Forms regression quantiles.

### Options

`Y` = variate |
Response variate |
---|---|

`DESIGNMATRIX` = matrix |
Design matrix for the regression model |

`TOLERANCE` = scalar |
Tolerance for the algorithm; default 10^{-12} |

### Parameters

`PRQUANTILE` = scalars |
Values for which to perform the quantile regressions |
---|---|

`RESIDUALS` = variates |
Parameter estimates from each quantile regression |

`ESTIMATES` = variates or matrices |
Estimates from each quantile regression, either a variate of estimates for a specific quantile or, if `PRQUANTILE` is set to a missing value, a matrix with a row of estimates for every cumulative probability value in the `CUMPROBABILITIES` variate |

`XBARQUANTILES = ` variates |
When `PRQUANTILE` is set to a missing value, saves the sum of the mean of each design column multiplied by its regression quantile for all the quantile solutions |

`CUMPROBABILITIES = ` variates |
When `PRQUANTILE` is set to a missing value, saves the cumulative probabilitiy values at which the estimated regression quantiles change |

`EXIT` = scalars |
Saves an exit code, with 0 to indicate success |

### Description

`FRQUANTILES`

calculates regression quantile statistics using the algorithm of Koenker & D’Orey (1987). The `Y`

option specifies the response variate, and the `DESIGNMATRIX`

option specifies the design matrix for the regression model to be fitted. The design matrix can be formed, for example, using the `TERMS`

directive.

The `PRQUANTILE`

parameter can be set to a scalar specifying the probability value whose quantiles are required. The `ESTIMATES`

parameter then saves the estimated regression quantile statistics, and the `RESIDUALS`

parameter saves the corresponding residuals.

Alternatively, if `PRQUANTILE`

parameter is set to a scalar containing a missing value, `FRQUANTILES`

forms the complete set of “solutions” by finding all the probability values at which the regression quantiles change. These cumulative probabilities can be saved in a variate, using the `CUMPROBABILITIES`

parameter, and the `ESTIMATES`

parameter then saves a matrix with a row of estimates for each cumulative probability. The `XBARQUANTILES`

parameter saves a variate containing the sum of the mean of each column of the `DESIGNMATRIX`

multiplied by its regression quantile for all the cumulative probabilities.

The `EXIT`

parameter can save a scalar containing an “exit” code, as follows:

0 | the algorithm was successful; |
---|---|

1 | the solution was not unique; |

2 | the algorithm failed. |

If `EXIT`

is set, no Genstat diagnostic is given if the algorithm fails, unless the failure arises from an incorrect option or parameter setting, or because Genstat has run out of workspace.

Options: `Y`

, `DESIGNMATRIX`

, `TOLERANCE`

.

Parameters: `PRQUANTILE`

, `RESIDUALS`

, `ESTIMATES`

, `CUMPROBABILITIES`

, `XBARQUANTILES`

, `EXIT`

.

### Method

For more details of quantile regression and of the estimation method, see Koenker (2005) and Koenker & D’Orey (1987).

### Action with `RESTRICT`

`FRQUANTILES`

takes account of restrictions on the `Y`

variate.

### References

Koenker, R. (2005). *Quantile Regression*. Cambridge University Press, New York.

Koenker, R.W. & D’Orey, V. (1987). Algorithm AS229 computing regression quantiles. *Applied Statistics*, 36, 383-393.

### See also

Procedures: `RQLINEAR`

, `RQNONLINEAR`

, `RQSMOOTH`

.

Function: `RQOBJECTIVE`

.

Commands for: Regression analysis, Calculations and manipulation.