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# How does the discordancy testing work?

A statistical discordancy test analysis two hypotheses; a working hypothesis and a different hypothesis. A working hypothesis, H, is a statement that the entire data set of n objects comes from an initial distribution model, F, i.e., H: o_{i} Î F, where i = 1, 2, n.

The hypothesis is retained if there is no statistically important evidence supporting its rejection. A discordancy test checks whether an object o_{i} is essentially large (or small) regarding the distribution F. Different test statistics have been proposed for use as a discordancy test, based on the available knowledge of the data.

Suppose that some statistic T has been selected for discordancy testing, and the value of the statistic for object o_{i} is v_{i}, then the distribution of T is constructed. Significance probability SP (v_{i}) = Prob (T > v_{i}) is evaluated.

If some SP (v_{i}) is sufficiently small, then o_{i} is discordant and the working hypothesis is rejected. An alternative hypothesis, which states that o_{i} appears from another distribution model, G, is adopted. The result is very much based on which F model is chosen because o_{i} can be an outlier under one model and a completely valid value under another.

The alternative distribution is very essential in deciding the power of the test, i.e. the probability that the working hypothesis is rejected when o_{i} is an outlier. There are several types of alternative distributions.

**Inherent alternative distribution** − In this case, the working hypothesis that all of the objects come from distribution F is rejected in favor of the alternative hypothesis that all of the objects increase from another distribution, G −

H: o_{i} Î G, where i = 1, 2, ..., n

F and G can be different distributions or differ only in parameters of the same distribution. There are constraints on the form of the G distribution in that it should have the potential to make outliers. For example, it can have a different mean or dispersion, or a long tail.

**Mixture alternative distribution** − The mixture alternative states that discordant values are not outliers in the F populations, but contaminates from some other populations. In this case, the alternative hypothesis is −

H: o_{i} Î (1 – l) F + lG, where i = 1, 2, ..., n

**Slippage alternative distribution** − This alternative states that all of the objects (apart from some prescribed small number) arise independently from the original model F with parameters m and s2, while the remaining objects are independent observations from a modified version of F in which the parameters have been changed.

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