Nonparametric sequential selection procedures

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General Introduction.- 1 Sequential Procedures for Selecting the Best of k ? 2 Binomial Populations Introduction.- 1 Selection Procedures with Unrestricted Patient Horizon.- 1 The Selection Model [2; PW; |SA?SB| = r].- 1.1 Derivation of the critical r-value.- 1.2 Derivation of the expectations.- 2 The Selection Model [2; VT; |SA?SB| = s].- 2.1 Derivation of the critical s-value.- 2.2 Derivation of the expectations.- 3 The Selection Model [2; PL; |FA?FB| = r].- 3.1 Derivation of the critical r-value.- 3.2 Derivation of the expectations.- 4 Comparison of the Selection Procedures No.1-No..- 4.1 Some general remarks.- 4.2 Comparison of PW- and VT-sampling.- 4.3 Comparison of PW- and PL-sampling.- 4.4 Comparison of PL- and VT-sampling.- 4.5 Recapitulation.- 4.6 Concluding remarks.- 4.7 Numerical results.- 5 Two-Stage Selection Procedures.- 5.1 The structure of two-stage selection procedures.- 5.2 Derivation of P(CS).- 5.3 Derivation of s(m,n,kc,M) and r(m,n,kc,M).- 5.4 Derivation of the expectations.- 5.5 Addition of PL-sampling.- 5.6 Concluding remarks.- 5.7 Numerical results.- 6 The Selection Model [2; PW; max{SA,SB} = r].- 6.1 Derivation of the critical r-value.- 6.2 Derivation of the expectations.- 6.3 Expectations for large r.- 7 The Selection Model [2; VT; max{SA,SB} = r].- 7.1 Derivation of the critical r-value.- 7.2 Derivation of the expectations.- 7.3 Expectations for large r.- 8 Comparison of the Selection Models No.6 and No.7 – Some Modifications of these Models.- 8.1 Comparison of the selection models no.6 and no.7.- 8.2 The selection model [2; PW; min{FA,FB} = r].- 8.3 The selection model [2; PL; max{FA,FB} = r].- 8.4 Concluding remarks.- 8.5 Numerical results.- 9 The Nature of Termination of a Classical Sequential Selection Procedure.- 9.1 Basic notions.- 9.2 The stopping-behaviour of selection model no.1.- 9.3 The stopping-behaviour of selection model no.2.- 9.4 The stopping-behaviour of the selection models no.3 and no.5.- 9.5 The stopping-behaviour of selection model no.6.- 9.6 The stopping-behaviour of selection model no.7.- 10 The Selection Model [k; PW; max{S1,...,Sk} = r].- 10.1 Introductory remarks.- 10.2 Derivation of the critical r-value.- 10.3 Derivation of the expectations.- 10.4 Expectations for large r.- 11 The Selection Model [k; VT; max{S1,...,Sk} = r].- 11.1 Derivation of the critical r-value.- 11.2 Derivation of the expectations.- 11.3 Expectations for large r.- 11.4 Concluding remarks.- 11.5 Numerical results.- 12 Expected Truncation Points.- 13 The Selection Model [2;PW;|SA-SB|=r or $$\left| {{{\hat p}_A} - {{\hat p}_B}} \right| \geqslant c/\left( {{F_A} + {F_B}} \right)$$].- 13.1 Introduction.- 13.2 Derivation of the critical r- and c-values.- 13.3 Numerical results.- 14 The Selection Model [2;VT;|SA-SB|=s or $$\left| {{{\hat p}_A} - {{\hat p}_B}} \right| \geqslant d/\left( {{F_A} + {F_B}} \right)$$].- 14.1 Derivation of the critical s- and d-values.- 14.2 Numerical results.- 15 The Selection Models [k;PW;el.Ai if Sj?Si=r] and [k;VT;e1.Ai if Sj?Si=s].- 15.1 The PW-elimination procedure.- 15.2 The VT-elimination procedure.- 15.3 Numerical results for the PW-procedure.- 15.4 Numerical results for the VT-procedure.- 15.5 Comparison of selection models.- 2 Selection Procedures with Restricted Patient Horizon.- 1 The Selection Model [2; PW; max{SA+FB, SB+FA} = r].- 1.1 Introduction.- 1.2 Derivation of the P (CS)-value.- 1.3 Determination of the LFC.- 1.4 Derivation of the critical r-value.- 1.5 Derivation of the expectations.- 1.6 Numerical results.- 2 The Selection Model [2; PW; max{SA,SB} = r or FA=FB = c].