Timeline for Likelihood Ratio Test for Comparing Two Poisson Distributions
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2023 at 4:13 | vote | accept | mathlover99999 | ||
| Jul 17, 2023 at 23:32 | answer | added | Ute | timeline score: 1 | |
| Jul 17, 2023 at 23:26 | comment | added | mathlover99999 | thank you so much for your help! @Ute | |
| Jul 17, 2023 at 21:18 | comment | added | Ute | This is actually not so easy, and I can understand that you struggle. The reason is that ugly compound hypothesis that makes it necessary to distinguish between cases $x_1>x_2$ and $x_1\leq x_2$. It helps a bit with notation to reparametrize, I will write something up | |
| Jul 17, 2023 at 18:59 | comment | added | mathlover99999 | Yes I know but I do not know how to find supremum of the numerator, can you please help me. @Ute | |
| Jul 17, 2023 at 18:55 | comment | added | Ute | OK, so you meant Wilks and not Wilson's theorem :-). Wilks theorem says something about the distribution of the test statistic, but the problem here is to actually find the test statistic. So you need the supremum of the numerator first. | |
| Jul 17, 2023 at 18:25 | comment | added | mathlover99999 | In reference to Wilk's theorem, it is known that $-2 \log{\lambda(X_1,…,X_n )} \rightarrow_D \chi_{\text{dim }Θ-\text{dim }H}^2$. Given this, if $-2 \log{\lambda(X_1,…,X_n )}$ exceeds $\chi_{\text{dim }Θ-\text{dim }H}^2$, we reject our null hypothesis $H$, where $H$ is $H: \lambda_1 > \lambda_2$. @Ute | |
| Jul 17, 2023 at 18:16 | comment | added | Ute |
I am actually wondering what you would prefer: slow guidance to working yourself towards the solution, or just a quick solution. If you want guidance, check out the self-study tag.
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| Jul 17, 2023 at 18:03 | comment | added | Ute | Very good so far! For the numerator, you have the restriction $(\lambda_1,\lambda_2)\in H$. Can you write down $H$? Btw, did you mean Wilks' theorem? That one does not apply here, because the problem wants you to specify the test statistic used in the LRT, that is, a function of $X_1$ and $X_2$. | |
| Jul 17, 2023 at 17:42 | comment | added | mathlover99999 | Here's what I've tried so far: I started by writing down the likelihood functions for $X_1$ and $X_2$ under the null and alternative hypotheses. I then tried to formulate the likelihood ratio, which can be written as: $\lambda = \frac{\sup_{\theta \in H} L(X_1,X_2; \theta)}{\sup_{\theta \in \Theta } L(X_1,X_2; \theta)}$ and that in the denominator we obtain the supremum at the values $\lambda_1 = X_1$ and $\lambda_2=X_2$. However, I do not know how to determine the supremum of the denominator. I also do not know if we can then apply Wilson's theorem here, as we have a small $n$ . @Ute | |
| Jul 17, 2023 at 8:12 | comment | added | Ute | welcome to cv! The likelihood in a Poisson distribution is the probability of your observatiins, since Poisson distribution is discrete. Do you know how to write that down? How far did you get on your own with the question, what have you tried? | |
| S Jul 17, 2023 at 6:52 | review | First questions | |||
| Jul 17, 2023 at 6:54 | |||||
| S Jul 17, 2023 at 6:52 | history | asked | mathlover99999 | CC BY-SA 4.0 |