# Generalized Beta Distribution of the Second Kind family function Maximum likelihood estimation of the 4-parameter generalized beta II distribution using Fisher scoring. ```r gen.betaIImr(lscale = "loglink", lshape1.a = "loglink", lshape2.p = "loglink", lshape3.q = "loglink", iscale = NULL, ishape1.a = NULL, ishape2.p = NULL, ishape3.q = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5), gshape1.a = exp(-5:5), gshape2.p = exp(-5:5), gshape3.q = exp(-5:5), probs.y = c(0.25, 0.50, 0.75), zero = "shape" ) ``` ## Arguments - `lscale, lshape1.a, lshape2.p, lshape3.q`: Parameter link functions applied to the shape parameter `a`, scale parameter `scale`, shape parameter `p`, and shape parameter `q`. All four parameters are positive. See `Links` for more choices. - `iscale, ishape1.a, ishape2.p, ishape3.q`: Optional initial values for `b`, `a`, `p` and `q`. Default is `NULL` for all of them, meaning initial values are computed internally. - `imethod`: Initializing method to internally compute the initial values. Currently, only `method = 1` is handled. - `gscale, gshape1.a, gshape2.p, gshape3.q`: Grid search initial values. See `CommonVGAMffArguments` for further information. - `zero`: Numeric or Character vector. Position(s) or name(s) of the parameters/linear predictors to be modeled as intercept-only. Details at `CommonVGAMffArguments`. - `lss, probs.y`: See `CommonVGAMffArguments` for fuller details. ## Details This distribution is most useful for unifying a substantial number of size distributions. For example, the Singh-Maddala, Dagum, Fisk (log-logistic), Lomax (Pareto type II), inverse Lomax, beta distribution of the second kind distributions are all special cases. Full details can be found in Kleiber and Kotz (2003), and Brazauskas (2002). The argument names given here are used by other families that are special cases of this family. Fisher scoring is used here and for the special cases too. The 4-parameter generalized beta II distribution has density $$ f(y) = a y^{ap-1} / [b^{ap} B(p,q) \{1 + (y/b)^a\}^{p+q}]%f(y) = a y^(ap-1) / [b^(ap) B(p,q) (1 + (y/b)^a)^(p+q)] $$ for $a > 0$, $b > 0$, $p > 0$, $q > 0$, $y >= 0$. Here $B$ is the beta function, and $b$ is the scale parameter `scale`, while the others are shape parameters. The mean is $$ E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(q - 1/a) /(\Gamma(p) \, \Gamma(q))%E(Y) = b gamma(p + 1/a) gamma(q - 1/a) / ( gamma(p) gamma(q)) $$ provided $-ap < 1 < aq$; these are returned as the fitted values. ## Warning `zero` can be a numeric or a character vector specifying the position(s) or the name(s) (partially or not) of the linear predictors modeled as intercept--only. Numeric values can be entered as usual. If names are used, note that the linear predictors in this family function are `c("scale", "shape1.a", "shape2.p", "shape3.q")`. For simplicity, using names rather than numeric vectors is recommended. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Brazauskas, V. (2002) Fisher information matrix for the Feller-Pareto distribution. **Statistics & Probability Letters**, 59 , 159--167. Kleiber, C. and Kotz, S. (2003) **Statistical Size Distributions in Economics and Actuarial Sciences**. Wiley Series in Probability and Statistics. Hoboken, New Jersey, USA. McDonald, J. B. and Xu, Y. J. (1995) A generalization of the beta distribution with applications. **Journal of Econometrics**. 