Topics on dimension reduction

In this thesis I present a computationally feasible test for the dimension of the central subspace based on sliced average variance estimation. I also provide a marginal coordinate test. Under the null hypothesis, both the test of dimension and the marginal coordinate test involve test statistics asymptotically have chi-squared distributions given normally distributed predictors, and have a distribution that is a linear combination of chi-squared distributions in general.;I also extend the sliced average variance estimation method to regressions with qualitative predictors and provide tests of dimension and a marginal coordinate hypothesis test. I provide tests of dimension and a marginal coordinate hypothesis test as well.;Finally, I show that under a linearity condition on the distribution of the predictors, the coefficient vector in a single-index regression can be estimated with the same efficiency as in the case when the link function is known. Thus, the linearity condition seems to substitute for knowing the exact conditional distribution of the response given the linear combinations of the predictors.