , Xn∗ , θ̂n ) based on bootstrap sample X1∗ , . . . , Xn∗ and
an estimator θ̂n of θ based on X1 , . . . , Xn .
ii. Understand the main idea of the bootstrap CI (i.e, re-creating sampling process resulting
in Tn and use distribution of bootstrap values Tn∗ to get an idea about the distribution
of Tn ; approximate P (Tn ≤ x) with P (Tn∗ ≤ x|X1 , . . . , Xn ) to calibrate CIs)
3. Evaluating Interval Estimators
(a) At a desired C.C. or confidence level, a “good” CI for a real-valued parameter θ, say C(X ) ≡
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IX = [L(X ), U (X )], should have a small expected length, i.e., smallest possible Eθ {U (X ) −
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L(X )}. (see examples from class)
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(b) If C(X ) is an estimator set (not necessarily an interval) for a parameter vector θ ∈ Rp , p > 1,
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then we want C(X ) to have small expected area or volume.
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