Analytic Error Estimates

Abstract

I present an analytic method for estimating the errors in fitting a distribution. A well-known theorem from statistics gives the minimum variance bound (MVB) for the uncertainty in estimating a set of parameters $ł_i$, when a distribution function $F(z;ł_1 ... ł_m)$ is fit to $N$ observations of the quantity(ies) $z$. For example, a power-law distribution (of two parameters $A$ and $\gaml$) is $F(z;A,\gaml) = A z^{-\gaml}$. I present the MVB in a form which is suitable for estimating uncertainties in problems of astrophysical interest. For many distributions, such as a power-law distribution or an exponential distribution in the presence of a constant background, the MVB can be evaluated in closed form. I give analytic estimates for the variances in several astrophysical problems including the gallium solar-neutrino experiments and the measurement of the polarization induced by a weak gravitational lens. I show that it is possible to make significant improvements in the accuracy of these experiments by making simple adjustments in how they are carried out or analyzed. The actual variance may be above the MVB because of the form of the distribution function and/or the number of observations. I present simple methods for recognizing when this occurs and for obtaining a more accurate estimate of the variance than the MVB when it does.