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Assume that I have ten measurements of a physical quantity that must be non-negative (e.g., mass). Because of measurement error, some of the results are nevertheless negative. Can someone suggest a way to calculate the mean of the 10 measurements so as to incorporate the constraint that the true measurements and therefore the true mean must be non-negative?

kjetil b halvorsen
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Thomas
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  • Are you looking for something more sophisticated than just replacing those entries with 0 (or some value you know to be the smallest possible measure)? I am sure you could model the distribution of the measurement error and probably end up with the same approach. – ilir Apr 23 '14 at 13:02
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    @ilir Your replacement proposal will bias the results as well as reduce estimates of variability. It is desirable to avoid both those errors. Thomas, I would like to suggest that *calculating* the mean of the measurements is not of concern. What you want is a good *estimator* of the true value of the physical quantity. Although it might seem strange, obtaining a negative mean for the data is valid. Its meaning can be appreciated when accompanied with an estimated standard error. For instance, $-0.12\pm 0.31$ would indicate your measurements cannot distinguish the quantity from a true zero. – whuber Apr 23 '14 at 13:32
  • @whuber I had not thought of it as a sum of two random variables, one with mean 0. If truncated it would bias the estimate of the mean of the other. Thanks for pointing it out. You may want to put that into an answer. – ilir Apr 23 '14 at 13:36
  • @ilir I believe the situation may be even simpler than that. Because the question refers to "a physical quantity," I assume the measurements are independently distributed with each distribution having a mean equal to that quantity. The objective is to estimate the mean. – whuber Apr 23 '14 at 13:54
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    @whuber You are right in saying that I want a good estimator of the true value. Typically the mean is taken to be that estimator. While a negative mean of the data is what it is, it can't be the best estimate of the true value. For example, in a Bayesian analysis the prior on the true value of the mean would be something like a gamma distribution that does not entertain negative values of the true mean. – Thomas Apr 23 '14 at 14:19
  • You could combine [truncated normal distributions](https://en.m.wikipedia.org/wiki/Truncated_normal_distribution) with $\mu $ equal to each measurement, then take the expected value (or the MLE). This should be equivalent to placing a Gaussian at each measurement, then computing the expected value conditioned to $x>0$. – Luca Citi May 28 '17 at 23:25

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