Download Chapter 5 Confidence Estimation

Chapter 5
Confidence Estimation
Let (X , BX , P ), P ∈ {Pθ |θ ∈ Θ} be sample space and {Θ, BΘ , µ} be measure
space with some σ−finite measure µ.
5.1
Uniformly Most Accurate Families of Confidence Sets
Def. 5.1.1: Let S : X → BΘ be a measurable function. A family {S(x)|x ∈
X }, where S(x) depends on x but not on θ, is called a family of random sets.
If Θ ⊂ IR and S(x) is an interval (θ(x), θ(x)), where θ and θ are functions
of x alone, then we call S(x) a random interval with θ(x) and θ(x) as lower
and upper bounds, respectively. θ(x) may be −∞ and θ(x) may be +∞.
The problem of confidence estimation consists in finding a function S : X →
BΘ such that for a given α
Pθ {S(X) 3 θ} ≥ 1 − α .
(5.1)
Def. 5.1.2: Let Θ ⊂ IR and 0 < α < 1. A function θ(X) satisfying
Pθ {θ(X) ≤ θ} ≥ 1 − α for all θ ∈ Θ.
(5.2)
is called a lower confidence bound for θ at confidence level 1 − α.
The quantity
inf Pθ {θ(X) ≤ θ}
(5.3)
θ∈Θ
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CHAPTER 5. CONFIDENCE ESTIMATION
is called confidence coefficient. A function θ(X) satisfying
Pθ {θ(X) ≥ θ} ≥ 1 − α for all θ ∈ Θ
is called an upper confidence bound for θ at confidence level 1 − α
and the quantity
inf Pθ {θ ≤ θ(X)}
θ∈Θ
confidence coefficient.
Def 5.1.3: A function θ(θ) that minimizes
Pθ {θ(X) ≤ θ0 } Pθ {θ0 ≤ θ(X)} for all θ0 < θ
(θ0 > θ)
subject to (5.2) is called a uniformly most accurate (UMA) lower (upper) confidence bound at confidence level 1 − α.
If S(X) is of the form S(X) = (θ(X, θ(X)) such that
Pθ {θ(X) ≤ θ ≤ θ(X)} ≥ 1 − α for all θ ∈ Θ,
we call it a confidence interval at confidence level 1 − α and
inf Pθ {θ(X) < θ < θ(X)}
θ
the confidence coefficient associated with the random interval.
Def. 5.1.4: A family of 1 − α level confidence sets {S(x)} is called a
UMA family of confidence sets at level 1 − α if
Pθ0 {S(X) contains θ} ≤ Pθ0 {S 0 (X) contains θ}
for all θ, θ0 ∈ Θ and 1 − α level family {S 0 (X} of confidence sets.
Theorem 5.1.1: Let Θ ⊂ IR be an interval and T (X; θ) be such that for
each θ, T as a measurable function is strictly monotone (antitone) in θ at
every x ∈ X . Let Λ be the range of T and for every λ ∈ Λ and x ∈ X let
the equation λ = T (x; θ) be solvable.
If the distribution of T (X; θ) is independent of θ, then one can construct a
confidence interval for θ at any level.
5.2. CONFIDENCE INTERVALS OF SHORTEST-LENGTH
5.2
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Confidence Intervals of Shortest-Length
Def. 5.2.1: A random variable T (X; θ), whose distribution is independent
of θ, is called a pivot.
Remark: An alternative to minimizing the length of a given confidence
interval consists in the minimization of the expected length Eθ [θ(X) − θ(X)].
In general there does not exist an element in the class of 1 − α confidence
sets which minimizes Eθ [θ(X) − θ(X)] for all θ ∈ Θ.
On the other hand, procedures based on a pivot are also applicable for finding
CI’s with minimal expected length.
