Maximum likelihood estimation in a Poisson convolution
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Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $nu$. The parameter $nu$ is unknown and we would like to estimate it. We only are given the single data point
$$
X-Y.
$$
What is the maximum likelihood estimator for $nu$?
I am unsure how to go about this, or even if it is analytically tractable, because I cannot calculate a simple expression for
$$
mathbb{P}[X-Y=k].
$$
Is it in fact possible to find a simple form for $mathbb{P}[X-Y=k]$ and then maximise this with respect to $nu$? Or a more general question: is it always necessary to explicitly compute a probability distribution before determining a maximum likelihood estimator?
Anyway, the most sensible estimator choice that I can naively envisage is
$$
widehat{nu}=(X-Y)^2,
$$
which is unbiased at least. Could this be the MLE?
poisson-distribution maximum-likelihood parameter-estimation
asked Nov 18 at 15:58
user301395
1789
add a comment |
up vote
0
down vote
favorite
Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $nu$. The parameter $nu$ is unknown and we would like to estimate it. We only are given the single data point
$$
X-Y.
$$
What is the maximum likelihood estimator for $nu$?
I am unsure how to go about this, or even if it is analytically tractable, because I cannot calculate a simple expression for
$$
mathbb{P}[X-Y=k].
$$
Is it in fact possible to find a simple form for $mathbb{P}[X-Y=k]$ and then maximise this with respect to $nu$? Or a more general question: is it always necessary to explicitly compute a probability distribution before determining a maximum likelihood estimator?
Anyway, the most sensible estimator choice that I can naively envisage is
$$
widehat{nu}=(X-Y)^2,
$$
which is unbiased at least. Could this be the MLE?
poisson-distribution maximum-likelihood parameter-estimation
asked Nov 18 at 15:58
user301395
1789
For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $nu$. The parameter $nu$ is unknown and we would like to estimate it. We only are given the single data point
$$
X-Y.
$$
What is the maximum likelihood estimator for $nu$?
I am unsure how to go about this, or even if it is analytically tractable, because I cannot calculate a simple expression for
$$
mathbb{P}[X-Y=k].
$$
Is it in fact possible to find a simple form for $mathbb{P}[X-Y=k]$ and then maximise this with respect to $nu$? Or a more general question: is it always necessary to explicitly compute a probability distribution before determining a maximum likelihood estimator?
Anyway, the most sensible estimator choice that I can naively envisage is
$$
widehat{nu}=(X-Y)^2,
$$
which is unbiased at least. Could this be the MLE?
poisson-distribution maximum-likelihood parameter-estimation
asked Nov 18 at 15:58
user301395
1789
Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $nu$. The parameter $nu$ is unknown and we would like to estimate it. We only are given the single data point
$$
X-Y.
$$
What is the maximum likelihood estimator for $nu$?
I am unsure how to go about this, or even if it is analytically tractable, because I cannot calculate a simple expression for
$$
mathbb{P}[X-Y=k].
$$
Is it in fact possible to find a simple form for $mathbb{P}[X-Y=k]$ and then maximise this with respect to $nu$? Or a more general question: is it always necessary to explicitly compute a probability distribution before determining a maximum likelihood estimator?
Anyway, the most sensible estimator choice that I can naively envisage is
$$
widehat{nu}=(X-Y)^2,
$$
which is unbiased at least. Could this be the MLE?
poisson-distribution maximum-likelihood parameter-estimation
poisson-distribution maximum-likelihood parameter-estimation
asked Nov 18 at 15:58
user301395
1789
asked Nov 18 at 15:58
user301395
1789
asked Nov 18 at 15:58
user301395
1789
asked Nov 18 at 15:58
user301395
1789
asked Nov 18 at 15:58
user301395
1789
1789
For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01
add a comment |
For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01
For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01
For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01
add a comment |
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For the difference of independently distributed Poisson variables, see Skellam distribution.
– StubbornAtom
Nov 18 at 19:01