Mathematical Methods for Physics and Engineering - Matematica.NET

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONSor all of the quantities a and they may be unknown. When this occurs, one mustsubstitute estimated values of any unknown quantities into the expression for σâin order to obtain an estimated standard error ˆσâ. One then quotes the result asa = â ± ˆσâ.◮Ten independent sample values x i , i =1, 2,...,10, are drawn at random from a Gaussiandistribution with standard deviation σ =1. The sample values are as follows (to two decimalplaces):2.22 2.56 1.07 0.24 0.18 0.95 0.73 −0.79 2.09 1.81Estimate the population mean µ, quoting the standard error on your result.We have shown in the final worked example of subsection 31.3.1 that, in this case, ¯x isa consistent, unbiased, minimum-variance estimator of µ and has variance V [¯x] =σ 2 /N.Thus, our estimate of the population mean with its associated standard error isˆµ = ¯x ± √ σ =1.11 ± 0.32.NIf the true value of σ had not been known, we would have needed to use an estimatedvalue ˆσ in the expression for the standard error. Useful basic estimators of σ are discussedin subsection 31.4.2. ◭It should be noted that the above approach is most meaningful for unbiasedestimators. In this case, E[â] =a and so σâ describes the spread of â-values aboutthe true value a. For a biased estimator, however, the spread about the true valuea is given by the root mean square error ɛâ, which is defined byɛ 2 â = E[(â − a)2 ]= E[(â − E[â]) 2 ]+(E[â] − a) 2= V [â]+b(a) 2 .We see that ɛ 2 âis the sum of the variance of â and the square of the bias and socan be interpreted as the sum of squares of statistical and systematic errors. Fora biased estimator, it is often more appropriate to quote the result asa = â ± ɛâ.As above, it may be necessary to use estimated values â in the expression for theroot mean square error and thus to quote only an estimate ˆɛâ of the error.31.3.4 Confidence limits on estimatorsAn alternative (and often equivalent) way of quoting a statistical error is with aconfidence interval. Let us assume that, other than the quantity of interest a, thequantities a have known fixed values. Thus we denote the sampling distribution1235