statistical inference may be divided into two major areas
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STATISTICAL INFERENCEMay be divided into two major
areas
PARAMETER
ESTIMATION
HYPOTHESIS
TESTING
POINT ESTIMATION
INTERVAL ESTIMATIONThe decision making procedure
about the hypothesis
CONFIDENCE INTERVALS
MEANS VARIANCES PROPORTIONS
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POINT ESTIMATIONA statistic used to estimate a population parameter is called a point estimator for and is denoted by .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
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. to be UNBIASED for 2. to have a small variance for large sample sizes In general, If X is a random variable with probability distribution , characterized by the unknown parameter , and if X1, X2, . . . . Xn is a random sample of size n from X, then the statistic is called a point estimator of . note that is a random variable, because it is a function of random variable
( ) ( )X Xf x or p x
1 2, , nh X X X
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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 If the estimator is not unbiased, then the difference is called the biased of the estimator
( ) .E
( )E
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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 ERRORDefinition : the mean square error of an- estimator
of the parameter is defined as :
2MSE E
The mean square error can be rewritten as follows :
2 2
MSE E E E
2MSE Var bias
The mean square error is an important criterion for comparing two estimators.
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Let be two estimators of the parameter , and let be the mean square error ofThen the relative efficiency of is defined as :
1 2and 1 2MSE and MSE
1 2.and 1 2to
If this relative efficiency is less than one, we would
conclude that is more efficient estimator of than 1
2
1
2
MSE
MSE
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 and a single observation from the sample, say, Xi,X
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Note, both and Xi are unbiased estimators of ; consequently, the MSE of both estimators is simply the variance.
X
21
22
1MSE nnMSE
2
We have MSE X Var Xn
Since for sample size n ≥ 2, we would conclude that the sample mean is a better
estimator of than a single observation Xi.
1 1n
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EXERCISES1. Suppose we have a random sample of size 2n from
a population denoted by X, and E(X) = and Var X = 2. Let
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 :
21 1
1 221 1
n n
i in ni i
X X and X X
1 2 7 1 6 41 2
..... 2;7 2
X X X X X X
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(a) Is either estimator unbiased?
(b) Which estimator is “best” ?
3. Suppose that are estimators of . We know
that
1 2 3, and
1 2 ,E E and
23 1 2 3, 12, 10 6E Var Var and E
Compare these three estimators. Which do you prefer? Why?
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5. Let X1, X2, X3 and X4 be a random sample of size 4 from a population whose distribution is exponential with unknown parameter .
11 2 2: X X
n np T and T
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
Determine and compare the MSE for T1 and T2
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b. Among the unbiased estimators of , determine the one with the smallest variance
1 11 1 2 3 46 3T X X X X
1 2 3 42 3 42 5
X X X XT 1 2 3 4
3 4X X X XT
a. Which of the following statistics are unbiased estimators
of ?
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