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# FNLINEAR procedure

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` Constant value for the function; default 0 Linear coefficients for the random variables in the function; no default – must be set

### Parameters

`ESTIMATES` = variates Estimated values of the random variables Variance-covariance matrix of the random variable estimates Saves the estimated value of the function Saves the standard error of the function estimate Saves new vectors of estimates, including the estimated value of the function 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 = a0 + a1 × f + a2 × g + a3 × h + …

where a0, a1, a2 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.

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;\