206 Introduction to Robust Estimation and Hypothesis Testing There remains the problem of approximating the null distribution of T and here a basic bootstrap-t method is used. To make sure the details are clear, the method begins by randomly sampling with replacement N = n + n 1 + n 2 pairs of observations from (X 1 , Y 2 ), . . . , (X N , Y N ) yielding (X 1 , Y 2 ), . . . , (X , Y N ). Based on this bootstrap sample, N compute the absolute value of the test statistic as just described and label the result T . Repeat this process B times and put the resulting T values in ascending order yielding T (1) · · · T (B) . Then an approximate 1 - confidence interval for t is ^ t ± T (c) ^ where c = (1 - )B rounded to the nearest integer. Method M2 Method M2 is based on the usual percentile bootstrap method. For the situation at hand, ~ ~ ~ generate a bootstrap sample using all N pairs of observations and let D t = X t - Y t , where ~ t is the trimmed mean based on all of the X values not missing and Y t is computed in a ~ X i ~ similar manner. Repeat this B times, put the resulting D t values in ascending order, and label ~ · · · D . Then an approximate 1 - confidence interval for µ t D is ~ the results D t (1) t (B) ~ ( D t ( +1) , ~ D t (u) ), where = B/2, rounded to the nearest integer, and u = B - . A p-value is computed in the ~ usual manner. That is, estimate p = P( µ D > 0) with p , the proportion of D t values greater ^ t ^ than 0. Then a (generalized) p-value is P = 2min( p , 1 - p ). ^ Method M3 Method M3 is based on D , the median of the distribution of D = X - Y . The method begins by forming all pairwise differences among all of the observed X and Y values. That is, compute D i j = X i - Y j (i = 1, . . . , N 1 ; j = 1, . . . , N 2 ) resulting in N 1 × N 2 D i j values. Then an estimate of D is obtained by computing the sample median of the D i j values. Again a basic percentile bootstrap method is used to make inferences about D . Generate a ^ bootstrap sample as done in Method M2 and let D be the resulting estimate of D . Repeat this ^ process B times yielding Db , b = 1, . . . , B. Next, put these B values in ascending order ^ · · · D(B) and let and u be defined as before. Then a 1 - confidence ^ yielding D(1) interval for D is ^ ^ ( D( +1) , D(u) ). www.elsevierdirect.com