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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
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3. Rank/ Tie Rank/ Rank Distribution/Mean/Variance/Covariance
11/22/2018 SA3202, Review of Part II
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Rank Correlation Coefficient
Samples Ranks (respectively) Mean Variance X1 X2…Xn R1 R2… Rn (n+1)/ (n^2-1)/12 Y1 Y2 …Yn S1 S2….Sn (n+1)/ (n^2-1)/12 11/22/2018 SA3202, Review of Part II
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5. Run/Run Length/Null Distribution/Mean/Variance
11/22/2018 SA3202, Review of Part II
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II. Some Basic Tests 1. Binomial Test Given X~Binom(n,p)
H0: p=p vs H1: p not equal to p0 H1: p>p0 H1: p<p0 H0: p<=p vs H1: p>p0 H0: p>=p vs H1:p<p0 11/22/2018 SA3202, Review of Part II
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2. Quantile Test and Quantile CI
11/22/2018 SA3202, Review of Part II
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3. Median Test and Estimation
2). Sign Test Di=Xi-a, U=#{i| Di>0}~Binom(n, p) p=P(D>0) H0: theta= 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
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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= vs H1: theta not equal 0 H1:theta>0 H1:theta<0 11/22/2018 SA3202, Review of Part II
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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
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IV Two Sample Tests Two Samples Pooled Ranks
X1 X2 …Xn Y1 Y2 …Yn 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
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V Multiple Sample Test Nonparametric ANOVA (Krushal-Wallis H Test)
Samples Ranks Total Average X11 X12 ….X1n R11 R12 …. R1n R A1=R1/n1 X21 X22…..X2n R21 R22 …..R2n R 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
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