Parametric estimation and tests through divergences and duality technique

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending maximum likelihood technique. An other view at the maximum likelihood approach, for estimation and test, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given both under the null and the alternative hypotheses, and approximation of the power functions is deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on $\chi ^{2}$-divergence on signed finite measures and duality technique

Likelihood-free hypothesis testing

Consider the problem of testing $Z \sim \mathbb P^{\otimes m}$ vs $Z \sim \mathbb Q^{\otimes m}$ from $m$ samples. Generally, to achieve a small error rate it is necessary and sufficient to have $m \asymp 1/\epsilon^2$, where $\epsilon$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions $\mathbb P$ and $\mathbb Q$ and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem, which we call likelihood-free (or simulation-based) hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ (which are a priori only known to belong to a large non-parametric family $\mathcal P$) is given through $n$ iid samples from each. We demostrate existence of a fundamental trade-off between $n$ and $m$ given by $nm \asymp n^2_\mathsf{GoF}(\epsilon,\mathcal P)$, where $n_\mathsf{GoF}$ is the minimax sample complexity of testing between the hypotheses $H_0: \mathbb P= \mathbb Q$ vs $H_1: \mathsf{TV}(\mathbb P,\mathbb Q) \ge \epsilon$. We show this for three non-parametric families $\cal P$: $\beta$-smooth densities over $[0,1]^d$, the Gaussian sequence model over a Sobolev ellipsoid, and the collection of distributions $\mathcal P$ on a large alphabet $[k]$ with pmfs bounded by $c/k$ for fixed $c$. The test that we propose (based on the $L^2$-distance statistic of Ingster) simultaneously achieves all points on the tradeoff curve for these families. In particular, when $m\gg 1/\epsilon^2$ our test requires the number of simulation samples $n$ to be orders of magnitude smaller than what is needed for density estimation with accuracy $\asymp \epsilon$ (under $\mathsf{TV}$). This demonstrates the possibility of testing without fully estimating the distributions.Comment: 48 pages, 1 figur