66 , p.133--152. McDonald, J. B. (1984) Some generalized functions for the size distribution of income. **Econometrica**, 52 , 647--663. ## Author(s) T. W. Yee and V. Miranda. ## Note Paramaters `"shape1.a", "shape2.p", "shape3.q"` are modeled as intercept only, by default. If the self-starting initial values fail, try experimenting with the initial value arguments, `iscale`, `ishape1.a`, `ishape2.p` and `ishape3.q` whose default is `NULL`. Also, the constraint $-ap < 1 < aq$ may be violated as the iterations progress so it is worth monitoring convergence, e.g., set `trace = TRUE`. Successful convergence depends on choosing good initial values. This process might be difficult for this distribution, since 4 parameters are involved. Presently, only `method = 1` is internally handled to set initial values. It involves **grid search**, an internal implementation of the well-known grid search algorithm for exhaustive searching through a manually specified subset of the hyperparameter space. Default value of `lss` is `TRUE` standing for the following order: location (b), shape1.a (a), shape2.p (p), shape3.q (q). In order to match the arguments of existing R functions, the option `lss = FALSE` might be set leading to switch the position of location (b) and shape1.a (a), only. ## See Also `betaff`, `betaII`, `dagum`, `sinmad`, `fisk`, `lomax`, `inv.lomax`, `paralogistic`, `inv.paralogistic`, `genbetaII`. ## Examples ```r #----------------------------------------------------------------------- # # An example.- In this data set parameters 'shape1.a' and 'shape3.q' are # generated in terms of x2. set.seed(1003) nn <- 200 gdata1 <- data.frame(x2 = runif(nn)) gdata <- transform(gdata1, y1 = rgen.betaII(nn, scale = exp(1.1), shape1.a = exp(1.2 + x2), shape2.p = exp(0.7) , shape3.q = exp(2.1 - x2)), y2 = rgen.betaII(nn, scale = exp(2.0), shape1.a = exp(1.8 + x2), shape2.p = exp(2.3) , shape3.q = exp(1.9 - x2)), y3 = rgen.betaII(nn, scale = exp(1.5), shape1.a = exp(1.8), shape2.p = exp(2.3) , shape3.q = exp(1.3))) #------------------------------------------------------------------------# # A single intercept-only model. No covariates. # Note the use of (optional) initial values. fit <- vglm(y2 ~ 1, #y3 ~ 1 gen.betaIImr(lss = TRUE, # OPTIONAL INITIAL VALUES #iscale = exp(1.5), #ishape1.a = exp(1.8), #ishape2.p = exp(2.3), #ishape3.q = exp(1.3), imethod = 1), data = gdata, trace = TRUE, crit = "loglik") Coef(fit) coef(fit, matrix = TRUE) summary(fit) #------------------------------------------------------------------------# # An intercept-only model. Two responses. fit1 <- vglm(cbind(y2, y2) ~ 1, # cbind(y1, y2) gen.betaIImr(lss = TRUE), data = gdata, trace = TRUE, crit = "loglik") Coef(fit1) coef(fit1, matrix = TRUE) summary(fit1) vcov(fit1, untransform = TRUE) #------------------------------------------------------------------------# # An example incorporating one covariate. Constraints are set accordingly. # x2 affects shape1.a and shape3.q. # Note that the first option uses 'constraints', whilst in the second # choice we use the argument 'zero' to 'set' the same constraints. ### Option 1. c1 <- rbind(0, 1, 0, 0) c2 <- rbind(0, 0, 0, 1) mycons <- matrix( c(c1, c2), nc = 2, byrow = FALSE) fit2 <- vglm(y1 ~ x2, gen.betaIImr(lss = TRUE, zero = NULL), data = gdata, trace = TRUE, crit = "loglik", constraints = list(x2 = mycons )) coef(fit2, matrix = TRUE) summary(fit2) vcov(fit2) constraints(fit2) ### Option 2. fit3 <- vglm(y1 ~ x2, gen.betaIImr(lss = TRUE, zero = c("scale", "shape2.p")), data = gdata, trace = TRUE, crit = "loglik") coef(fit3, matrix = TRUE) summary(fit3) vcov(fit3) constraints(fit3) ```

gen.betaIImr function