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However, there are good reasons for using S as an estimator of in samples from normal distributions, as we will see in the next three chapters when are discuss con dence intervals and hypothesis testing.Barcode barcode library with .netgenerate, create bar code none on .net projectsCHAPTER 7 POINT ESTIMATION OF PARAMETERS Control qr barcode data on c#to paint qr codes and qr codes data, size, image with c#.net barcode sdkSometimes there are several unbiased estimators of the sample population parameter. For example, suppose we take a random sample of size n 10 from a normal population and obtain the data x1 12.8, x2 9.4, x3 8.7, x4 11.6, x5 13.1, x6 9.8, x7 14.1, x8 8.5, x9 12.1, x10 10.3. Now the sample mean is x 12.8 11.04 the sample median is ~ x 10.3 2 11.6 10.95 9.4 8.7 11.6 13.1 10 9.8 14.1 8.5 12.1 10.3Control quick response code size for .net quick response code size with .netand a 10% trimmed mean (obtained by discarding the smallest and largest 10% of the sample before averaging) is xtr1102 8.7 10.98 We can show that all of these are unbiased estimates of . Since there is not a unique unbiased estimator, we cannot rely on the property of unbiasedness alone to select our estimator. We need a method to select among unbiased estimators. We suggest a method in Section 7-2.3. 9.4 9.8 10.3 11.6 8 12.1 12.8 13.1Deploy qr codes with vbusing .net framework tocompose qr codes for asp.net web,windows application7-2.2.NET Crystal bar code integrating on .netgenerate, create barcode none in .net projectsProof That S is a Biased Estimator of Linear implement for .netgenerate, create linear barcode none in .net projects(CD Only)Assign ean / ucc - 13 with .netusing barcode creation for .net vs 2010 crystal control to generate, create gs1-128 image in .net vs 2010 crystal applications.7-2.3 Variance of a Point Estimator PDF 417 generator on .netuse .net barcode pdf417 implementation torender pdf417 for .netSuppose that 1 and 2 are unbiased estimators of . This indicates that the distribution of each estimator is centered at the true value of . However, the variance of these distributions may be different. Figure 7-1 illustrates the situation. Since 1 has a smaller variance than 2, the estimator 1 is more likely to produce an estimate close to the true value . A logical principle of estimation, when selecting among several estimators, is to choose the estimator that has minimum variance. De nition If we consider all unbiased estimators of , the one with the smallest variance is called the minimum variance unbiased estimator (MVUE).Identcode barcode library with .netgenerate, create identcode none in .net projects^ Distribution of 1 Control data matrix 2d barcode data for visual basic.netto print barcode data matrix and datamatrix 2d barcode data, size, image with visual basic barcode sdkFigure 7-1 The sampling distributions of two unbiased estimators 1 and 2 . WinForms Crystal qr code generator on c#generate, create qr code 2d barcode none on .net c# projects^ Distribution of 2 Control barcode 128 size on .netto paint code 128 barcode and barcode standards 128 data, size, image with .net barcode sdk7-2 GENERAL CONCEPTS OF POINT ESTIMATION Control pdf-417 2d barcode data on .netto create pdf 417 and barcode pdf417 data, size, image with .net barcode sdkIn a sense, the MVUE is most likely among all unbiased estimators to produce an estimate that is close to the true value of . It has been possible to develop methodology to identify the MVUE in many practical situations. While this methodology is beyond the scope of this book, we give one very important result concerning the normal distribution.Control pdf417 2d barcode data for c#.net pdf417 2d barcode data in visual c#.netTheorem 7-1Control code128 data with office excelcode 128 data on excel spreadsheetsIf X1, X2, p , Xn is a random sample of size n from a normal distribution with mean and variance 2 , the sample mean X is the MVUE for .Control code 128b image in office wordgenerate, create code 128a none on word documents projectsIn situations in which we do not know whether an MVUE exists, we could still use a minimum variance principle to choose among competing estimators. Suppose, for example, we wish to estimate the mean of a population (not necessarily a normal population). We have a random sample of n observations X1, X2, p , Xn and we wish to compare two possible estimators for : the sample mean X and a single observation from the sample, say, Xi. Note that both X and Xi are unbi2 n from Equation 5-40b and the ased estimators of ; for the sample mean, we have V1X 2 2 variance of any observation is V1Xi 2 . Since V1X 2 V1Xi 2 for sample sizes n 2, we would conclude that the sample mean is a better estimator of than a single observation Xi.Insert qr code 2d barcode with .netusing ssrs toadd qr on asp.net web,windows application7-2.4 Standard Error: Reporting a Point Estimate When the numerical value or point estimate of a parameter is reported, it is usually desirable to give some idea of the precision of estimation. The measure of precision usually employed is the standard error of the estimator that has been used. De nition The standard error of an estimator is its standard deviation, given by 2V1 2 . If the standard error involves unknown parameters that can be estimated, substitution of those values into produces an estimated standard error, denoted by .Sometimes the estimated standard error is denoted by s or se1 2 . Suppose we are sampling from a normal distribution with mean and variance 2 . Now the distribution of X is normal with mean and variance 2 n, so the standard error of X is 1nIf we did not know but substituted the sample standard deviation S into the above equation, the estimated standard error of X would be