Nonparametric Statistics

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Nonparametric Statistics Review of Part II Nonparametric Statistics Some Basic Concepts: 1. Order statistics/Distribution/Density 2. Quantile/Quartile/Decile/Percentile/Median/Quantile Distribution 11/22/2018 SA3202, Review of Part II

3. Rank/ Tie Rank/ Rank Distribution/Mean/Variance/Covariance 11/22/2018 SA3202, Review of Part II

Rank Correlation Coefficient Samples Ranks (respectively) Mean Variance X1 X2…Xn R1 R2… Rn (n+1)/2 (n^2-1)/12 Y1 Y2 …Yn S1 S2….Sn (n+1)/2 (n^2-1)/12 11/22/2018 SA3202, Review of Part II

5. Run/Run Length/Null Distribution/Mean/Variance 11/22/2018 SA3202, Review of Part II

II. Some Basic Tests 1. Binomial Test Given X~Binom(n,p) H0: p=p0 vs H1: p not equal to p0 H1: p>p0 H1: p<p0 H0: p<=p0 vs H1: p>p0 H0: p>=p0 vs H1:p<p0 11/22/2018 SA3202, Review of Part II

2. Quantile Test and Quantile CI 11/22/2018 SA3202, Review of Part II

3. Median Test and Estimation 2). Sign Test Di=Xi-a, U=#{i| Di>0}~Binom(n, p) p=P(D>0) H0: theta=0 vs H1: theta not equal 0 H1:theta>0 H1:theta<0 3). Wilcoxon Signed Rank Test Di=Xi-a, Rank |Di| Sp= Sm= Sp+Sm=n(n+1)/2 Under H0, E(Sp)=E(Sm)=n(n+1)/4 Var(Sp)=Var(Sm)=n(n+1)(2n+1)/24 11/22/2018 SA3202, Review of Part II

III Paired Sample Tests 1. The Location Test Problem (X1,Y1), (X2,Y2),….,(Xn,Yn) 2. Sign Test Di=Xi-Yi, U=#{i| Di>0}~Binom(n, p) p=P(D>0) H0: theta=0 vs H1: theta not equal 0 H1:theta>0 H1:theta<0 11/22/2018 SA3202, Review of Part II

3. Wilcox on Signed Rank Test Di=Xi-Yi, Rank |Di| Sp= Sm= Sp+Sm=n(n+1)/2 Under H0, E(Sp)=E(Sm)=n(n+1)/4 Var(Sp)=Var(Sm)=n(n+1)(2n+1)/24 11/22/2018 SA3202, Review of Part II

IV Two Sample Tests Two Samples Pooled Ranks X1 X2 …Xn1 Y1 Y2 …Yn2 R1 R2 …Rn1 S1 S2….Sn2 1. Wilcoxon Rank Sum Test Wa=R1+R2+…+Rn1 , Wb=S1+S2+…+Sn2, Wa+Wb=n(n+1)/2, where n=n1+n2 Under H0, E(Wa)=n1(n+1)/2, E(Wb)=n2(n+1)/2, Var(Wa)=Var(Wb)=n1n2(n+1)/12 2. Mann-Whitney U Test Ua+Ub=n1n2, E(Ua)=E(Ub)=n1n2/2, Var(Ua)=Var(Ub)=n1n2(n+1)/12 Ua=n1n2+n1(n1+1)/2-Wa 11/22/2018 SA3202, Review of Part II

V Multiple Sample Test Nonparametric ANOVA (Krushal-Wallis H Test) Samples Ranks Total Average X11 X12 ….X1n1 R11 R12 …. R1n1 R1 A1=R1/n1 X21 X22…..X2n2 R21 R22 …..R2n2 R2 A2=R2/n2 ………………………………………………………………………………….. Xk1 Xk2……Xknk Rk1 Rk2 ……Rknk Rk Ak=Rk/nk V= H=V/(n(n+1)/12) 11/22/2018 SA3202, Review of Part II