Transcript POINT ESTIMATION - Middle East Technical University

STATISTICAL INFERENCE
PART I
POINT ESTIMATION
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STATISTICAL INFERENCE
• Determining certain unknown properties of
a probability distribution on the basis of a
sample (usually, a r.s.) obtained from that
distribution
ˆ

Point Estimation: ˆ  5

(
)
Interval Estimation: 3    8
Hypothesis Testing: H 0 :   5
H1 :   5
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STATISTICAL INFERENCE
• Parameter Space (): The set of all
possible values of an unknown parameter,
; .
• A pdf with unknown parameter: f(x; ), .
• Estimation: Where in ,  is likely to be?
{ f(x; ),  }
The family of pdfs
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STATISTICAL INFERENCE
• Statistic: A function of rvs (usually a
sample rvs in an estimation) which does not
contain any unknown parameters.
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X , S , etc
• Estimator of an unknown parameter :ˆ
A statistic used for estimating .
ˆ : estimator  U  X 1 , X 2 , , X n 
X : Estimator

An observed value
x : Estimate : A particular value of an estimator
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METHODS OF ESTIMATION
Method of Moments Estimation,
Maximum Likelihood Estimation
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METHOD OF MOMENTS
ESTIMATION (MME)
• Let X1, X2,…,Xn be a r.s. from a population
with pmf or pdf f(x;1, 2,…, k). The MMEs
are found by equating the first k population
moments to corresponding sample moments
and solving the resulting system of
equations.
Population Moments
k  E  X k 
Sample Moments
1n k
Mk   Xi
n i1
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METHOD OF MOMENTS
ESTIMATION (MME)
1  M 1
1n
E  X    Xi
n i1
2  M 2
3  M 3
1n 2
E  X    Xi
n i1
1
E  X    X i3
n i1
2
3
so on…
n
Continue this until there are enough equations
to solve for the unknown parameters.
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EXAMPLES
• Let X~Exp().
• For a r.s of size n, find the MME of .
• For the following sample (assuming it is
from Exp()), find the estimate of :
11.37, 3, 0.15, 4.27, 2.56, 0.59.
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EXAMPLES
• Let X~N(μ,σ²). For a r.s of size n, find the
MMEs of μ and σ².
• For the following sample (assuming it is
from N(μ,σ²)), find the estimates of μ and
σ²:
4.93, 6.82, 3.12, 7.57, 3.04, 4.98, 4.62,
4.84, 2.95, 4.22
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DRAWBACKS OF MMES
• Although sometimes parameters are
positive valued, MMEs can be negative.
• If moments does not exist, we cannot find
MMEs.
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MAXIMUM LIKELIHOOD
ESTIMATION (MLE)
• Let X1, X2,…,Xn be a r.s. from a population
with pmf or pdf f(x;1, 2,…, k), the
likelihood function is defined by
L 1 , 2 ,
, k x1 , x2 ,
, xn   f  x1 , x2 ,
, xn ;1 , 2 ,
  f  xi 1 , 2 ,
, k 
  f  xi ;1 , 2 ,
, k 
n
i 1
n
i 1
, k 
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MAXIMUM LIKELIHOOD
ESTIMATION (MLE)
• For each sample point (x1,…,xn), let
ˆ1  x1 ,..., xn  , ,ˆk  x1 ,..., xn 
be a parameter value at which
L(1,…, k| x1,…,xn) attains its maximum as a
function of (1,…, k), with (x1,…,xn) held fixed. A
maximum likelihood estimator (MLE) of
parameters (1,…, k) based on a sample (X1,…,Xn)
is
ˆ1  x1 ,..., xn  , ,ˆk  x1 ,..., xn 
• The MLE is the parameter point for which the
observed sample is most likely.
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EXAMPLES
• Let X~Bin(n,p). One observation on X is available,
and it is known that n is either 2 or 3 and p=1/2
or 1/3. Our objective is to estimate the pair (n,p).
x
0
(2,1/2)
1/4
(2,1/3)
4/9
(3,1/2)
1/8
(3,1/3)
8/27
Max. Prob.
4/9
1
1/2
4/9
3/8
12/27
1/2
2
1/4
1/9
3/8
6/27
3/8
3
0
0
1/8
1/27
1/8
(2,1/ 3) if x  0
(2,1/ 2) if x  1

