Maximum Spacing Estimation - Goodness of Fit

Goodness of Fit

The statistic S_n(θ) is also a form of Moran or Moran-Darling statistic, M(θ), which can be used to test goodness of fit. It has been shown that the statistic, when defined as

$S_n(\theta) = M_n(\theta)= -\sum_{j=1}^{n+1}\ln{D_j(\theta)},$

is asymptotically normal, and that a chi-squared approximation exists for small samples. In the case where we know the true parameter, Cheng & Stephens (1989) show that the statistic has a normal distribution with

$\begin{align} \mu_M & \approx (n+1)(\ln(n+1)+\gamma)-\frac{1}{2}-\frac{1}{12(n+1)},\\ \sigma^2_M & \approx (n+1)\left ( \frac{\pi^2}{6} -1 \right ) -\frac{1}{2}-\frac{1}{6(n+1)}, \end{align}$

where γ is the Euler–Mascheroni constant which is approximately 0.57722.

The distribution can also be approximated by that of, where

$A = C_1 + C_2\chi^2_n \,$ ,

in which

$\begin{align} C_1 &= \mu_M - \sqrt{\frac{\sigma^2_Mn}{2}},\\ C_2 &= {\sqrt\frac{\sigma^2_M}{2n}},\\ \end{align}$

and where follows a chi-squared distribution with degrees of freedom. Therefore, to test the hypothesis that a random sample of values comes from the distribution, the statistic can be calculated. Then should be rejected with significance if the value is greater than the critical value of the appropriate chi-squared distribution.

Where θ₀ is being estimated by, Cheng & Stephens (1989) showed that has the same asymptotic mean and variance as in the known case. However, the test statistic to be used requires the addition of a bias correction term and is:

$T(\hat\theta) = \frac{M(\hat\theta)+\frac{k}{2}-C_1}{C_2},$

where is the number of parameters in the estimate.

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