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I am using a minimum chi-square estimator technique to estimate a set of parameters using some sample data (essentially, finding what set of model parameters minimize the difference between observed and expected values, wherein the expected values are calculated through a series of equations). The approach is similar to maximum likelihood estimation, but uses a chi-square objective function instead of a likelihood function.

My question is what is the proper approach for estimating standard errors from this method? With maximum likelihood estimation, one can use the inverse of the Hessian matrix evaluated at the estimated values. Is there a similar technique for minimum chi-square estimation? Can an inverse of the Hessian be used for both? Or is there something else?

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    $\begingroup$ Could you elaborate on what a "chi-square objective function" might mean? And since you ask about estimating standard errors, you must have a probability model in mind: please tell us what it is. $\endgroup$
    – whuber
    Jun 24 at 18:57
  • $\begingroup$ I'm using a population projection model that takes a set of parameters (starting population size, average annual survival rate, birth rate, etc.) and uses those values to project a population into the future. It then estimates the expected number of animals that should be seen in a study of that population. The objective function then compares that expected number to the number we actually observed in the field using the standard (O - E)^2 / E objective function. I'm using numerical optimization to determine which set of starting parameters produce the smallest value of that objective function $\endgroup$ Jun 27 at 14:29
  • $\begingroup$ That likely places your results too much at the mercy of the observations with tiny counts. Also, by not being equivalent to any particular probability model, you can't expect it to supply standard errors: those would have to be obtained in a separate way, such as by bootstrapping. Ordinarily one would use a Poisson model (a generalized linear regression) in this setting. That will yield theoretically valid standard errors. $\endgroup$
    – whuber
    Jun 27 at 14:38
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    $\begingroup$ I'm not sure if I agree. This method has been used extensively over the past 20 years, in both management papers and refereed journal articles. The observations are not tiny, and there are numerous ways of incorporating auxiliary data to further stabilize parameter estimates. There is a version of this approach that uses maximum likelihood estimations instead of chi-square estimation, and that is able to supply standard errors quite easily through the use of the Hessian matrix. So, I'm not sure why you believe there isn't a way to get standard errors without bootstrapping here. $\endgroup$ Jun 27 at 18:25

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There are various Chi-square minimization methods described in the literature.

All of them boil down to minimizing a sum (or average) of normalized squared residuals (or Chi-square distances).

The normalization constant is usually the predicted value or an average of the predicted value and the observed value.

Provided the data generating process satisfies some mild hypotheses (to be checked), all of these estimators are M-estimators, hence extremum estimators.

Extremum estimators are asymptotically normal and their asymptotic covariance matrix is easy to compute (see, the popular Econometrics book by Fumio Hayashi). The asymptotic covariance can be used to compute standard errors.

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  • $\begingroup$ Are there any packages in R that can calculate such a covariance matrix? $\endgroup$ Jun 27 at 14:31

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