This thesis consists two chapters. The first chapter proposes a class of minimum distance tests for fitting a parametric regression model to a regression function when some responses are missing at random. These tests are based on a class of minimum integrated square distances between a kernel type estimator of a regression function and the parametric regression function being fitted. The estimators of the regression function are based on two completed data sets constructed by imputation and inverse probability weighting methods. The corresponding test statistics are shown to have asymptotic normal distributions under null hypothesis. Some simulation results are also presented.The second chapter considers the problem of testing the equality of two nonparametric regression curves against a one-sided alternatives based on two samples with possibly distinct design and error densities, when responses are missing at random. This chapter proposes a class of tests using imputation and covariate matching. The asymptotic distributions of these test statistics are shown to be Gaussian under null hypothesis and a class of local nonparametric alternatives. The consistency of these tests against a large class of fixed alternatives is also established. This chapter also includes a simulation study, which assesses the finite sample behavior of a member of this class of tests.
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