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Top2. Problem Definition
The problem is to generate random samples from a discrete probability distribution, P(x),that are selected to be as representative as possible, given the constraint of small sample size, N. This situation arises in stochastic optimization problems; where there is a need to generate too many sequences of the samples. For example, such sequences (of the samples)can be used to represent a random variable in a stochastic simulation model, which is embedded in the fitness function of a genetic algorithm. In such situations, computational time considerations will limit the sample size. The generated random samples must have two important properties: fitting the required probability distribution and independence (i.e. each random sample must be an independent sample drawn from the defined distribution). To check whether these desirable properties have been achieved, a number of tests can be performed. The tests can be placed into two categories, according to the properties of interest: goodness of fit, and independence.
Failure to reject the null hypothesis means that the evidence of lack of fit has not been detected by this test. The Kolmogorov – Smirnov test compares the CDF, F(x), of the required distribution with the empirical CDF, SN(x), of the generated samples. It is based on calculating the largest absolute deviation between the actual CDF and the empirical CDF. Therefore, the test is based on the statistic:
D = max |F(x) – S
N(x)|
(2)The sampling distribution of D is known as a function of N.
Autocorrelation test: Ljung-Box test(Cryer & Chan, 2008; Ljung & Box, 1978)is used to test the overall randomness between the generated samples, where the hypotheses are as follows:H0: xi~ independently (Null hypothesis) (3) H1: xi !~ independently
Failure to reject the null hypothesis means that the evidence of dependence has not been detected by this test. The Ljung-Box test is based on the test statistic:
(4) where: