91 research outputs found

### Goodness-of-fit testing and quadratic functional estimation from indirect observations

We consider the convolution model where i.i.d. random variables $X_i$ having
unknown density $f$ are observed with additive i.i.d. noise, independent of the
$X$'s. We assume that the density $f$ belongs to either a Sobolev class or a
class of supersmooth functions. The noise distribution is known and its
characteristic function decays either polynomially or exponentially
asymptotically. We consider the problem of goodness-of-fit testing in the
convolution model. We prove upper bounds for the risk of a test statistic
derived from a kernel estimator of the quadratic functional $\int f^2$ based on
indirect observations. When the unknown density is smoother enough than the
noise density, we prove that this estimator is $n^{-1/2}$ consistent,
asymptotically normal and efficient (for the variance we compute). Otherwise,
we give nonparametric upper bounds for the risk of the same estimator. We give
an approach unifying the proof of nonparametric minimax lower bounds for both
problems. We establish them for Sobolev densities and for supersmooth densities
less smooth than exponential noise. In the two setups we obtain exact testing
constants associated with the asymptotic minimax rates.Comment: Published in at http://dx.doi.org/10.1214/009053607000000118 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Sharp minimax tests for large covariance matrices and adaptation

We consider the detection problem of correlations in a $p$-dimensional
Gaussian vector, when we observe $n$ independent, identically distributed
random vectors, for $n$ and $p$ large. We assume that the covariance matrix
varies in some ellipsoid with parameter $\alpha >1/2$ and total energy bounded
by $L>0$. We propose a test procedure based on a U-statistic of order 2 which
is weighted in an optimal way. The weights are the solution of an optimization
problem, they are constant on each diagonal and non-null only for the $T$ first
diagonals, where $T=o(p)$. We show that this test statistic is asymptotically
Gaussian distributed under the null hypothesis and also under the alternative
hypothesis for matrices close to the detection boundary. We prove upper bounds
for the total error probability of our test procedure, for $\alpha>1/2$ and
under the assumption $T=o(p)$ which implies that $n=o(p^{ 2 \alpha})$. We
illustrate via a numerical study the behavior of our test procedure. Moreover,
we prove lower bounds for the maximal type II error and the total error
probabilities. Thus we obtain the asymptotic and the sharp asymptotically
minimax separation rate $\tilde{\varphi} = (C(\alpha, L) n^2 p )^{- \alpha/(4
\alpha + 1)}$, for $\alpha>3/2$ and for $\alpha >1$ together with the
additional assumption $p= o(n^{4 \alpha -1})$, respectively. We deduce rate
asymptotic minimax results for testing the inverse of the covariance matrix. We
construct an adaptive test procedure with respect to the parameter $\alpha$ and
show that it attains the rate $\tilde{\psi}= ( n^2 p / \ln\ln(n
\displaystyle\sqrt{p}) )^{- \alpha/(4 \alpha + 1)}$

### Sharp detection of smooth signals in a high-dimensional sparse matrix with indirect observations

We consider a matrix-valued Gaussian sequence model, that is, we observe a
sequence of high-dimensional $M \times N$ matrices of heterogeneous Gaussian
random variables $x_{ij,k}$ for $i \in\{1,...,M\}$, $j \in \{1,...,N\}$ and $k
\in \mathbb{Z}$. The standard deviation of our observations is \ep k^s for
some \ep >0 and $s \geq 0$.
We give sharp rates for the detection of a sparse submatrix of size $m \times
n$ with active components. A component $(i,j)$ is said active if the sequence
$\{x_{ij,k}\}_k$ have mean $\{\theta_{ij,k}\}_k$ within a Sobolev ellipsoid of
smoothness $\tau >0$ and total energy $\sum_k \theta^2_{ij,k}$ larger than
some r^2_\ep. Our rates involve relationships between $m,\, n, \, M$ and $N$
tending to infinity such that $m/M$, $n/N$ and \ep tend to 0, such that a
test procedure that we construct has asymptotic minimax risk tending to 0.
We prove corresponding lower bounds under additional assumptions on the
relative size of the submatrix in the large matrix of observations. Except for
these additional conditions our rates are asymptotically sharp. Lower bounds
for hypothesis testing problems mean that no test procedure can distinguish
between the null hypothesis (no signal) and the alternative, i.e. the minimax
risk for testing tends to 1

### Detection of a sparse submatrix of a high-dimensional noisy matrix

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+\xi_{ij}$ with $\xi_{ij}\sim
{\mathcal {N}}(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in \mathbb {R}$. We test the
null hypothesis $s_{ij}=0$ for all $i,j$ against the alternative that there
exists some submatrix of size $n\times m$ with significant elements in the
sense that $s_{ij}\ge a>0$. We propose a test procedure and compute the
asymptotical detection boundary $a$ so that the maximal testing risk tends to 0
as $M\to\infty$, $N\to\infty$, $p=n/N\to0$, $q=m/M\to0$. We prove that this
boundary is asymptotically sharp minimax under some additional constraints.
Relations with other testing problems are discussed. We propose a testing
procedure which adapts to unknown $(n,m)$ within some given set and compute the
adaptive sharp rates. The implementation of our test procedure on synthetic
data shows excellent behavior for sparse, not necessarily squared matrices. We
extend our sharp minimax results in different directions: first, to Gaussian
matrices with unknown variance, next, to matrices of random variables having a
distribution from an exponential family (non-Gaussian) and, finally, to a
two-sided alternative for matrices with Gaussian elements.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ470 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Quadratic functional estimation in inverse problems

We consider in this paper a Gaussian sequence model of observations $Y_i$,
$i\geq 1$ having mean (or signal) $\theta_i$ and variance $\sigma_i$ which is
growing polynomially like $i^\gamma$, $\gamma >0$. This model describes a large
panel of inverse problems. We estimate the quadratic functional of the unknown
signal $\sum_{i\geq 1}\theta_i^2$ when the signal belongs to ellipsoids of both
finite smoothness functions (polynomial weights $i^\alpha$, $\alpha>0$) and
infinite smoothness (exponential weights $e^{\beta i^r}$, $\beta >0$, $0<r \leq
2$). We propose a Pinsker type projection estimator in each case and study its
quadratic risk. When the signal is sufficiently smoother than the difficulty of
the inverse problem ($\alpha>\gamma+1/4$ or in the case of exponential
weights), we obtain the parametric rate and the efficiency constant associated
to it. Moreover, we give upper bounds of the second order term in the risk and
conjecture that they are asymptotically sharp minimax. When the signal is
finitely smooth with $\alpha \leq \gamma +1/4$, we compute non parametric upper
bounds of the risk of and we presume also that the constant is asymptotically
sharp

- …