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I am trying to do an excerise on likelihood functions and score vectors and I am having some trouble extending it to parameter vectors. The question is stated below.

Suppose we have two independent samples: $X_{1}, ..., X_{n}$ are iid Exponential random variables with mean $θ_{1}$ and $Y_{1}, . . . , Y_{n}$ are iid Exponential random variables with mean $θ_{2}$.

Find the likelihood and log likelihood functions for the parameter vector $θ = (θ_{1}, θ_{2})^{T}. $

In this case would the likelihood function be $$\prod{θ e^{-θ x_{i}} = θ ^ne^{-θ\sum x_{i}}}$$ (same as one with one parameter)? Or would it be the product of the two functions as shown below: $$\prod{θ_{1} e^{-θ_{1} x_{i}} θ_{2}e^{-θ_{2}y_{i}} = θ_{1} ^ne^{-θ_{1}\sum x_{i}}} θ_{2}^{n}e^{-θ_{2}\sum y_{i}}$$

Or would it be something different?

Thanks!

monkey328
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  • Is this a Bayesian question? If it is then you may need to provide a prior to update with your observations. If not, then the parameters are what they are (albeit unknown) and do not have a pdf – Henry May 28 '21 at 00:41
  • This is not a Bayesian question. In this case, do you know how I would be able to determine the likelihood functions for the parameter vector? – monkey328 May 28 '21 at 00:43
  • I'm not sure what you're asking here. Are you asking about the pdf of the data given the parameters? You may be interested in [What is the reason that a likelihood function is not a pdf?](https://stats.stackexchange.com/questions/31238/what-is-the-reason-that-a-likelihood-function-is-not-a-pdf). Otherwise, Henry is right—you'd need a prior. – Arya McCarthy May 28 '21 at 02:35
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    I think that I may be asking the wrong question here. My final goal is to find a likelihood function for the parameter vector and I assumed that a pdf would be required to do so. – monkey328 May 28 '21 at 02:57
  • Part of the confusion may be that you have not explicitly stated what you are trying to do. Are you testing $H_0: \theta_1 = \theta_2$ against $H_a: \theta_1 \ne \theta_2.$ Are you seeking a 95% Ci for $\theta_1 - \theta_2.$ Both of these? Something else entirely? – BruceET May 28 '21 at 03:05
  • Are you looking for _the likelihood of the parameters_? You've assumed an exponential distribution; its likelihood function can be looked up. – Arya McCarthy May 28 '21 at 03:17
  • Ok, you’ve changed the question enough to be answerable—thanks! Why do you think the likelihood function would exclude the $y$s and $\theta_2$, as in your first equation? – Arya McCarthy May 28 '21 at 15:41
  • Thanks for replying again! I don't think it should exclude the $y$s and $θ2$ but the textbook I'm reading is slightly confusing and made it seem that it would use the same function – monkey328 May 28 '21 at 15:49

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