  In many fields, and especially in the medical and social sciences and in
recommender systems, data are gathered through clinical studies or targeted
surveys. Participants are generally reluctant to respond to all questions in a
survey or they may lack information to respond adequately to some questions.
The data collected from these studies tend to lead to linear regression models
where the regression vectors are only known partially: some of their entries
are either missing completely or replaced randomly by noisy values. In this
work, assuming missing positions are replaced by noisy values, we examine how a
connected network of agents, with each one of them subjected to a stream of
data with incomplete regression information, can cooperate with each other
through local interactions to estimate the underlying model parameters in the
presence of missing data. We explain how to adjust the distributed diffusion
through (de)regularization in order to eliminate the bias introduced by the
incomplete model. We also propose a technique to recursively estimate the
(de)regularization parameter and examine the performance of the resulting
strategy. We illustrate the results by considering two applications: one
dealing with a mental health survey and the other dealing with a household
consumption survey.
