Sunday, 13 December 2009

Jacknife Test

Miller (1968) introduced a test based on the jacknife technique for testing the null hypothesis (F 2.1) in two-sample examples. This test used the value clip_image002which is defined as:

clip_image004 (F 2.13)

In the equation above, clip_image006 is defined as follows (being the ith sample mean excluding the jth observation) :

clip_image008

Based on the conjecture by Turkey (1962) that clip_image010are distributed approximately independently, this lead Miller to use the values clip_image010[1]which are defined as:

clip_image012 (F 2.14)

Miller proposed testing the null hypothesis (F 2.1) using a two-sample t-test derived from the clip_image010[2]. Miller demonstrated that:

clip_image014 (F 2.15)

And that the convergence in probability can be shown to be,

clip_image016 (F 2.16)

Miller demonstrates that this test asymptotically robust. The Monte Carlo simulations have validated that Miller’s Jacknife robust with a diverse assortment of population distributions.

Layard (1973) proposed a K-sample generalisation of the Miller two-sample jacknife test based on a one-way analysis F test. This test was derived using the clip_image018 values as follows:

clip_image020 (F 2.17)

Using the Layard variation of the Jacknife test, the null hypothesis (F 2.1) will be rejected were the J statistic exceeds the clip_image022percentile of an F-distribution with (K-1) and (n-K) degrees of freedom.

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