I have normally-distributed processes from which I get small samples (n typically 10-30) that I want to use to estimate variance. But frequently the samples are so close together that we can't measure individual points near the center.

I have this vague understanding that we should be able to construct an efficient estimator using ordered samples: E.g., if I know the sample contains 20 points, and that 10 are clustered near the center too tightly to measure individually, but I have discrete measurements of 5 on either tail, is there a standard/formulaic approach for estimating the process variance that makes optimal use of such samples?

(Note that I don't think I can just weight the center average. For example, it's possible for 7 samples to cluster tightly while another three are asymmetrically skewed to one side but close enough we can't tell that without more tedious single sampling.)

If the answer is complicated any tips on what I should be researching would be appreciated. E.g., is this an order-statistic problem? Is there likely to be a formulaic answer, or is this a computational problem?

Updated detail: The application is analysis of shooting targets. A single underlying sample is the point of impact (x,y) of a single shot on the target. The underlying process has a symmetric bivariate normal distribution but there is no correlation between axes, so we are able to treat the {x} and {y} samples as independent draws from the same normal distribution. (We could also say the underlying process is Rayleigh-distributed, but we can't measure the sample Rayleigh variates because we can't be certain of the coordinates of the "true" center of the process, which for small n can be significantly distant from the sample center ($\bar{x}$, $\bar{y}$).)

We are given a target and the number of shots fired into it. The problem is that for n>>3 precise guns will typically shoot a "ragged hole" surrounded by distinct shots. We can observe the x- and y-width of the hole, but we don't know where in the hole the non-distinct shots impacted.

Here are some examples of more problematic targets:

[Sample target with n=10]

Sample target with n=100

(Granted, in an ideal world we would change/switch targets after each shot and then aggregate the samples for analysis. There are a number of reasons that's often impractical, though it is done when possible.)

Further Notes following WHuber's clarifications in the comments: Shots produce target holes that are of uniform and known diameter. When a shot is outside any "ragged group" we know the projectile radius and so we can measure the precise center $x_i$. In each "ragged group" we can discern some number of peripheral "balls" and again mark the precise center of those outer shots based on the known projectile radius. It is the remaining "center-censored" shots that we only know impacted somewhere in the interior of a "ragged group" (which is typically -- and if necessary let us assume -- one per target).

To facilitate solution I believe it will be easiest to reduce this to a set of one-dimensional samples from the normal, with a central interval of width w > d, where d is the projectile diameter, containing c < n "censored" samples.

  • $\begingroup$ (1) Is the Normal distribution an assumption or do you have good evidence in support of it? (2) Is the problem that you cannot accurately count the data near the center? (That would be different than the usual meaning of "censoring," which is that you can count those data but you know only that their values lie within certain intervals.) $\endgroup$
    – whuber
    Nov 14, 2013 at 7:38
  • $\begingroup$ @whuber: Yes, we have both fundamental and empirical evidence the process is normally distributed. And yes we know the exact count of points in the total group, and we can observe the interval(s) where too many samples lie to determine individual values. $\endgroup$
    – feetwet
    Nov 14, 2013 at 13:15
  • $\begingroup$ Thanks, that's helpful. The nature of the uncertainty is still unclear, though, and a good model for it could motivate a good solution. Could you perhaps provide an illustration or example or at least describe the measurement process in a little more detail? $\endgroup$
    – whuber
    Nov 14, 2013 at 15:20
  • $\begingroup$ @whuber: Updated. If it will help I will also work on posting links to some real samples. $\endgroup$
    – feetwet
    Nov 14, 2013 at 16:56
  • $\begingroup$ Very interesting problem! I think it will take some creative thought to derive a good solution. Would it be fair to say you are considering the centers of each shot, $x_i,$ as an iid sample of a bivariate Normal$(\mu, \sigma^2)$ distribution; you wish to estimate $\sigma$; but all you can observe--with some imprecision--is $\cup_i B(x_i, r)$ (where $r$ is the known common radius of each projectile and $B(x,r)$ is the ball of radius $r$ around $x$)? $\endgroup$
    – whuber
    Nov 14, 2013 at 18:01

2 Answers 2


That is an interesting problem. First, I would not make the assumption of a normal distribution. It seems what you are really looking for is some estimate of dispersion that you apply fairly to many different shooters or guns or ammo or whatever.

