fstretch {EQUIVNONINF}

Critical constants and power of the UMPI (uniformly most powerful invariant) test for dispersion equivalence of two Gaussian distributions

Description

The function computes the critical constants defining the optimal test for the problem \sigma^2/\tau^2 \le \varrho_1 or \sigma^2/\tau^2 \ge \varrho_2 versus \varrho_1 < \sigma^2/\tau^2 < \varrho_2, with (\varrho_1,\varrho_2) as a fixed nonempty interval around unity.

Usage

fstretch(alpha,tol,itmax,ny1,ny2,rho1,rho2) 

Arguments

alpha

significance level

tol

tolerable deviation from \alpha of the rejection probability at either boundary of the hypothetical equivalence interval

itmax

maximum number of iteration steps

ny1

number of degrees of freedom of the estimator of \sigma^2

ny2

number of degrees of freedom of the estimator of \tau^2

rho1

lower equivalence limit to \sigma^2/\tau^2

rho2

upper equivalence limit to \sigma^2/\tau^2

Value

alpha

significance level

tol

tolerable deviation from \alpha of the rejection probability at either boundary of the hypothetical equivalence interval

itmax

maximum number of iteration steps

ny1

number of degrees of freedom of the estimator of \sigma^2

ny2

number of degrees of freedom of the estimator of \tau^2

rho1

lower equivalence limit to \sigma^2/\tau^2

rho2

upper equivalence limit to \sigma^2/\tau^2

IT

number of iteration steps performed until reaching the stopping criterion corresponding to TOL

C1

left-hand limit of the critical interval for

T = \frac{n-1}{m-1} \sum_{i=1}^m (X_i-\overline{X})^2 / \sum_{j=1}^{n-1} (Y_j-\overline{Y})^2

C2

right-hand limit of the critical interval for

T = \frac{n-1}{m-1} \sum_{i=1}^m (X_i-\overline{X})^2 / \sum_{j=1}^{n-1} (Y_j-\overline{Y})^2

ERR

deviation of the rejection probability from \alpha under \sigma^2/\tau^2 = \varrho_1

POW0

power of the UMPI test against the alternative \sigma^2/\tau^2 = 1

Note

If the two independent samples under analysis are from exponential rather than Gaussian distributions, the critical constants computed by means of fstretch with \nu_1 = 2m, \nu_2 = 2n, can be used for testing for equivalence with respect to the ratio of hazard rates. The only difference is that the ratio of sample means rather than variances has to be used as the test statistic then.

Author(s)

Stefan Wellek <[e-mail]>
Peter Ziegler <[e-mail]>

References

Wellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall/CRC Press, 2010, \S 6.5.

Examples

fstretch(0.05, 1.0e-10, 50,40,45,0.5625,1.7689)