n
,
p
x

ˆ
ˆ
   
(3,1/ 2) if x  2
(3,1/ 2) if x  3
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MAXIMUM LIKELIHOOD
ESTIMATION (MLE)
• It is usually convenient to work with the logarithm
of the likelihood function.
• Suppose that f(x;1, 2,…, k) is a positive,
differentiable function of 1, 2,…, k. If a
supremum ˆ1 ,ˆ2 , ,ˆk exists, it must satisfy the
likelihood equations
 ln L 1 , 2 ,
, k ; x1 , x2 ,
 j
, xk 
 0, j  1, 2,..., k
• MLE occurring at boundary of  cannot be
obtained by differentiation. So, use inspection.
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MLE
• Moreover, you need to check that you are
in fact maximizing the log-likelihood (or
likelihood) by checking that the second
derivative is negative.
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EXAMPLES
1. X~Exp(), >0. For a r.s of size n, find the
MLE of .
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EXAMPLES
2. X~N(,2). For a r.s. of size n, find the
MLEs of  and 2.
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EXAMPLES
3. X~Uniform(0,), >0. For a r.s of size n,
find the MLE of .
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INVARIANCE PROPERTY OF THE
MLE
• If ˆ is the MLE of , then for any function
(), the MLE of () is  ˆ .
Example: X~N(,2). For a r.s. of size n, the
MLE of  is X . By the invariance property
of MLE, the MLE of 2 is X 2 .
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ADVANTAGES OF MLE
• Often yields good estimates, especially for
large sample size.
• Usually they are consistent estimators.
• Invariance property of MLEs
• Asymptotic distribution of MLE is Normal.
• Most widely used estimation technique.
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DISADVANTAGES OF MLE
• Requires that the pdf or pmf is known except the
value of parameters.
• MLE may not exist or may not be unique.
• MLE may not be obtained explicitly (numerical or
search methods may be required.). It is sensitive
to the choice of starting values when using
numerical estimation.
• MLEs can be heavily biased for small samples.
• The optimality properties may not apply for small
samples.
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SOME PROPERTIES OF
ESTIMATORS
• UNBIASED ESTIMATOR (UE): An
estimator ˆ is an UE of the unknown
parameter , if
E ˆ    for all   
Otherwise, it is a Biased Estimator of .
Bias ˆ   E ˆ   

ˆ   0.

If ˆ is UE of , Bias


Bias of
ˆ for estimating 
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SOME PROPERTIES OF
ESTIMATORS
• ASYMPTOTICALLY UNBIASED
ESTIMATOR (AUE): An estimator ˆ is
an AUE of the unknown parameter , if
ˆ  0
Bias ˆ   0 but lim
Bias


n


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SOME PROPERTIES OF
ESTIMATORS
• CONSISTENT ESTIMATOR (CE): An
estimator ˆ which converges in probability
to an unknown parameter  for all  is
called a CE of .
p
ˆ
 
 .
For large n, a CE tends to be closer to
the unknown population parameter.
• MLEs are generally CEs.
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EXAMPLE
For a r.s. of size n,
E  X     X is an UE of  .
By WLLN,
p
X 

 X is a CE of  .
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MEAN SQUARED ERROR (MSE)
• The Mean Square Error (MSE) of an
estimator ˆ for estimating  is

MSE ˆ   E   ˆ   Var ˆ   Bias ˆ 
2



2
If MSE ˆ  is smaller, ˆ is the better

estimator of .
For two estimators, ˆ1 and ˆ2 of  , if
MSE ˆ1   MSE ˆ2  , 


ˆ1 is better estimator of  .
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MEAN SQUARED ERROR
CONSISTENCY
• Tn is called mean squared error
consistent (or consistent in quadratic
mean) if E{Tn}20 as n.
Theorem: Tn is consistent in MSE iff
i) Var(Tn)0 as n.
ii ) lim E Tn    .
n 
• If E{Tn}20 as n, Tn is also a CE of .
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EXAMPLES
X~Exp(), >0. For a r.s of size n, consider
the following estimators of , and discuss
their bias and consistency.
n
n
 Xi
T1  i 1
n
 Xi
,
T2  i 1
n 1
Which estimator is better?
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