STATISTICAL INFERENCE
May be divided into two major areas
PARAMETER
ESTIMATION
HYPOTHESIS
TESTING
POINT ESTIMATION
INTERVAL ESTIMATION
CONFIDENCE
MEANS
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INTERVALS
The decision making
procedure about the
hypothesis
VARIANCES PROPORTIONS
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POINT ESTIMATION
A statistic used to estimate a population parameter  is
called a point estimator for  and is denoted by ˆ .
The numerical value assumed by this statistic when
evaluated for a given sample is called a point estimate for .
There is a difference in the terms :
ESTIMATOR
and
ESTIMATE
is the statistic used to
generate the estimate ;
it is a random variable
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is a number
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We want, the estimator to generate estimates that can be
expected to be close in value to .
We would like :
1. ˆ to be UNBIASED for 
2. ˆ to have a small variance for large sample sizes
In general, If X is a random variable with probability
distribution f X ( x) or pX ( x) , characterized by the unknown
parameter , and if X1, X2, . . . . Xn is a random sample of
size n from X, then the statistic   h  X 1 , X 2 , X n  is called
a point estimator of . note that  is a random variable,
because it is a function of random variable
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Definition : A point estimate of some population parameter ,
is a single numerical value  of a statistic 
Definition : The point estimator  is an unbiased estimator for
the parameter  if E()   .
If the estimator is not unbiased, then the
difference E()  is called the biased of the
estimator 
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VARIANCE AND MEAN SQUARE ERROR OF
A POINT ESTIMATOR
A logical principle of estimation, when selecting
among several estimator, is to chose the
estimator that has minimum variance.
Definition : If we consider all unbiased estimator of , the one
with the smallest variance is called the minimum
variance unbiased estimator (MVUE).
Some times the MVUE is called the UMVUE, where the first
U represents “Uniformly”, meaning “for all ”
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MEAN SQUARE ERROR
Definition : the mean square error of an- estimator  of the
parameter  is defined as :
  
MSE   E   

2
The mean square error can be rewritten as follows :
 
 
 
2
MSE   E   E      E  

 

 
MSE   Var    bias 
2
2
The mean square error is an important criterion for
comparing two estimators.
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Let 1 and 2 be two estimators of the parameter , and let
MSE 1 and MSE 2 be the mean square error of 1 and 2 .
 
 
Then the relative efficiency of 1 to 2 is defined as :
 
MSE   
MSE 1
2
If this relative efficiency is less than
one, we would conclude that 1 is more
efficient estimator of  than 2
Example :
Suppose we wish to estimate the mean  of a population. We
have a random sample of n observations X1, X2, …..Xn and we
wish to compare two possible estimator for  : the sample mean X
and a single observation from the sample, say, Xi,
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Note, both X and Xi are unbiased estimators of  ;
consequently, the MSE of both estimators is simply the
variance.
2
We have MSE X  Var X 
n
2

MSE 1
1
n
 2 

n
MSE  2
 
 
 
Since 1n  1 for sample size n ≥ 2, we would conclude that
the sample mean is a better estimator of  than a single
observation Xi.
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EXERCISES
1. Suppose we have a random sample of size 2n from a
population denoted by X, and E(X) =  and Var X = 2.
2n
n
Let
X1 
1
2n
 X i and X 2 
i 1
1
n
X
i 1
i
be two estimator of . Which is the better estimator of  ?
Explain your choice.
2. Let X1, X2, . . . , X7 denote a random sample from a
population having mean  and variance 2. Consider the
following estimator of  :
X 1  X 2  .....  X 7
1 
;
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2 X1  X 6  X 4
2 
2
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(a) Is either estimator unbiased?
(b) Which estimator is “best” ?
3. Suppose that 1, 2 and 3 are estimators of . We know
that E 1  E 2 and   ,
 
   
2
E 3   , Var 1  12, Var 2  10 and E 3     6


Compare these three estimators. Which do you prefer?
Why?
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4. In a Binomial experiment exactly x successes are
observed in n independent trials. The following two
statistics are proposed as estimators of the proportion
parameter
p : T1 
X
n
and T2
X 1
n 2
Determine and compare the MSE for T1 and T2
5. Let X1, X2, X3 and X4 be a random sample of size 4 from
a population whose distribution is exponential with
unknown parameter .
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a. Which of the following statistics are unbiased
estimators of  ?
T1  16  X1  X 2   13  X 3  X 4 
T2 
T3 
 X1  2 X 2 3 X 3  4 X 4 
5
 X1  X 2  X 3  X 4 
4
b. Among the unbiased estimators of , determine the
one with the smallest variance
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