A Note on a Biased Bias Estimate

The procedure of not making bias corrections to a measurement process unless statistical tests indicate a non-zero bias results in long-term biased bias estimates unless the long-term bias is zero. The two cases considered are when there are and are not short-term variations in the bias. A Note on a Biased Bias Estimate Introduction This particular subject is difficult to discuss since it concerns the bias of a bias estimator. This is the reason for the somewhat sparse accompanying narrative and comments on the result are saved for later in the text. /N Suppose that 9 is an unbiased estimate of 9, the bias in a measurement process. It is assumed that 9 is normally distributed with a standard deviation ag, i.e., § ^ N(9, as). Bias corrections are made to the measurement process by subtracting 9 from the observations if [§| > z 100(l-a/2)% tile point of the N(0,l) distribution. Otherwise the process is not corrected for bias. If |e| is near will be made about half the time. Let 9 denote a variable such that 1_a/2ae, where z^a/? is theZ]-a./2aQ> bias correction^ if e = o if |e| < The average value of 6 is the long-term estimate of 6. Since 6 is unbiased, it appears that 6 is biased unless 6 = 0 , i.e., unless the measurement process itself is unbiased. 9, Consider the procedure which eventuates in Since 9 is unbiased -za /°°edF(e) za where za = z,_ , a 2a and dF(9) = exp - -5- But if 0 is deemed to be 0 when 0 falls in the interval (-za, za), the bias in 9 as an estimate of 9 is B = E(o) - 9, or fi-u n /*ZCT B = -/ SdF(O) = -/ (o-O)dF(G) •^zo J-ZQ r - 9/ dF(0). zo J-7n Upon evaluation B becomes B = - ag | p [*(z-p)-*(-z-p)] (1) where 6 = pa~, and where respectively, the zero mean, unit variance cumulative and density functions of the normal distribution. Figures 1 and 2 give, respectively, B and E(§) as functions of z-|_ $(y) and <J>(y) are,a/2 and 9 = Pzl-ci/2-