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Is it possible to generate numbers from Gamma distribution (with parameters shape=10, scale=15, say) which also follow a AR(1) process, simultaneously? If it's possible, than how to do that?

Janak
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I don't think this sort of thing will generally be possible without changing some aspects of the usual definition of an AR.

[most of original answer removed, since I have better information]

Richard Hardy points out in the comments below that probably meant a Gamma marginal distribution (... and rereading your question, that makes sense); Richard thought to use an innovation-approach to create the model. While it's not possible to pick just any distribution for innovations, there is at least one such innvation-model that does work (see the Lawrance reference below).

Here's some references that will prove useful:

Grunwald G.K., Hyndman R.J., Tedesco, L. and Tweedie, R. L. (2000)
"Non-Gaussian Conditional Linear AR(1) Models,"
Australian & New Zealand Journal of Statistics, 42:4 (Dec) p479-495 DOI: 10.1111/1467-842X.00143 (Working paper here: http://robjhyndman.com/papers/clar.pdf)

This and the research report below discuss a variety of approaches to non-Gaussian AR(1) models. [The above paper gives a general formulation of non-Gaussian AR(1) models that includes nearly all published non-Gaussian AR(1) models, while the report below is partly a survey paper as well as beginning the synthesis of the paper above.]

Grunwald, G.K., Hyndman, R.J. and Tedesco, L.M. (1995),
"A unified view of linear AR(1) models,"
Research Report, Department of Statistics, University of Melbourne
(http://robjhyndman.com/papers/ar1.pdf)

Lawrance, A.J. (1982),
"The innovation distribution of a gamma distributed autoregressive process,"
Scand. J. Statist., 9, 234–236

[You could perhaps more readily construct an AR in the log of a gamma. Fitted to log-data, that would correspond to a shifting scale in the conditional distribution for the gamma.]

Glen_b
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    The $\varepsilon_t$ does not belong in the formula since its expectation is zero. Also, could you elaborate a little on how an AR(1) process seems to imply a need for a location-family? – Richard Hardy Nov 04 '15 at 09:11
  • @Richard Thanks for pointing out the epsilon. The effect of the previous observation shifts the mean linearly with $y_{t-1}$ but leaves the conditional variance unaltered. Under the condition that one also wants to have a particular distribution family (as was asked for in the question, which to begin with I interpreted as a requirement on the conditional distribution) the need to be able to shift the mean without changing the variance withing a distribution family suggests a location family for that conditional distribution (which would only change the location). – Glen_b Nov 04 '15 at 11:05
  • Similar, but slightly different issues arise if we try to deal with the unconditional distribution being gamma. – Glen_b Nov 04 '15 at 11:20
  • I thought the unconditional distribution had to be gamma, and I was interested how one could reject this intuitively, i.e. intuitively motivate why outcomes of AR(1) could not be unconditionally Gamma-distributed.. – Richard Hardy Nov 04 '15 at 11:51
  • @Richard Hmm. Are you anticipating that the shape of the conditional distribution changes from observation to observation in order to end up with a Gamma? (without that, there's a pretty obvious problem) – Glen_b Nov 04 '15 at 12:02
  • I thought maybe an AR(1) process with a fixed error distribution (thus it is always Normal or always Student or always something else) can give an unconditional Gamma distribution. I might very well be wrong, but I wonder what an intuitive reason could be for rejecting this possibility. – Richard Hardy Nov 04 '15 at 12:16
  • @Richard We can certainly reject both of those because they give nonzero probabilities of a negative observation. which can't then be gamma. Similarly if we restrict ourselves to a distribution that's bounded on the left, we still have problems (e.g. because of the various ways negatives can still arise) unless you make the distribution change. – Glen_b Nov 04 '15 at 13:00
  • @RIchardHardy I've added some references; the specific one by Lawrance seems to addresses the particular approach you were envisioning, where in that case the innovation has a compound Poisson distribution. – Glen_b Nov 04 '15 at 13:29
  • Thank you very much for the answer and the references. It helped me to understand the issue more clearly. – Janak Nov 12 '15 at 11:13
  • Hi @Glen_b, thanks a lot for this answer, and these two papers that you cited; they gave me a new perspective to think about AR processes as a process characterized by conditional and marginal distributions—just super nice!

    Though, can you elaborate your latter remark on the log of gamma and constructing AR(1) on log of the data?

    I want to use GAR(1) (of Gaver and Lewis, 1980; Lawrance 1982) to generate, date and my supervisors find it too complicated.

     – psyguy May 30 '21 at 16:18