  This article introduces a new instrumental variable approach for estimating
unknown population parameters with data having nonrandom missing values. With
coarse and discrete instruments, Shao and Wang (2016) proposed a semiparametric
method that uses the added information to identify the tilting parameter from
the missing data propensity model. A naive application of this idea to
continuous instruments through arbitrary discretizations is apt to be
inefficient, and maybe questionable in some settings. We propose a
nonparametric method not requiring arbitrary discretizations but involves
scanning over continuous dichotomizations of the instrument; and combining scan
statistics to estimate the unknown parameters via weighted integration.
  We establish the asymptotic normality of the proposed integrated estimator
and that of the underlying scan processes uniformly across the instrument
sample space. Simulation studies and the analysis of a real data set
demonstrate the gains of the methodology over procedures that rely either on
arbitrary discretizations or moments of the instrument.
