Estimates linear functions of one or more random variables, and calculates their variances and covariances (S.A. Gezan).

### Options

`PRINT` = string token |
Output required (`summary` ); default `summ` |
---|---|

`CONSTANTVALUE` = scalar |
Constant value for the function; default 0 |

`COEFFICIENTS` = scalar |
Linear coefficients for the random variables in the function; no default – must be set |

### Parameters

`ESTIMATES` = variates |
Estimated values of the random variables |
---|---|

`VCOVARIANCE` = symmetric matrices |
Variance-covariance matrix of the random variable estimates |

`FUNCTIONESTIMATE` = scalars |
Saves the estimated value of the function |

`SE` = scalars |
Saves the standard error of the function estimate |

`NEWESTIMATES` = variates |
Saves new vectors of estimates, including the estimated value of the function |

`NEWVCOVARIANCE` = symmetric matrices |
Saves variance-covariance matrices for the `NEWESTIMATES` |

### Description

`FNLINEAR`

estimates linear functions of one or more random variables. The estimated values of the random variables, from which the function value is calculated, are supplied (in a variate) by the `ESTIMATES`

parameter. Their variances and covariances are supplied (in a symmetric matrix) by the `VCOVARIANCE`

parameter. The linear coefficients for the function are supplied (again in a variate) by the `COEFFICIENTS`

, and the constant is supplied (in a scalar) by the `CONSTANTVALUE`

option. So the function value is given by

`SUM(ESTIMATES * COEFFICIENTS) + CONSTANTVALUE`

The value can be saved by the `FUNCTIONESTIMATE`

parameter, and its standard error can be saved by the `SE`

option (both in scalars). The `NEWESTIMATES`

parameter can save a new variate of estimates, containing the original `ESTIMATES`

variate and then the function value inserted at the end. The corresponding variance-covariance matrix can be saved (in a symmetric matrix) by the `NEWVCOVARIANCE`

parameter.

Options: `PRINT`

, `CONSTANTVALUE`

, `COEFFICIENTS`

.

Parameters: `ESTIMATES`

, `VCOVARIANCE`

, `FUNCTIONESTIMATE`

, `SE`

, `NEWESTIMATES`

, `NEWVCOVARIANCE`

.

### Method

The linear function *w* of the random variables *f*, *g*, *h* etc. is defined by the expression:

*w* = *a*_{0} + *a*_{1} × *f* + *a*_{2} × *g* + *a*_{3} × *h* + …

where *a*_{0}, *a*_{1}, *a*_{2} etc. are (known) coefficients. The estimated means and variances of the random variables, supplied by the `ESTIMATES`

and `VCOVARIANCE`

parameter, are used to calculate the estimated value of the function *w* and to calculate its variance. If the original random variables are Normally distributed, the random variable *w* is also Normally distributed and the variance calculation is exact.

### Action with `RESTRICT`

Any restrictions are ignored.

### See also

Procedures: `FNCORRELATION`

, `FNPOWER`

.

Commands for: Calculations and manipulation.

### Example

CAPTION 'FNLINEAR example'; STYLE=meta VARIATE [VALUES=4.01,19.63,13.65] means SYMMETRICMATRIX [ROWS=3; VALUES=150.40,-31.85,161.13,0.93,-9.32,23.31] vcov PRINT means,vcov FNLINEAR [PRINT=summary; CONSTANTVALUE=2; COEFFICIENTS=!(1,2,0)]\ ESTIMATES=means; VCOVARIANCE=vcov; FUNCTIONESTIMATE=est; SE=se;\ NEWESTIMATES=newmeans; NEWVCOVARIANCE=newvcov PRINT est,se & newmeans,newvcov