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# Or Classical Decision Theory5 - Chapter 3 Classical...

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Classical Decision theory Test of Hypothesis – Nonparametric Method Chapter - 3

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1 2 A multinomial trial with parameters , , , is a trial that can result in exactly one of the possible outcomes. The probability that outcome will occur on a given trial is for 1,2, , . Cons k i p p p k i p i k = 1 2 1 2 ider an experiment consisting of independent and identical multinomial trials with parameters , , , . If is the number of trials that result in outcome for 1,2, , , the - tuple ( , , , k i n p p p O i i k k O O O = 1 2 ) is called a multinomial random variable with parameters , , , , . k k n p p p
Nonparametric methods require no assumptions about the population probability distributions. Nonparametric methods are often the only way to analyze nominal or ordinal data and draw statistical conclusions. Nonparametric methods are often called distribution-free methods . Nonparametric Methods

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Advantages of Nonparametric Tests 1.Used with all scales 2.Easier to compute Developed originally before wide Computer Use 3.Make fewer assumptions 4.Need not involve population parameters 5.Results may be as exact as parametric procedures
Disadvantages of Nonparametric Tests 1.May waste information If data permit using parametric procedures Example: converting data from ratio to ordinal Scale 2.Difficult to compute by hand for large samples 3.Tables not widely available

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Classification of typical problems of Nonparametric Methods Tests of goodness of fit Chi square test Kolmogorov-Smirnov test Test of independence Chi square test Test of homgeneity Chi square test
The Chi – Square Test for Goodness of Fit Here the purpose is to test a null hypothesis that a given set of observations is drawn from or “fits” a specified probability distribution. H 0 : The random variable X has certain specified distribution i.e. F = F 0 F unknown cdf of X F 0 the cdf of the conjectured distribution F 0 is completely specified cdf in its form as well as its parameter H 1 : H 0 is false. Divide the range of X into k disjoint subsets called cells or categories or classes.

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If X is continuous r. v. the cells could be intervals I 1 , I 2 , …, I k , where I 1 = ( - ∞, a 1 ], I j = ( a j -1 , a j ], j = 1, 2, …, k -1; I k = ( a k -1 , ∞) where a 1 < a 2 < … < a k -1. If X is discrete r. v. Then cell may contain one or more values of r. v. X .
We denote the classes by C j , j = 1, 2, …, k. E j the event that X C j , j = 1, 2, …, k Obviously one and only one of the event E j can occur. p j the probability that E j occurs, j = 1, 2, …, k .

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An observation on X gives rise to a multinomial trial which can result in one and only one way of the k mutually exclusive and exhaustive event E j with probability p j , j = 1, 2, …, k .
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Or Classical Decision Theory5 - Chapter 3 Classical...

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