5.3
Relation between Confidence Estimation
Hypotheses Testing
In the sequel we consider only nonrandomized tests and introduce the short
hand notation H0 (θ0 ) for H0 : θ = θ0 and H1 (θ1 ) for the alternative, which
may be one or two-sided.
Theorem 5.3.1: Let A(θ0 ), θ0 ∈ Θ, denote the region of acceptance of
an α−level test of H0 (θ0 ), and for each x ∈ X let
S(x) = {θ|x ∈ A(θ), θ ∈ Θ},
(A(θ0 ) ∈ BX , S(x) ∈ BΘ ). Then S(x) is a family of confidence sets for
θ at confidence level 1 − α. If, moreover, A(θ0 ) is UMP for the problem
(α, H0 (θ0 ), H1 (θ0 )), then S(X) minimizes
Pθ {S(X) 3 θ0 } for all θ ∈ H1 (θ0 ),
among all 1 − α level families of confidence sets.
Remark: In view of Def. 5.1.4, the family of confidence sets associated
with a UMP acceptance region is a UMA family at level 1 − α.
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5.4
CHAPTER 5. CONFIDENCE ESTIMATION
Unbiased Confidence Sets
Def. 5.4.1: A family {S(x)} of confidence sets for a parameter θ is said to
be unbiased at confidence level 1 − α if
(i) Pθ {S(X) contains θ} ≥ 1 − α and
(ii) Pθ0 {S(X) contains θ} ≤ 1 − α for all θ, θ0 ∈ Θ.
If S(X) is an interval satisfying (i) and (ii), we call it a 1 − α level unbiased
confidence interval. If a family of unbiased 1 − α confidence sets is UMA
in the class of all 1 − α level unbiased confidence sets, we call it a UMA
unbiased (UMAU) family of confidence sets at level 1 − α.
If S ∗ (X) satisfies (i) and (ii) and minimizes Pθ {S(X) contains θ0 } for all
θ, θ0 ∈ Θ among all unbiased families of confidence sets at level 1 − α , then
S ∗ (X) is a UMAU family of confidence sets at level 1 − α.
According to the above definition a family S(X) of confidence sets for a parameter θ is unbiased at level 1 − α if S(X) traps the true parameter value
with probability at least 1 − α and S(X) traps a false parameter value with
a probability at most 1 − α. Hence un unbiased set traps a true parameter
value more often than it does a false one.
Theorem 5.4.1: Let A(θ0 ) be the acceptance region of a UMPU size α
test of H0 (θ0 ) : θ = θ0 against H1 (θ0 ) : θ 6= θ0 for each θ0 . Then S(x) = {θ ∈
Θ : x ∈ A(θ)} is a UMAU family of confidence sets at level 1 − α.
Remarks
(1) The concept of unbiasedness is most suitable in situations where UMP
tests do not exist. This is, e.g., the case, if Θ consists of points (θ, τ ),
both unknown, and one is interested in obtaining confidence sets for θ
alone. The parameter τ is usually refered to as a nuisance parameter.
(2) Shortest-length confidence intervals do not exist for most commonly
used distributions. The restriction to the unbiased family of confidence
intervals makes it often possible ot obtain 1−α level confidence intervals
5.4. UNBIASED CONFIDENCE SETS
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that have uniformly minimum expected length among all 1 − α level
unbiased confidence intervals.
Theorem 5.4.2: Let Θ ⊂ IR be an interval, let fθ be a µ−density for
P = {Pθ |θ ∈ Θ} µ and let S(X) = (θ(X), θ(X)) be a family of 1 − α level
confidence intervals with stochastically finite length, i.e. Pθ (θ(X) − θ(X) <
∞) = 1. Then
Z
[(θ(x) − θ(x)]fθ (x)µ(dx) =
Z
θ0 6=θ
Pθ {S(X) contains θ0 }dθ0
for all θ ∈ Θ.
The Theorem says that the expected length of the confidence interval is
the probability that S(X) includes θ, averaged over all false values of θ.