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# Point Estimation

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Point Estimation

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### Point Estimation

1. 1. 2.1 Point Estimation<br />
2. 2. POINT ESTIMATION<br />POINT ESTIMATE :-<br />An estimate of a population parameter given by a <br /> single number is called point estimate<br />POINT ESTIMATOR :-<br />A point estimator is a statistic for Estimating the <br /> population Parameter ө and will be denoted by ө*<br />
3. 3. Example<br />Problem of point estimation of the population mean µ :-<br />The statistic chosen will be called a point estimator for µ<br />Logical estimator for µ is the Sample mean <br />Hence µ* = <br />
4. 4. UNBIASED ESTIMATOR<br />Unbiased Estimator:-<br />If the mean of sampling distribution of a Statistic equals<br /> the corresponding Population Parameter,the Statistic is<br /> called an Unbiased Estimator of the Parameter<br /> i.e E(ө*) = ө<br />Biased Estimator:-<br /> If E(ө*)≠ ө<br /> i.e Estimator is not Unbiased.<br />Bias Of Estimator<br /> Bias of Estimator = E(ө*) - ө<br />
5. 5. STANDARD ERROR OF THE MEAN<br /> Let denote the Sample mean based on a Sample of size n drawn from a distribution with standard deviation σ.The Standard deviation of is given by<br />σ /<br />and is called standard error of the mean<br />
6. 6. METHODS FOR FINDINGS ESTIMATORS:-<br />METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />METHOD OF MOMENTS<br />
7. 7. METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />LIKELIHOOD FUNCTION:-<br />Let x1,x2,….xn be a random sample of size n from a population with density function f(x) and parameter ө.Then the likelihood function of the sample value x1,x2,…..xn is denoted by L , is their joint density function given by<br />L(ө)= f(x1) f(x2)….. f(xn)<br />
8. 8. METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />Principal of Maximum likelihood consist in finding an<br /> estimator (of the parameter) which maximize L. thus if their<br /> exist function<br />ө*=ө*(x1,x2,x3,….xn)<br />Of the sample values which maximizes L then ө* is taken <br />as an Estimator of ө.<br />
9. 9. METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />Thus ө* is the solution ,if any of<br />The eqn (1) can be rewritten as<br />
10. 10. METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />Since L &gt;0, so is Log L which shows that L and Log L<br /> attains its extreme values at the same value of ө* which is <br />called maximum likelihood estimator.<br />Note:-<br />Eqn (3) is more convenient from practical point of view<br />
11. 11. METHOD OF MAXIMUM LIKELIHOOD ESTIMATION<br />The likelihood equation for estimating λis<br />Thus the M.L.E for λ is the sample mean.<br />
12. 12. METHOD OF MOMENTS<br />METHOD<br /> Let f(x,ө1,ө2,…..өk) be the density function of the parent<br /> population with k parameter<br />If µr’ denotes r th moment about origin then<br />
13. 13. STEPS OF METHOD OF MOMENTS<br />Let x1,x2,……,xn be random sample of size n from<br /> the given population<br />Step 1:-<br />solve k equations (1)<br />for ө1,….,өk in terms of µ1’,……,µk’<br />Step2:-<br />Replace these moments<br />µr’ r =1,2,….,k by the sample moments m1’,m2’,….,mk’.<br />i.e if өi*= өi(µ1’*,µ2’*,……,µk’*)<br /> =өi(m1’,m2’, …..,mk’)<br /> i=1,2,…,k<br />Step3:-<br />ө1*,ө2*,……,өk* are the required estimators<br />
14. 14. ERROR OF ESTIMATE<br />When we use a sample mean to estimate the population mean, we know<br /> that although we are using a method of estimation which has certain<br /> desirable properties, the chances are slim, virtually nonexistent, that the<br /> estimate will actually equal to population mean .<br />Error of estimate is the difference between the estimator and the quantity it is<br /> supposed to estimate.<br />is t.he error of estimate for population mean<br />To examine this error, let us make use of the fact that for large n<br />is a random variable having approximately the standard normal distribution <br />
15. 15. STEPS OF METHOD OF MOMENTS<br />
16. 16. STEPS OF METHOD OF MOMENTS<br />Figure: The large sample <br />distribution of <br />1- <br />/2<br />/2<br />z/2<br />- z/2<br />0<br />As shown in Figure, we can assert with probability 1 -  that the<br />the inequality <br />will be satisfied or that <br />where z/2 is such that the normal curve area to its right equals /2.<br />
17. 17. Determination of sample size:<br />Suppose that we want to use the mean of a large random sample to estimate the mean of population and we want to be able to assert with probability 1 -  that the error will be at most prescribed quantity E. The sample size can be determined by <br />