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Say we are fitting a model to count data (if needed we can assume it follows zero inflated Poisson model).

To be specific, for each sample we have multivariate factors $\boldsymbol{x}$, the (unknown) mean number of counts $y \sim \boldsymbol{x}$, and the counts $c \sim y$. I want to fit an arbitrary positive function $f$ with parameters $\theta$ to predict $y$ given $\boldsymbol{x}$.

Mathematically speaking, does the $\theta$ fitted to $(c,\boldsymbol{x})$ always converge to the $\theta$ fitted to $(y,\boldsymbol{x})$ as the sample size increases? Is there a theory that guarantees this for some class of models/estimators?

kjetil b halvorsen
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user157969
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    Please note that normally in statistics you'd used a *Greek* letter for the unknown mean and a *Roman* letter for the observed count (e.g. most people here will expect to see something like $E(Y|X=x)=\mu(x)$, where you hypothesize that $\mu(x)=f(x)$, and where you observe $y_i,\mathbf{x}_i$). This inversion of common statistical convention is likely to lead to a good deal of confusion. [Note for example you have mixed Greek and Roman letters both for data values and for unobserved (functions of) parameters.] You can leave it this way; at least my comment might lead people to beware. – Glen_b Jul 03 '15 at 02:35
  • You haven't specified how this fitting is to be achieved; are you using ML, say, or at the least can we guarantee we're using a consistent estimator? – Glen_b Jul 03 '15 at 02:44
  • OK thanks, I've changed the notation - I don't remember why I used $\mu$! It is OK to assume ML, I was just wondering what the most 'general' theory was able to show. Thanks again! – user157969 Jul 03 '15 at 08:56
  • In the class of GLMs, under certain conditions there's a [unique global maximum](https://stats.stackexchange.com/questions/87615/does-log-likelihood-in-glm-have-guaranteed-convergence-to-global-maxima). However this doesn't necessarily guarantee convergence in practice; I've occasionally run into issues with models even in canonical form. – Glen_b Nov 18 '17 at 00:24

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