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sufficient statistic for a parameter

创建日期：2024-05-02 10:14:34

所属分类：统计学statistics

标签： statistic

sufficient statistic for a parameter

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Ch 7. Sufficiency

Ch 7. Sufficiency
Ch 7.1 A sufficient statistic for a parameter, p433
- Example 7.1
▶ Experiment A: Observe IID sample X1 and X2 from b(1, θ),
0 < θ < 1
▶ Experiment B: Observe X1 and X1 +X2 where
X1, X2
iid∼ b(1, θ)
▶ Experiment C
1st step: Observe Y = X1 +X2
2nd step: Observe X1 given Y = y
Ch 7. Sufficiency
▶ Probability Structure of Experiment A
(x1, x2) (0, 0) (0, 1) (1, 0) (1, 1)
P (x1, x2) (1− θ)2 (1− θ)θ θ(1− θ) θ2
▶ Probability Structure of Experiment B
(x1, x2) (0, 0) (0, 1) (1, 1) (1, 2)
P (x1, x2) (1− θ)2 (1− θ)θ θ(1− θ) θ2
Ch 7. Sufficiency
▶ Experiment C
1st step: Observe Y = X1 +X2
2nd step: Observe X1 given Y = y
▶ Probability Structure of Experiment C
1. P ((X1, X2) = (0, 0)|Y = 0)) = 1
2. P ((X1, X2) = (0, 1)|Y = 1)) = P ((X1, X2) = (1, 0)|Y =
1)) = 0.5
3. P ((X1, X2) = (1, 1)|Y = 2)) = 1
▶ From this probability structure, we can see that the
conditional probability of (X1, X2) given Y does not depend
on θ. This implies that if Y is given, knowing (X1, X2) does
not give any useful information for the inference of θ.
Ch 7. Sufficiency
Definition
Let X1, . . . , Xn be a random sample from pdf f(x; θ), θ ∈ Ω.
T (X) = T (X1, . . . , Xn) is called sufficient statistic for θ ∈ Ω if
(i) Y is a statistic
(ii) Pθ [(X1, . . . , Xn) ∈ A|T (X) = y] does not depend on θ for
each y and all A.
Ch 7. Sufficiency
- Example 7.2 (p.434)
For X1, . . . , Xn
iid∼ b(1, θ), Y = X1 + . . .+Xn is a sufficient
statistic for θ.
Ch 7. Sufficiency
Theorem (Factorization theorem, p436)
Let X1, . . . , Xn be a random sample from pdf f(x; θ), θ ∈ Ω.
Y = T (X) is a sufficient statistic if and only if we can find two
nonnegative function k1 and k2 such that
L(θ) =
n∏
i=1
f(xi; θ) = k1(y, θ)k2(x)
Ch 7. Sufficiency
- Example 7.2 (revisited): X1, . . . , Xn
iid∼ b(1, θ)
Ch 7. Sufficiency
- Example 7.3: X1, . . . , Xn
iid∼ N(µ, 1)
Ch 7. Sufficiency
- Example 7.4 (p.438): X1, . . . , Xn
iid∼ Beta(θ, 1)
Ch 7. Sufficiency
- Example 7.5: X1, . . . , Xn
iid∼ U [0, θ]
Ch 7. Sufficiency
- Fact: one-to-one function of a sufficient statistic for θ is also a
sufficient statistic for θ.
Ch 7. Sufficiency
Ch 7.2 Properties of a sufficient statistic
Theorem (Rao-Blackwell, p.441)
Let X1, . . . , Xn be a random sample from pdf f(x; θ), θ ∈ Ω. If
Y = T (X) is a sufficient statistic for θ, and θˆ1 is an unbiased
estimator of θ, then θˆ2 = E(θˆ1|T (X)) = E(θˆ1) = θ with
V (θˆ2) ≤ V (θˆ1).
Ch 7. Sufficiency
Remarks
▶ For any unbiased estimator θˆ1 for θ, we can always find a
better unbiased estimator θˆ2 = E(θˆ1|u1(X)) whe a sufficient
statistic u1(X) is given.
▶ Although SS is not unique, we can know that a desirable
estimator should be at least a function of a sufficient statistic.
Ch 7. Sufficiency
Theorem
If a unique MLE of θˆ exists, θˆ should be a function of SS.
Ch 7. Sufficiency
Ch 7.3 Completeness and uniqueness
- Fact: MVUE with finite variance is unique when it exists.
Proof.
Let θˆ1 and θˆ2 be MVUE’s of θ. We will show that θˆ1 = θˆ2 or
P (θˆ1 = θˆ2) = 1. For this, define θˆ3 = (θˆ1 + θˆ2)/2. Then θˆ3 is
unbiased and this means V (θˆ3) ≥ V (θˆ1) = V (θˆ2). Hence,
Cov(θˆ1, θˆ2) ≥ V (θˆ1) and this implies V (θˆ1 − θˆ2) = 0.
Ch 7. Sufficiency
Definition
X1, . . . , Xn: random sample from pdf f(x; θ), θ ∈ Ω.
(a) Y = u(X) is a complete statistic for θ if and only if
Eθ(g(Y )) = 0 for all θ ∈ Ω implies Pθ(g(Y ) = 0) = 1 for all θ ∈ Ω
(b) Y = u(X) is a complete sufficient statistic (CSS) for θ if Y is
complete and sufficient for θ.
Ch 7. Sufficiency
Theorem (Lehmann & Scheffe, p.446)
If T is CSS for θ and E(ϕ(T )) = θ, then ϕ(T ) is the MVUE of θ.
- Why this theorem works?
▶ From Rao-Blackwell, we know that a good estimator should be a
