Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSSuppose, we wish to estimate the value of one of the quantities a 1 ,a 2 ,...,whichwe will denote simply by a. Since the sample values x i provide our only source ofinformation, any estimate of a must be some function of the x i , i.e. some samplestatistic. Such a statistic is called an estimator of a and is usually denoted by â(x),where x denotes the sample elements x 1 ,x 2 ,...,x N .Since an estimator â is a function of the sample values of the random variablesx 1 ,x 2 ,...,x N , it too must be a random variable. In other words, if a number ofrandom samples, each of the same size N, are taken from the (one-dimensional)population P (x|a) then the value of the estimator â will vary from one sample tothe next and in general will not be equal to the true value a. This variation in theestimator is described by its sampling distribution P (â|a). From section 30.14, thisis given byP (â|a) dâ = P (x|a) d N x,where d N x is the infinitesimal ‘volume’ in x-space lying between the ‘surfaces’â(x) =â and â(x) =â + dâ. The form of the sampling distribution generallydepends upon the estimator under consideration and upon the form of thepopulation from which the sample was drawn, including, as indicated, the truevalues of the quantities a. It is also usually dependent on the sample size N.◮The sample values x 1 ,x 2 ,...,x N are drawn independently from a Gaussian distributionwith mean µ and variance σ. Suppose that we choose the sample mean ¯x as our estimatorˆµ of the population mean. Find the sampling distributions of this estimator.The sample mean ¯x is given by¯x = 1 N (x 1 + x 2 + ···+ x N ),where the x i are independent random variables distributed as x i ∼ N(µ, σ 2 ). From ourdiscussion of multiple Gaussian distributions on page 1189, we see immediately that ¯x willalso be Gaussian distributed as N(µ, σ 2 /N). In other words, the sampling distribution of¯x is given byP (¯x|µ, σ) =[1√2πσ2 /N exp −Note that the variance of this distribution is σ 2 /N. ◭](¯x − µ)2. (31.13)2σ 2 /N31.3.1 Consistency, bias and efficiency of estimatorsFor any particular quantity a, we may in fact define any number of differentestimators, each of which will have its own sampling distribution. The qualityof a given estimator â may be assessed by investigating certain properties of itssampling distribution P (â|a). In particular, an estimator â is usually judged onthe three criteria of consistency, bias and efficiency, each of which we now discuss.1230