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?


1 Answer 1


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

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.]

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    $\begingroup$ 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? $\endgroup$ Nov 4, 2015 at 9:11
  • $\begingroup$ @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). $\endgroup$
    – Glen_b
    Nov 4, 2015 at 11:05
  • $\begingroup$ Similar, but slightly different issues arise if we try to deal with the unconditional distribution being gamma. $\endgroup$
    – Glen_b
    Nov 4, 2015 at 11:20
  • $\begingroup$ 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.. $\endgroup$ Nov 4, 2015 at 11:51
  • $\begingroup$ @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) $\endgroup$
    – Glen_b
    Nov 4, 2015 at 12:02

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