Estimating  0

1. Estimating 0 Estimating the proportion of true null hypotheses with the method of moments By Jose M Muino. Email: jmui@igr.poznan.pl

2. 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

3. 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

4. Define: Then: The idea

5. The estimators Assumed known

6. Any moment Because: Then:

7. Estimators Test value level Sample level

8. 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

9. Properties Assuming independence

10. Properties Assuming independence

11. Properties Assuming independence

12. Numerical Simulations m0=450,m1=50, H0->N(0,1), H1->N(1,1) (2000 simulations) 5000 simulations

13. Numerical Simulations 5000 simulations

14. From moments to quantiles • A family of distributions (eg: Pearson family) can be used to calculate the quantiles 100 simulations

15. From moments to quantiles

16. 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

17. 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

18. 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”)