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## Estimating 0

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**Estimating 0**Estimating the proportion of true null hypotheses with the method of moments By Jose M Muino. Email: jmui@igr.poznan.pl**The objective**• Objective • To obtain some information (0 and moments) to help in the construction of the critical region in a multiple hypotheses problem • The situation: • Low sample size • The distribution under the null hypothesis is unknown • But the expectation of the null distribution is known**t**Definitions Let Ti, i = 1, . . . ,m, be the test statistics fortesting null hypotheses H0,i based on observable random variables. Assume thatH0,i is true with probability 0 and false with probability (1- 0) Assume Ti follows a density function f0(T) under H0,iand f1(T) if H0,iis false. Assume that the first m0 =m*0H0,i are true, and the next m1 =m*(1-0) H0,i are false**Define:**Then: The idea**The estimators**Assumed known**Any moment**Because: Then:**Estimators**Test value level Sample level**Example: The mean value as test statistic**• The properties of the estimators can be studied by taking Taylor series. • The properties will be illustrated with the example of the mean value as test statistic • Testing m hypotheses regarding m observed samples xi,j, i=1,…m, j=1,…n, using as test statistic the mean of the observations**Properties**Assuming independence**Properties**Assuming independence**Properties**Assuming independence**Numerical Simulations**m0=450,m1=50, H0->N(0,1), H1->N(1,1) (2000 simulations) 5000 simulations**Numerical Simulations**5000 simulations**From moments to quantiles**• A family of distributions (eg: Pearson family) can be used to calculate the quantiles 100 simulations**Advantages**• Combine information from sample and test level. • No assumptions about the shape of the distribution (finite moments required) • Analytical solution • Properties can be obtained • Estimator can be improved**Disadvantages**• Different estimator for different test statistic • Estimators of the central moments of the test statistics are required • The estimation can be outside of the parameter space**Questions?, Suggestions?**Thanks for your attention!! Or write me at: jmui@igr.poznan.pl Work funded by Marie Curie RTN: “Transistor” (“Trans-cis elements regulating key switches in plant development”)