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# FRQUANTILES directive

Forms regression quantiles.

### Options

`Y` = variate Response variate Design matrix for the regression model Tolerance for the algorithm; default 10-12

### Parameters

`PRQUANTILE` = scalars Values for which to perform the quantile regressions Parameter estimates from each quantile regression 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 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 When `PRQUANTILE` is set to a missing value, saves the cumulative probabilitiy values at which the estimated regression quantiles change 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; the solution was not unique; 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.

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.

Directives: `FIT`, `TERMS`.
Procedures: `RQLINEAR`, `RQNONLINEAR`, `RQSMOOTH`.
Function: `RQOBJECTIVE`.