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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 ?