Efficient and robust tests for semiparametric models

In this paper, we investigate a hypothesis testing problem in regular semiparametric models using the Hellinger distance approach. Specifically, given a sample from a semiparametric family of $$\nu $$ ν -densities of the form $$\{f_{\theta ,\eta }:\theta \in \Theta ,\eta \in \Gamma \},$$ { f θ , η : θ ∈ Θ , η ∈ Γ } , we consider the problem of testing a null hypothesis $$H_{0}:\theta \in \Theta _{0}$$ H 0 : θ ∈ Θ 0 against an alternative hypothesis $$H_{1}:\theta \in \Theta _{1},$$ H 1 : θ ∈ Θ 1 , where $$\eta $$ η is a nuisance parameter (possibly of infinite dimensional), $$\nu $$ ν is a $$\sigma $$ σ -finite measure, $$\Theta $$ Θ is a bounded open subset of $$\mathbb {R}^{p}$$ R p , and $$\Gamma $$ Γ is a subset of some Banach or Hilbert space. We employ the Hellinger distance to construct a test statistic. The proposed method results in an explicit form of the test statistic. We show that the proposed test is asymptotically optimal (i.e., locally uniformly most powerful) and has some desirable robustness properties, such as resistance to deviations from the postulated model and in the presence of outliers.