# What is Mu in zero/one beta inflated models? (gamlss (BEINF))

I am estimating a zero/one inflated beta regression model with gamlss (family BEINF). My dependent variable is [0,1] with a lot of 0s, quite some 1s, and some values in between. This means I assume that my model consists of three submodels: Reference formula: Rigby & Stasinopoulos (2010), A Flexible regression approach using GAMLSS in R, section 10.8.2, p 215.

With regard to link functions used, this source gives a nice table (see BEINF) In R, the coefficients of these submodels are addressed by Nu (zero-probability), Mu and Tau (one-probability)?

gam<-gamlss(y_proportion ~ x1+x2, nu.formula=~x1+x2, tau.formula=~x1+x2, family= BEINF,data=Alldata)
### find raw probabilities
prob.of.0.raw <- predict(gam, what="nu", type="response")
prob.of.1.raw <- predict(gam, what = "tau", type = "response")
prob.of.mu <- predict(gam, what="mu", type="response")


My question is related to Mu. What exactly is this?

• Some people tell me this is the "mean". But do they mean the "mean" of the entire model (all three of them together) and is it therefore comparable with mean(y).
• Or is it only the mean of the middle model (and is it therefore not taking into account 0 and 1)? I am assuming it is the last answer but I am confused. Thanks for your help!
• Might be easier to answer if you gave clearer references. The first formula (from where?) uses a completely different set of parameters on the left side to on the right side! What does thegamlss documentation say? – Scortchi - Reinstate Monica Dec 29 '16 at 20:37
• The definitions of the parameters $\mu$ &c. are given immediately below Equation 10.47. I'd suggest you include them in your question, as they;re essential to answering it. – Scortchi - Reinstate Monica Dec 29 '16 at 20:48
• The table's of the link functions used. – Scortchi - Reinstate Monica Dec 29 '16 at 20:51
• Thanks for the suggestions/clarifications. I edited the question. Hope it is clear now! – user33125 Dec 29 '16 at 20:56
• Thank you. It's clear, I believe. Briefly, $\mu$ is the mean of the beta distribution, the middle model. The mean of the entire model is given by $\operatorname{E}{y}=\frac{\tau + \mu}{1+\nu+\tau}$. An answer will go into more detail. – Scortchi - Reinstate Monica Dec 29 '16 at 21:16