Fit a Student \(t\) linear regression to astronomy data from Rousseeuw (1991) relating to the star cluster CYG OBI, which contains 47 stars in the direction of Cygnus. The predictor ( \(\mathrm{x}[\mathrm{i}]\) in the data in Exercise 4.2. odc) is the logarithmic temperature at the star surface and \(y_{i}\) is the log of the star's light intensity. Compare the LPML of a model with \(v\) set at 5, and a model with \(v\) as an unknown. For example, \(v\) may be assigned an exponential prior (Geweke, 1993), or a uniform prior assumed on \(1 / u\), such as \(1 / v \sim U(1,0.01)\) so that \(v\) will be between 1 and 100 .

Data from Example 4.2

rate among a minority of subjects.

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Consider data on health care use from the German Socioeconomic Panel (SOEP), as analyzed by Riphahn et al. (2003). The data considered here are for n = 14243 male subjects, with the response variable doctor visits. The analysis includes coefficients (81, ..., 67} for time dummy variables at year points (1984, 1985, 1986, 1987, 1988, 1991, 1994), with a corner constraint 6 = 0 on the first year. The other regression effects are for 15 predictors denoted (Age; Agesq; HSat; Handdum; Handper; Married; Educ; Hhninc; Hhkids; Self; Civil; Bluec; Working; Public; AddOn) in the BUGS code. Details on predictors are presented in Riphahn et al. (2003, p 393). The generalized Negative Binomial NBR with unknown power R on i in the conditional variance (Greene, 2008) is applied using the Poisson-Gamma equivalence to the Negative Binomial. A Ga(1,0.001) prior is assumed on a, and an exponential E(0.5) prior on R. It may be noted that the NB2 may be estimated in INLA as well as BUGS, and so assist in setting initial values. The code (including predictor recodes) is require (INLA) D = read.table("docvis.txt", header=T)