Maximum Spacing Estimation - Goodness of Fit

Goodness of Fit

The statistic Sn(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.

Read more about this topic:  Maximum Spacing Estimation

Famous quotes containing the words goodness of, goodness and/or fit:

    “Thank God, I never was cheerful. I come from the happy stock of the Mathers, who, as you remember, passed sweet mornings reflecting on the goodness of God and the damnation of infants.”
    —Henry Brooks Adams (1838–1918)

    “Dear and most respected bookcase! I welcome your existence, which has for over one hundred years been devoted to the radiant ideals of goodness and justice.”
    —Anton Pavlovich Chekhov (1860–1904)

    “Life admits not of delays; when pleasure can be had, it is fit to catch it: every hour takes away part of the things that please us, and perhaps part of our disposition to be pleased.”
    —Samuel Johnson (1709–1784)