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# FNPOWER procedure

Estimates products of powers of two 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 Specifies the powers of the two random variables Specifies the locations of the random variables corresponding to the elements of the `POWERS` variate Whether to apply an additional correction to the variance of a product, using terms from the second-order approximation; default `no`

### 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 new vestors (including the function estimate)

### Description

`FNPOWER` estimates products of powers of two random variables. The estimated values of the random variables, from which the function estimate 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 positions of the random variables in the `ESTIMATES` variate are specified by the `INDEXES` option, and their powers are specified by the `POWERS` option (both in variates of length two).

The estimate 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 estimate inserted at the end. The corresponding variance-covariance matrix can be saved (in a symmetric matrix) by the `NEWVCOVARIANCE` parameter.

The variance and covariances are calculated using a first-order Taylor expension. You can obtain a more accurate value for the variance of an ordinary product by setting option `CORRECTION=yes`. (`FNPOWER` then uses a second-order Taylor expansion.)

Options: `PRINT`, `CONSTANTVALUE`, `POWERS`, `INDEXES`, `CORRECTION`.

Parameters: `ESTIMATES`, `VCOVARIANCE`, `FUNCTIONESTIMATE`, `SE`, `NEWESTIMATES`, `NEWVCOVARIANCE`.

### Method

The power function w, of the random variables f and g, is defined by the expression:

w = fp × gq

for the real-valued coefficients p and q (defined by the `POWERS` parameter). The functions that can be defined thus include:

    single power w = fp (i.e. q = 0), w = √f (i.e. p = 0.5, q = 0), w = f × g (i.e. p = q = 1), w = f / g (i.e. p = q = -1).

The variances and covariances of the function are approximated using a first-order Taylor series expansion (i.e. the delta method); see Kendall & Stuart (1963). For example the expressions for the variance of the product and ratio functions are as follows:

    product var(w) = var(f × g) = E(f)2 × var(g) + E(g)2 × var(f) + 2 × E(f) × E(g) × cov(f,g) var(w) = var(f / g) = (1 / E(g)2) × { var(f) – 2 × E(w) × cov(f,g) + E(w)2 × var(g) }

The quality of this approximation depends on the linearity of the function near the estimate. For a product, you can request an additional correction for the product function based on a second-order Taylor expansion. A correction factor cf is then added to the expression above, where

cf = var(f) × var(g) + cov(f,g)2.

### Reference

Kendall, M. & Stuart, A. (1963). The Advanced Theory of Statistics, Volume 1. Griffin, London.

Procedures: `FNCORRELATION`, `FNLINEAR`.

Commands for: Calculations and manipulation.

### Example

```CAPTION 'FNPOWER 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
FNPOWER [PRINT=summary; CONSTANTVALUE=0; POWERS=!(1,-1); INDEXES=!(1,3)]\
ESTIMATES=means; VCOVARIANCE=vcov; FUNCTIONESTIMATE=est;\