- 2.1 Derivation of the P (CS)-value.- 2.2 Derivation of the critical r- and c-values.- 2.3 Derivation of the expectations.- 2.4 Numerical results.- 3 The Selection Model [2; VT; max{SA,SB} = r or min{FA,FB} = c].- 3.1 Derivation of the P (CS)-value.- 3.2 Derivation of the critical r- and c-values.- 3.3 Derivation of the expectations.- 3.4 Numerical results.- 4 The Selection Model [2; VT; max{SA,SB} = r or max{FA,FB} = c].- 4.1 Derivation of the P (CS)-value.- 4.2 Derivation of the critical r- and c-values.- 4.3 Derivation of the expectations.- 4.4 Numerical results.- 5 The Selection Model [2; PW; |SA?SB| = r or FA+FB = s].- 5.1 Derivation of the P (CS)-value.- 5.2 Derivation of the critical r- and s-values.- 5.3 Derivation of the expectations.- 5.4 Numerical results.- 6 The Selection Model [k;PW;max{S1,...,Sk}=r or min{F1,...,Fk}=c].- 6.1 Introductory remarks.- 6.2 Derivation of the P (CS)-value.- 6.3 Derivation of the critical r- and c-values.- 6.4 Derivation of the expectations.- 6.5 Numerical results.- 7 The Selection Model [k;VT;max{S1,...,Sk}=r or min{F1,...,Fk}=c].- 7.1 Derivation of the P (CS)-value.- 7.2 Derivation of the critical r- and c-values.- 7.3 Derivation of the expectations.- 7.4 Numerical results.- 8 The Selection Model [k;VT;max{S1,...,Sk}=r or el.Ai if Fi=c].- 8.1 Derivation of the P (CS)-value.- 8.2 Derivation of the critical r- and c-values.- 8.3 Numerical results.- 9 Further Elimination Procedures.- 9.1 The selection model [k;PW;e1.Ai if Sj-Si=r or if Fi=c].- 9.2 The selection model (k;VT;el.Ai if Sj-Si=r or if Fi=c].- 9.3 The selection model [k;PW;e1.Ai if Sj-Si=r or stop if F1+...+Fk=s].- 9.4 The selection model [k;PW;el.Ai if Sj-Si=r or el.Ai if $${\hat p_j} - {\hat p_i} \geqslant c/\left( {{F_i} + {F_j}} \right)$$].- 9.5 The selection model [k;VT;el.Ai if Sj-Si=r or el.Ai if $${\hat p_j} - {\hat p_i} \geqslant d/\left( {{F_i} + {F_j}} \right)$$].- 9.6 Numerical results.- 9.7 Comparison of selection models.- 9.8 Further selection procedures.- 3 Selection Procedures with Fixed Patient Horizon.- 1 Historical Remarks.- 2 The Zelen Selection Model.- 2.1 Definition of the model.- 2.2 Comparison with a VT-sampling procedure.- 2.3 Determination of the optimal value of n.- 3 The Selection Models [2;PW;fixed N] and [2;VT;fixed N].- 3.1 Introduction.- 3.2 Comparison of the P (CS)-values.- 3.3 Comparison of the expectations.- 3.4 Exact and asymptotic formulae for E (NB).- 3.5 Extension of the selection models to odd N.- 3.6 Numerical results.- 3.7 Equivalence to Hoel’s selection model.- 4 The Selection Models [2;PW;fixed N] and [2;VT;fixed N] with Curtailment.- 4.1 Introductory remarks.- 4.2 The PW-sampling procedure with curtailment.- 4.3 The VT-sampling procedure with curtailment.- 4.4 Numerical results.- 5 The Selection Model [2;VT;fixed N] with Truncation Based on |SA?SB|.- 5.1 Description of the model.- 5.2 Derivation of the P (CS)-value.- 5.3 Derivation of the probability of declaring the two treatments equal.- 5.4 Derivation of an upper bound for E (NB).- 5.5 Derivation of the truncation points and of the patient horizon N.- 5.6 Numerical results.- 6 The Selection Model [2;PW;fixed N] with Truncation Based on |SA?SB|.- 6.1 Description of the model.- 6.2 Derivation of the P (CS)-value.- 6.3 Derivation of the probability of declaring the two treatments equal.- 6.4 Derivation of E (NB).- 6.5 Numerical results.- 6.6 Comparison of the selection models.- 7 Selection Models Based on the Randomized Play-the-Winner Rule.- 7.1 Introductory remarks.- 7.2 Expected number of patients on the better treatment within n trials.- 7.3 Derivation of the P(CS)-values.- 7.4 Derivation of the expectations.- 7.5 Numerical results.