I'd try to turn this around. You don't know exactly where all the bullets went unless you see 10 separate holes (assuming 10 shots). But you do know where they didn't go. This could be used to constrain the distribution assuming Bayesian statistics if you want to start with a distribution.

An idea that might be best here is to stop trying to do it mathematically and just do something sensible like this. Take the target and run an image processing routine to mark the shot through area which may be unconnected. Measure the mean and second moment of this and use these are an estimator. If you want to go a little further and try to Gaussianize it, you can run simple monte carlo experiment to get a calibration factor.

  • $\begingroup$ Let me explain a little more. Let's say you have 10 shots and there are 6 clear holes where you know where the bullets went. First take these points and use them to constraint the Gaussian width. Following the usual routine, this constrains the sigma of the Gaussian sigma (to be some known distribution. cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf $\endgroup$
    – Dave31415
    Mar 4, 2014 at 17:18
  • $\begingroup$ Now, once you have done that, you want to consider the 4 bullets that didn't make new holes. Since the bullets are independent, this new likelihood (on the Gaussian sigma) can simply be multiplied. So basically for each of the 4 bullets, you want to multiply by the probability that they don't make a new hole. $\endgroup$
    – Dave31415
    Mar 4, 2014 at 17:22
  • $\begingroup$ A simple way of doing this with monte carlo is to draw a set of sigma from your constrained distribution and using this sigma, calculate the chance of not making a new hole. Thus, draw many simulated shots from this and count what fraction don't make new holes. This can then be used to update the likelihood. Then move on to the next one and do the same. Now you have your final likelihood. $\endgroup$
    – Dave31415
    Mar 4, 2014 at 17:22
  • $\begingroup$ Last comment. From a practical point of view, the estimate of the sigma shouldn't really be affected that much by where exactly the unseen bullets went as long as you assume they went through previous holes. It will mostly be constrained by the ones that you can see that define the edge. That's because the chance of a bullet going through a hole twice that is far from the center is very low. So even a crude monte carlo will get you very close to the optimal estimator. $\endgroup$
    – Dave31415
    Mar 4, 2014 at 17:28
  • $\begingroup$ If we don't assert a normal (or other) distribution then it seems unlikely we can say anything more than to put an upper or lower bound on what's going on in the censored region. In the 1-dimensional case where we have n shots censored a lower-bound on variance is to assume they all hit the same interior point closest to the mean, and (assuming the mean is centered in the interior) an upper-bound is to assume the censored points are equally distributed on the periphery of the interior. But if we assume the underlying process is normal it seems like we should be able to do something better. $\endgroup$
    – feetwet
    Mar 4, 2014 at 22:37

From another vantage point, one could view this in the light of the field of Spatial Statistics, which has created an assortment of metrics, many of which have been placed in toolboxes (see, for example, ArcGIS).

Wikipedia (link: http://en.m.wikipedia.org/wiki/Spatial_descriptive_statistics ) actually has a good introductory page discussing such concepts as measures of spatial central tendency and spatial dispersion. To quote Wikipedia on the latter:

For most applications, spatial dispersion should be quantified in a way that is invariant to rotations and reflections. Several simple measures of spatial dispersion for a point set can be defined using the covariance matrix of the coordinates of the points. The trace, the determinant, and the largest eigenvalue of the covariance matrix can be used as measures of spatial dispersion. A measure of spatial dispersion that is not based on the covariance matrix is the average distance between nearest neighbors.

Related concepts include measures of spatial homogeneity, Ripley's K and L functions, and perhaps most relevant for the analysis of bullet clusters, the Cuzick–Edwards test for clustering of sub-populations within clustered populations. The latter test is based on the comparison (using "nearest-neighbor" analyses to tabulate statistics) to a control population, which in the current context could be based on actual observed targets classified as not displaying clustering, or per a theoretical simulation, from say the Rayleigh distribution.


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