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I'm trying to arrive at a time series of optimized parameter values $Z_t$ that maximizes the likelihood of occurrence of a specific time series $Y_t$. There is a subsample within the sample that requires greater accuracy. So, I have 7 samples within a time series $Y_i (Y_1,Y_2,...,Y_7) $, all of which follow a binomial distribution. The MLE function used assumes all the distributions to be binomial and iid. I am particularly interested in assuring that $Y_7$ is more accurate/ there are lesser errors around the series.

My teacher suggested weighted MLEs and weighting the subsample at a weight greater than 1 to reduce the errors around the particular subsample. The way we could go about it is either by increasing the number of occurrences of the particular sample by n, or by manually increasing the negative likelihood value of $Y_7$ before summing it up.

However, I am a bit confused. Wouldn't this manually increase the probability of occurrence of $Y_7$ in our sample? Shouldn't we use some correction factor to correct its probability once we have calculated it?

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    $\begingroup$ $Y_7$ is a sample. What do you mean by assuring that it is more accurate? this makes no sense. Please clarify your question $\endgroup$
    – J. Delaney
    Commented Jun 6, 2022 at 18:35
  • $\begingroup$ @J.Delaney - I think he means that the final estimate should fit $y_7$ better than if all the samples were just lumped together and estimated jointly. Fitting $y_7$ well is more important than fitting the others well. This can be achieved by weighting the likelihood in any of several ways. $\endgroup$
    – jbowman
    Commented Jun 6, 2022 at 23:40
  • $\begingroup$ @jbowman The $Y_i$ are assumed to be binomial random variables according to the question, so presumably you would want the estimate to fit to underlying mean of the distribution and not $Y_7$ itself. But it's not clear what is the model, which parameters are being estimated and so on ... $\endgroup$
    – J. Delaney
    Commented Jun 7, 2022 at 10:58
  • $\begingroup$ @J.Delaney - my belief, at the moment at any rate, is that if, for example, the observed frequencies of success were $0.2, 0.3, 0.1, 0.2, 0.2, 0.3, 0.8$, and all the sample sizes were the same, the OP would prefer an estimated probability that is weighted more heavily towards the seventh number than the equally-weighted probability estimate that would occur if you just fit the sample mean of 0.3 - which would be a better estimate if we assumed that all the underlying probabilities were in fact the same. $\endgroup$
    – jbowman
    Commented Jun 7, 2022 at 13:39

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