# ao: Transformations In marco-geraci/Qtools: Utilities for Quantiles

## Description

Functions used in quantile regression transformation models

## Usage

 ```1 2 3 4 5 6 7 8``` ```ao(theta, lambda, symm = TRUE, omega = 0.001) invao(x, lambda, symm = TRUE, replace = TRUE) bc(x, lambda) invbc(x, lambda, replace = TRUE) mcjI(x, lambda, symm = TRUE, dbounded = FALSE, omega = 0.001) invmcjI(x, lambda, symm = TRUE, dbounded = FALSE) mcjII(x, lambda, delta, dbounded = FALSE, omega = 0.001) invmcjII(x, lambda, delta, dbounded = FALSE) ```

## Arguments

 `x, theta` numeric vector of singly (`x`) or doubly (`theta`) bounded observations; `theta` must be between 0 and 1 (see `map` to map generic [a,b] intervals to [0,1]). `lambda, delta` transformation parameters. `symm` logical flag. If `TRUE` (default) a symmetric transformation is used. `dbounded` logical flag. If `TRUE` the argument `x` is assumed to be bounded between 0 and 1. `omega` small constant to avoid numerical problems when `theta` is exactly 0 or 1. `replace` logical flag. If `TRUE` (default), values that are outside the admissible range after the Box-Cox or the Aranda-Ordaz back-transformations are replaced by the range bounds.

## Details

These functions transform (back-transform) `x` or `theta` conditional on the parameters `lambda` and `theta`, using the Box–Cox (`bc`), Aranda-Ordaz (`ao`), Proposal I (`mcjI`) and Proposal II (`mcjII`) transformations.

## Value

Transformed or back-transformed values.

Marco Geraci

## References

Aranda-Ordaz FJ. On two families of transformations to additivity for binary response data. Biometrika 1981;68(2):357-363.

Box GEP, Cox DR. An analysis of transformations. Journal of the Royal Statistical Society Series B-Statistical Methodology 1964;26(2):211-252.

Dehbi H-M, Cortina-Borja M, and Geraci M. Aranda-Ordaz quantile regression for student performance assessment. Journal of Applied Statistics. 2016;43(1):58-71.

Geraci M and Jones MC. Improved transformation-based quantile regression. Canadian Journal of Statistics 2015;43(1):118-132.

Jones MC. Connecting distributions with power tails on the real line, the half line and the interval. International Statistical Review 2007;75(1):58-69.

`tsrq`, `tsrq2`, `rcrq`, `nlrq2`