- 8 Supplementary Investigations—Topics Requiring Further Research.- 2 Continuous Response Selection Models Introduction.- 1 Subset-Selection Procedures Based on Linear Rank-Order Statistics.- 1 Linear Rank-Order Statistics and their Asymptotic Distributions.- 1.1 The general linear rank-order statistic.- 1.2 Some special linear rank-order statistics.- 1.3 The joint asymptotic distribution of the vector of rank-order statistics (S1,...,Sk) based on joint ranks.- 1.4 The treatment of ties.- 2 Two Subset-Selection Procedures in One-Factor-Designs Including the General Behrens-Fisher-Problem.- 2.1 The selection rule R1.- 2.2 The infimum of the probability P(CS | R1).- 2.3 The asymptotic distributions of two special rank-order statistics in case of consistent estimation of the unknown parameters.- 2.4 The probability P(CS | R1) in the LFC.- 2.5 A numerical example.- 2.6 Some Monte-Carlo studies.- 3 The Selection Rule R1 in the Case of Equal Scale-Parameters.- 3.1 The probability P(CS | R1) in the LFC.- 3.2 The Haga-statistic.- 3.3 A numerical example.- 3.4 Some Monte-Carlo studies.- 4 A Further Class of Subset-Selection Procedures in One-Factor Designs.- 4.1 The selection rule R2.- 4.2 The infimum of the probability P(CS | R2).- 4.3 Exact and asymptotic distribution of max Sj?S1 for identically distributed populations.- 4.4 A numerical example.- 4.5 Some Monte-Carlo studies.- 5 Some Properties of Optimality and a Brief Comparison of the Procedures No.2 – No.4.- 5.1 Local optimality of selection rule R1.- 5.2 Influence of the scorefunctions on the efficiency of procedures based on rules R1 and R2.- 5.3 Comparison of the procedures given in sections 2,3,and 4.- 6 The Selection Rules R1 and R2 in Case of Randomized-Block-Designs.- 6.1 Modified definition of ranks and the distribution of the resul-ting rank-order statistics.- 6.2 The rules R1 and R2.- 6.3 The asymptotic and the exact distribution of max Sl?S1 for identical parameters.- 6.4 A numerical example.- 6.5 Some Monte-Carlo studies.- 2 Asymptotic Distribution-Free Sequential Selection Procedures Based on an Indifference-Zone Model.- 1 Introduction.- 2 A Class of Estimators of the Functions fi(?1,...,?k).- 2.1 General one-sample rank-order statistics.- 2.2 The one-sample rank-order statistics based on median-scores.- 2.3 The general Hodges-Lehmann-estimator.- 2.4 A class of compatible estimators of the functions fi(?1,...,?k).- 3 Several Strongly Consistent Estimators.- 3.1 An estimator of (B2(G))-1.- 3.2 An estimator of (g(0)2)-1.- 3.3 Two estimators of ?j(G) and G*(0,0).- 4 A Class of Sequential Selection Procedures.- 4.1 Definition of the selection procedures.- 4.2 Some important properties of the sequential selection procedures.- 5 A Numerical Example and some Remarks Concerning the Practical Working with the Sequential Procedures.- 5.1 An example.- 5.2 The implementation of the procedures.- 6 Another Class of Sequential Selection Procedures.- 7 Asymptotic Efficiency and some Monte-Carlo Studies.- 7.1 The asymptotic efficiency of the procedures of section 4 with respect to the procedures of section 6.- 7.2 Some Monte-Carlo studies of the procedures given in section 4 and 6.- 7.3 Some remarks concerning the application of general scorefunctions.- 3 Methods for Selecting an Optimal Scorefunction.- 1 The Basic Idea.- 2 Two Statistics for Characterizing a Distribution.- 2.1 An estimator for the skewness of some distribution.- 2.2 An estimator for the peakedness of some distribution.- 3 Two Methods for Selecting a Scorefunction.- 3.1 Selection based on the joint sample.- 3.2 Selection based on the single samples.- Appendix 1.- Appendix 2.- Appendix 3.- Appendix 4.- Appendix 5.- Appendix 6.- Abbreviations.- References.