Easily applied tests of fit for the Rayleigh distribution

Abstract

In this note two easily applied moment tests of the Rayleigh hypothesis are compared with the Anderson-Darling test, some smooth tests and tests based on the empirical Laplace transform.

Appendix

Asymptotically U √(n/0.37183) has a normal distribution with mean √(n/0.37183)E[U] and variance V. For the null case E[U] = 0 and V = 1. Asymptotically, for any alternative and using the delta method,

$$ \matrix{ {E\left[ U \right] = {\mu_2} - \left( {4/\pi - 1} \right){\mu^2}{\hbox{ and}}} \cr {V = f_x^2{\mu_2} + {\mu_4} - \mu_2^2 + 2{f_x}{\mu_3}.} \cr }<!end array> $$

Here ${f_x} = - \left( {{{8} \left/ {\pi } \right.} - 2} \right)\mu$, μ is the mean and μ _r , r = 2, 3, ... are the central moments of the alternative distribution. The asymptotic power of U for test size α = 0.05 and an alternative with E[U] = 0 is

$$ P\left( {Z > 1.645/\surd \left( {V/0.37183} \right)\left| {Z\;{\hbox{is}}\;{\hbox{N}}\left( {0,1} \right)} \right.} \right). $$

Suppose the alternative is a rectangular distribution with support (a, b), R(a, b) say, in which $a = \surd \left( {\pi /2} \right) - \surd \left\{ {3\left( {2 - \pi /2} \right)} \right\}$ and $b = 2\surd \left( {\pi /2} \right) - a$. Then E[R] = √(π/2) and var(R) = 2−π/2, equal to the corresponding moments of the Rayleigh distribution with θ = 1, so that asymptotically E[U] = 0 and V = 0.3487. Thus the asymptotic power of U is

$$ P\left( {Z > 1.699\left| {Z\;{\hbox{is}}\;{\hbox{N}}\left( {0,1} \right)} \right.} \right) = 0.045. $$

Thus asymptotically the test based on U is biased, albeit that the bias is only slight.

The following table shows how the power of the test based on U approaches its asymptotic limit as n→∞. The finite sample powers for the test based on U in Table 5 are calculated by parametric bootstrap with 1,000 simulations in both the inner and outer loops.

Table 5 Power of the test based on U for various n and α = 0.05