Given $Y_1, Y_2,...,Y_n$ i.i.d random variables, where $Y_i|x_i \sim N(\mu_i, \sigma^2)$ and $\mu_i = \beta_0 + \log(\beta_1 + \beta_2x_i)$.

How do I proof that the distribution is a GLM (general linear model)?

My first guess was to proof that the distribution is from the EF (exponential family), becuse if it is from EF we know that is GLM.

In Cassela, from Statistical Inference, the quote says:

A family of pdfs, or pmfs, is called an exponential family if it can be expressed as

$$f(x|\theta) = h(x) \; c(\theta) \; exp( \; \sum_{i=1}^{k}w_i(\theta)\;t_i(x))$$

So I tried to apply the definition in the conditional distribution where:

$$Y_i|x_i = \frac{1}{\sqrt{2\pi\sigma^2}} \; exp \left \{{-\frac{(x-\mu)^2}{2\sigma^2}} \right \}$$

Step 1) I took the $ln$ and replace the $\mu_i = \beta_0 + log(\beta_1 + \beta_2x_i)$.

$$ln(Y_i|x_i) = -\frac{1}{2}\;\ln(2\pi\sigma^2) - (x_i -\beta_0 + \ln(\beta_1 + \beta_2x_i))^2$$

Step 2) I took the exponential of the function.

$$Y_i|x_i = \exp \left \{ - \frac{1}{2} \; \ln(2\pi\sigma^2) - (x_i - \beta_0 + \ln(\beta_1 + \beta_2x_i))^2 \right \}$$

From here I got stuck. Usually I would take the square and tried to find the terms of E.F, but I'm not sure if I'm going to the right path.

How do I proof that the distribution is a GLM?

  • 2
    $\begingroup$ There some confusion in your question between generalized linear models and general linear models, which are not the same thing, although the latter is a special case of the former. Are you familiar with the difference? Anyway, the model you have proposed is not either of those things because it isn't linear: your $\mu$ is a non-linear function of the $\beta_j$s. So you will not be able to prove what you seeking to. $\endgroup$ – Gordon Smyth Apr 23 '19 at 7:07
  • $\begingroup$ If you really are trying to understand generalized linear models, then I would point out that checking whether the response distribution belongs to an exponential family is neither necessary or sufficient for a GLM. For known dispersion, the response distribution needs to be a linear exponential family (more special than an EF). The dispersion parameter needs to obey certain rules but doesn't not need to be an EF parameter (more general than an EF). Putting these things together, the response distribution family actually needs to be an exponential dispersion model. $\endgroup$ – Gordon Smyth Apr 23 '19 at 7:22
  • $\begingroup$ I think you have a fundamental misunderstanding of what a model is and a distribution is. I'd suggest you look up the definitions of these terms to start. There is no such thing a s "general linear model" distribution. A GLM is a type of model or simplified way to describe and understand your data. The model itself make distributional assumptions about your data or parameters that describe your model. I'm not sure why you are trying to find the distribution, when you've told us what the distribution is in your opening sentence. $\endgroup$ – StatsStudent Apr 24 '19 at 4:07
  • $\begingroup$ @StatsStudent my distribution is a funcion, in the exemple above the normal distribution. I want to proof that function is from Exponential Family. The function that explains my data is a model, right? Maybe I'm making confusion with the formal aspects of the question. $\endgroup$ – Arduin Apr 24 '19 at 17:19

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