1.5. Let Y yij be a data matrix and let M mij be the corresponding...

1.5. Let Y ¼ ðyijÞ be a data matrix and let M ¼ ðmijÞ be the corresponding missingdata indicator matrix, where mij ¼ 1 indicates missing and mij ¼ 0 indicates present.

(a) Propose situations where two values of mij are not sufficient. (Hint: see Heitjan and Rubin, 1991).

(b) Nearly always it is assumed that M is fully observed. Describe a realistic case when it may make sense to regard part of M itself as missing. (Hint: can you think of a situation where the meaning of a ‘‘blank’’ is unclear?)

(c) Suppose mij ¼ 1 or mij ¼ 0. When attention is focused only on the units that fully respond, the conditional distribution of yi given mi ¼ ð0; 0; ... ; 0Þ is being estimated, where yi and mi are the ith rows of Y and M, respectively. Propose situations where it makes sense to define the conditional distribution of yi given other missing-data patterns. Propose situations where it makes no sense to define these other distributions.

(d) Express the marginal distribution of yi in terms of the conditional distributions of yi given the various missing-data patterns and their probabilities.