Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS Consistency An estimatoraˆisconsistentif its value tends to the true valueain the lar ...

STATISTICS and describes the spread of valuesaˆaboutE[aˆ] that would result from a large number of samples, each of sizeN. An es ...

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS and, on differentiating twice with respect toμ, we find ∂^2 lnP ∂μ^2 =− N σ^2 . This ...

STATISTICS is trivially extended to multiple integrals and shows that for two real functions,g(x)and h(x), (∫ g^2 (x)dNx )(∫ h^2 ...

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS or all of the quantitiesaand they may be unknown. When this occurs, one must substitu ...

STATISTICS aˆ P(aˆ|a) aˆα(a) aˆβ(a) α β Figure 31.2 The sampling distributionP(aˆ|a)ofsomeestimatorˆafor a given value ofa. The ...

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS aˆ P(ˆa|a−) P(aˆ|a+) aˆobs α β Figure 31.3 An illustration of how the observed value ...

STATISTICS 31.3.5 Confidence limits for a Gaussian sampling distribution An important special case occurs when the sampling dist ...

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS For the 90% central confidence interval, we requireα=β=0.05. From table 30.3, we find ...

STATISTICS where in the last line we have used the fact that ∑ i(xi−x ̄) = 0. Hence, for given values ofμandσ, the sampling dist ...

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS several estimators, however, it is usual to quote their full covariance matrix. This ...

STATISTICS a regionRˆinˆa-space, such that ∫∫ Rˆ P(ˆa|a)dMaˆ=1−α. A common choice for such a region is that bounded by the ‘surf ...

31.4 SOME BASIC ESTIMATORS a 1 a 2 ˆa 1 aˆ 2 atrue atrue aˆobs ˆaobs (a) (b) Figure 31.4 (a) The ellipseQ(aˆ,a)=cinaˆ-space. (b) ...

STATISTICS % 99 95 10 5 2.5 1 0.5 0.1 n=1 1 .57 10−^43 .93 10−^3 2.71 3.84 5.02 6.63 7.88 10.83 2 2 .01 10−^2 0.103 4.61 5.99 7. ...

31.4 SOME BASIC ESTIMATORS exact expressions, valid for samples of any sizeN, for the expectation value and variance of ̄x. From ...

STATISTICS whereμris therth population moment. Sinceσ̂^2 is unbiased andV[σ̂^2 ]→0asN→∞, showing that it is also a consistent es ...

31.4 SOME BASIC ESTIMATORS wheres^4 is given by s^4 = [∑ ix 2 i N − (∑ ixi N ) 2 ]^2 = ( ∑ ix 2 i) 2 N^2 − 2 ( ∑ ix 2 i)( ∑ ixi) ...

STATISTICS where in the last line we have used again the fact that, since the population mean is zero, μr=νr. However, result (3 ...

31.4 SOME BASIC ESTIMATORS the form σˆ= ( N N− 1 ) 1 / 2 s, wheresis the sample standard deviation. The expectation value of thi ...

STATISTICS However, since the sample valuesxiare assumed to be independent, we have E[xrixrj]=E[xri]E[xrj]=μ^2 r. (31.52) The nu ...