Circular Error Probable

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Estimating the Circular Error Probable (CEP)

The Circular Error Probable \(CEP(p)\) for \(p \in [0,1)\) is the estimated radius of the smallest circle that is expected to cover proportion \(p\) of the shot group. Several different methods for estimating \(CEP(p)\) have been proposed which are based on different assumptions about the underlying distribution of coordinates.

Common CEP estimators

The following approaches are frequently encountered (see the CEP literature overview for references):

  • The general correlated normal estimator DiDonato (2007) is based on the assumption of bivariate normality of the shot coordinates. It allows the \(x\)- and \(y\)-coordinates to be correlated and have different variances. This estimator is based on the exact quantile function of radial error in the correlated bivariate normal distribution re-written in polar coordinates (radius and angle). Its calculation is difficult and requires numerical approaches only available in specialized software.
  • The Grubbs (1964) estimator shares its assumptions with the general correlated normal estimator. It approximates the true cumulative distribution function of radial error with a central \(\chi^{2}\)-distribution. This approach has the advantage that its calculation is much easier than the exact distribution and does not require special software. For \(p \geq 0.5\), the approximation to the true cumulative distribution function is very close but diverges from it for \(p < 0.5\).
  • The Rayleigh estimator uses the Rayleigh quantile function for radial error. It assumes an uncorrelated bivariate normal process equal variances and is easy to calculate with standard software (e.g., spreadsheets).
  • The Ethridge (1983) estimator is not based on the assumption of bivariate normality of \((x,y)\)-coordinates but uses a robust unbiased estimator for the median radius. This estimator "assumes that the square root of the radial miss distances follows the logarithmic generalized exponential power distribution." (Williams, 1997). It is only available for \(p = 0.5\).
  • The modified RAND R-234 estimator is an early example of CEP and is based on lookup tables that have later been fitted with a regression model. It assumes a mostly circular distribution of \((x,y)\)-coordinates. In its original form it was only available for \(p = 0.5\), but McMillan & McMillan (2008) proposed an extension to levels \(p = 0.9\) and \(p = 0.95\) based on numerical simulations.

Comparing CEP estimators

When comparing the relative merits of different CEP estimators, it is important to distinguish two situations:

  • The true variances of x- and y-coordinates as well as their true correlation are known. In practice, this is never the case.
  • The variances and the correlation of \((x,y)\)-coordinates have to be estimated from a sample with limited number of observations.

With known variances and correlation, the CEP estimator based on the exact bivariate normal distribution is superior to the Grubbs estimator (as this is only an approximation), to the Rayleigh estimator (as this is a special case of the general bivariate normal estimator with stricter assumptions), and to the RAND estimator. When its assumptions are met, the only downside of the general bivariate normal estimator is the difficulty of computation without specialized software. The Ethridge estimator stands out because it does not require bivariate normality of the \((x,y)\)-coordinates.

When variances and correlation have to be estimated from limited samples, the situation is different. While it seems common to plug in the sample estimates into the formula for the distribution of radial error when all parameters are known, the resulting distribution is not strictly valid anymore. The reason is that even if the estimates are unbiased, the uncertainty from estimating the variances and correlation are not reflected in the distribution formulae (compare with the case of the t-test vs. the z-test). With growing sample-size, the so-called sampling distribution of radial error will be asymptotically the same as with the case of know variances/correlation.

For small samples the question therefore is which estimator has the best characteristics, i.e., gives a good approximation to the true \(CEP(p)\). This question has been studied, e.g., by Williams (1997). A related question is which estimator is most robust to a very small number of outliers (fliers) that may result from clear operator error.