Maximum spacing estimation: Difference between revisions

Revision as of 08:02, 14 November 2018

Revision as of 18:18, 29 January 2019

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Calculating the geometric mean and then taking the logarithm, statistic ''S''_''n'' will be equal to

: <math>

S_n(a,b) = \tfrac{1}{n+1}\ln(x_{(1)}-a) + \tfracsum_{1i=2}{^n+1} \ln(b-x_{(ni)}) - \lnx_{(bi-a1)}) + \sum_tfrac{i=21}^{n +1}\ln(b-x_{(in)}) -x_{ \ln(ib-1)}a)

</math>

Here only the first three terms depend on the parameters ''a'' and ''b''. Differentiating with respect to those parameters and solving the resulting linear system, the maximum spacing estimates will be

: <math alt="MS estimator of a is the minimal x minus the sample range divided by n−1; MS estimator of b is the maximal x plus the sample range divided by n−1">

\hat{a} = \frac{nx_{(1)} - x_{(n)}}{n-1},\ \ \hat{b} = \frac{nx_{(n)}-x_{(1)}}{n-1}.