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#### I. Introduction Pages 1-7

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 From page 1...... The problem raised by the IRS provides a strikingly simple example of data from a nonstandard distnbudon for which statistical methodologies have ordy recently begun to be developed, and for which much additional research is needed. The example is of such national importance, both for government agencies and for business and industry, Cat it is He primary focus of this report. Read the entire page → From page 2...... Since ~en, there have been many other papers dealing win particular mixture models. The published research primarily deals with mixtures of distnbutions of similar types, such as mixtures of nominal distr~bution-s, mixtures of chi-square distnbuiions, mixtures of exponential distributions, mixtures of binomial distnbutions, and so on. Read the entire page → From page 3...... Nevertheless, when mixture models are formulated reasonably, they can provide useful tools for statistical analysis. There is by now a large literature pertaining to statistical analyses of mixtures of distnbudons; for a source of references, see Tittenngton, Smith and Makov (19851. Read the entire page → From page 4...... The distribution of Eros can, therefore, be Rewed as a mixture of two distinguishable distributions, one win a discrete probability mass at zero and He over a continuous distribution of non-zero positive and/or negative error amounts. The main statistical objective in this auditing problem is to provide a statistical bound for Be total error amount in the population. Read the entire page → From page 5...... The distribution of this positive random variable usually looks reasonably smooth, but in fact is itself a complex mixture arising from many different types of events. Read the entire page → From page 6...... 7. In series of genetic bird defects, children can be characterized by two vanates, a discrete or categoncal vanable to indicate if one is not affected, affected and born dead, or affected and born alive, and a continuous variable measuring me survival time of affected children born alive. Read the entire page → From page 7...... , a zero measurement can occur because the subject has totally ceased the behavior, or because the subject has reduced the behavior to such a low average level that the time of observation is insufficient to observe the behavior. This indecision might also occur in the example concerning tumor measurement or in rainfall measurement. Read the entire page →

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