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Consider multiple observations $x[n]$ for an integer parameter $A$ under White Gaussian Noise $w[n]$:

$x[n]=A+w[n]; \quad$ $n=0,1,...,N−1$ with $w[n] \sim N(0,σ^2)$.

Is it possible to have an minimum variance unbiased estimator for the integer parameter $A$?

Stochastic
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Thiruppathirajan
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  • I have reopened it, but suspect most readers--like me--won't have a clue what "DC level A in WGN" means. It doesn't seem like it adds any useful information to the question, though. – whuber Nov 27 '19 at 16:57
  • @whuber: Reference: How to find Maximum Likelihood estimates of an *integer* parameter? – Thiruppathirajan Nov 27 '19 at 19:07
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    One more question: in the situation you posit, there is a well-known MVUE for $A$ regardless of what value $A$ might have: namely, the mean of the data. *A fortiori,* this works when $A$ is integral. Are you perhaps trying to ask whether there is an MVUE estimator of $A$ *whose values are always integers*? – whuber Nov 27 '19 at 19:23
  • @whuber yes, A is always an integer – Thiruppathirajan Nov 27 '19 at 19:25
  • Right--but that doesn't matter. In that case, as I pointed out, your question has a well-known answer. Are you supposing the *estimator* must always have integer values? – whuber Nov 27 '19 at 19:28
  • @whuber When we take mean of the observations, the resultant estimate *A^* is not necessarily an integer. One may apply rounding to nearest integer for the mean value *A^*. If so, will the estimator be MVUE? – Thiruppathirajan Nov 27 '19 at 19:30
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    Obviously not, because it is a worsened version of the MVUE estimator. But conceivably it is MVUE *among the class of integer-valued estimators.* Maybe that's your question? If so, you need to edit your post to clarify these points. – whuber Nov 27 '19 at 19:33
  • @whuber Can you clarify by proving that "an Estimator which computes mean value of the observations and then rounding the mean value to nearest integer" is an MVUE. In this case, how to find CRLB. Kindly excuse since I have not come across any literature for finding CRLB for an integer-valued parameter. – Thiruppathirajan Nov 27 '19 at 19:40
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    I can't clarify a statement that is exactly the opposite of what I stated! It's still unclear what you're even trying to ask. – whuber Nov 27 '19 at 19:42
  • @whuber: From your statement "because it is a worsened version of the MVUE estimator. But conceivably it is MVUE among the class of integer-valued estimators.", I understand that an Estimator which computes mean value of the observations and then rounding the mean value to nearest integer is still an MVUE. Kindly clarify. – Thiruppathirajan Nov 27 '19 at 19:46
  • "Worsened version" means it is *not* MVUE. – whuber Nov 27 '19 at 19:48
  • @Whuber Thanks for your valuable comments. I understood that the integer-valued estimator which we discussed is not an MVUE. – Thiruppathirajan Nov 27 '19 at 19:51

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