function of SS.
▶ If a given sufficient statistic T is also complete, we can say that
h(T ) = 0 whenever Eθ(h(T )) = 0 for all θ ∈ Ω.
▶ When T is CSS for θ, suppose that there are two unbiased
estimators for θ: ϕ(T ) and ψ(T ). Then the completeness of T
implies that ϕ(T ) = ψ(T ).
▶ This means that there is only one unbiased estimator which is a
function of CSS.
▶ How to show the completeness? No general way exists..
Ch 7. Sufficiency
- Example 7.5 (revisited): X1, . . . , Xn
iid∼ U [0, θ], θ > 0
→ From Example 7.5, we know that Yn = max(X1, . . . , Xn) is a
sufficient statistic for θ.
1. Yn is also a complete statistic?
(i) Assume Eθ(g(Yn)) = 0 for all θ > 0
(ii) Show that g(y) = 0
Ch 7. Sufficiency
- Example 7.5 (revisited): X1, . . . , Xn
iid∼ U [0, θ], θ > 0
→ From Example 7.5, we know that Yn = max(X1, . . . , Xn) is a
sufficient statistic for θ.
1. Yn is also a complete statistic? Yes!
2. What is the MVUE of θ?
3. Is there any easy way to show completeness? For some family
of distributions, we can show the completeness.
Ch 7. Sufficiency
- Example 7.6: X1, . . . , Xn
iid∼ Gamma(1, θ), θ > 0
Definition (Laplace transform)
The Laplace transform of f(x) is
Lf(x) =
∫ ∞
0
f(t)e−txdt
provided with
∫∞
0 f(t)e
−txdt <∞
Theorem (Uniqueness of Laplace transform)
Lf(x) = Lg(x) if and only if f(x) = g(x)
Ch 7. Sufficiency
- Example 7.6: X1, . . . , Xn
iid∼ Gamma(1, θ), θ > 0
→ ∑ni=1Xi is a complete and sufficient statistic
Ch 7. Sufficiency
Definition (Exponential family of distributions, p. 449)
f(x; θ) is said to be a member of the regular exponential family if
(i) Support of f(x; θ) does not depend on θ.
(ii) f(x; θ) = exp(p(θ)K(x) +H(x) + q(θ)), where p(θ) is a
nontrivial continuous function of θ.
(iii) p(θ) is a nontrivial continuous function of θ, and K(x) is a
nontrivial function of x
Ch 7. Sufficiency
- Examples
▶ Binomial distributions
▶ Gamma distributions
▶ Normal distributions
▶ Beta distributions
▶ Uniform distributions → NOT exponential family!
Ch 7. Sufficiency
Theorem (p. 450)
If f(x; θ) belongs to the exponential family
(i.e. f(x; θ) = exp(p(θ)K(x) +H(x) + q(θ)), then
(i) q(θ) = − log (∫ exp(p(θ)K(x) +H(x))dx)
(ii) E(K(X)) = −q′(θ)/p′(θ)
(iii)
V (K(X)) = −p
′′(θ)E(K(X)) + q′′(θ)
(p′(θ))2
=
p′′(θ)q′(θ)− q′′(θ)p′(θ)
(p′(θ))3
If p(θ) = θ, then E(K(X)) = −q′(θ) and V (K(X)) = −q′′(θ)
Proof.
Note that
d
dθ
f(x; θ) = (p′(θ)K(x) + q′(θ))f(x; θ)
d2
dθ2
f(x; θ) = (p′′(θ)K(x) + q′′(θ))f(x; θ) + (p′(θ)K(x) + q′(θ))2f(x; θ)
Ch 7. Sufficiency
Ch 7. Sufficiency
- Example 7.7
(i) X ∼ b(n, θ)
Ch 7. Sufficiency
- Example 7.7
(ii) X follows exponential distribution having mean λ.
Ch 7. Sufficiency
Theorem (p.451)
For X1, . . . , Xn
iid∼ f(x; θ), if f(x; θ) is a member of exponential
family, then
∑
K(Xi) is CSS
Ch 7. Sufficiency
- Example 7.7 (revisited)
(i) If X1, . . . , Xn
iid∼ b(1, θ), find the MVUE of θ.
Ch 7. Sufficiency
- Example 7.7(revisited)
(ii) If X1, . . . , Xn is a random sample from the exponential
distribution having mean λ, find the MVUE of λ.
Ch 7. Sufficiency
- Example 6.9 (revisited)
If X1, . . . , Xn
iid∼ Beta(θ, 1), find the MVUE of θ.
In Example 6.12, the MLE of θ is θˆMLE = − n∑ logXi and
E(θˆMLE) = nθn+1 . What is the MVUE of θ?
Ch 7. Sufficiency
▶ Rao-Blackwell+Lehmann-Scheffe
If X1, . . . , Xn
iid∼ f(x; θ), θˆ1 is unbiased, and Y is CSS, then
θˆ2 = E(θˆ1|Y ) is the MVUE of θ.
▶ Remark
1. If your estimator is a function of CSS and unbiased, then your
estimator is the unique MVUE.
2. If you have an unbiased estimator θˆ which is not a function of
CSS, θˆ∗ = E(θˆ|Y ) is the MVUE.
Ch 7. Sufficiency
- Example 7.8
If X1, . . . , Xn
iid∼ b(1, p), find the MVUE of p.
Ch 7. Sufficiency
- Example 7.9
If X1, . . . , Xn is a random sample from the exponential distribution
having mean θ, find the MVUE of η = e−1/θ = P (X1 > 1).
Ch 7. Sufficiency

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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