Description Usage Arguments Details References
betapr_flexfit is used to fit a Beta Prime distribution to a strictly positive response variable. The shape parameter may be specified either as a function of covariates or as a constant estimated using the response variable alone. If the shape parameter is specified to be a function of covariates, the canonical log link function is used.
1 2 |
formula |
An object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. |
data |
An optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which betapr_flexfit is called. |
weights |
An optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector. |
subset |
An optional vector specifying a subset of observations to be used in the fitting process. |
ownstart |
An optional list containing starting values for the maximum likelihood estimation procedure. If a model with an intercept has been specified, the list must be of the form ownstart = list(beta = , beta0 = , beta1 =, …); if a mode with no intercept has been specified the list must be of the form ownstart = list(beta = , beta1 = , …); if the rate parameter is not a function of covariates the list must be of the form ownstart = list(alpha = , beta = ). It is important that the list have as many elements as there are parameters in the model, and that these be supplied in the order set out above. |
key |
A logical parameter dictating whether a key is produced alongside the model’s output. |
warnings |
A logical parameter dictating whether warnings from the maximum likelihood estimation procedure are produced alongside the model’s output. |
... |
Additional arguments to be passed to the function optim within the maximum likelihood estimation procedure. Useful arguments include the gradient descent algorithm to be used and bounds on parameter values; see the stats package. |
This function uses the two parameter parametrization of the Beta Prime distribution is used in Johnson and Kotz (1995). The tow parameter distribution ins a special case of the three parameter distribution, with σ = 1. Starting values for the maximum likelihood estimation procedure are obtained by solving for the roots of a quadratic obtained by finding the ratio of the first and second moment. The probability probability density function is used is:
f(y) = [y^α-1 (1+y)^-(α+β)]/Β(α,β)
α is the first shape parameter and β is the second shape parameter.
When the argument formula specifies a full model with an intercept, α takes the following form and is estimated via a two step (least squares and maximum likelihood) procedure:
α = exp(β0 + β1x1 + …. + βkxk)
When the formula argument specifies a model without an intercept, α takes the bellow form and is estimated via a two step (least squares and maximum likelihood) procedure. Unless theory suggests that an intercept should not be used, users are advised to use a model with an intercept as the maximum likelihood estimation procedure is more stable.
α = exp(β1x1 + …. + βkxk)
When a null model is specified (formula = y ~ 0) α is not estimated as a function of covariates. The starting value for the maximum likelihood estimation procedure is obtained by calling the function betapr_shape1.
Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley.
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