Given noisy samples of a signal, the problem of testing whether the signal belongs to a restricted class of parametrically specified class of signals is considered. Using a nonparametric kernel-based approach for reconstructing the signal we compare the parametric signal class to a broad alternative of signal classes that cannot be parametrized. For such a setup, we introduce testing procedures relying on nonparametric kernel-type sampling reconstruction algorithms properly adjusted for noisy data. First, we propose the testing procedures utilizing the L2-distance between the kernel estimate and signals from the target parametric class. Then, we make use of the maximum likelihood ratio test extended to the posed nonparametric signal specification problem. Central limit theorems of the test statistics are derived yielding consistent testing methods. Hence, we obtain testing procedures that keep asymptotically the desired level of the probability of false alarm and showing a power tending to one as the number of samples increases. The finite sample size properties of the introduced tests are also given.

The paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the Whittaker-Shannon (WS) sampling interpolation formula. A class of band-limited signals as well as signals which are non-band-limited are taken into consideration. The proposed reconstruction method is a smooth post-filtering correction of the classical WS interpolation series. We assess both the point-wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes. This includes short and long-memory stationary errors being independent of the sampling rate. We also examine a class of noise processes for which the correlation function depends on the sampling rate. Whereas the short-memory errors have relatively small influence on the reconstruction accuracy, the long-memory errors can greatly slow down the convergence rate. In the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed. We give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero. We argue that the obtained rates are close to be optimal. In fact, in a number of special cases they agree with known optimal min-max rates. The problem of the limit distribution of the L2distance of the proposed reconstruction algorithm is also addressed. This result allows us to tackle an important problem of designing non- parametric lack-of-fit tests. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case.

The paper examines the impact of the additive correlated noise on the accuracy of a signal reconstruction algorithm originating from the Whittaker-Shannon sampling interpolation formula. The proposed reconstruction method is a smooth post-filtering correction of the classical Whittaker-Shannon interpolation series. We assess the global accuracy of the proposed reconstruction algorithm for long memory stationary errors being independent on the sampling rate. We also examine a class of long memory noise processes for which the correlation function depends on the sampling rate. Exact rates at which the reconstruction error tends to zero are evaluated. We apply our theory to the problem of designing non-parametric lack-of-fit tests for verifying a parametric assumption on a signal. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case

The post-filtering modification of the Whittaker-Shannon interpolation series in the presence of long-memory errors is examined. The pointwise and global rates of convergence of the reconstruction algorithm are evaluated.

The problem of recovering a signal in the class of band-limited functions is studied. We consider a situation when discrete data points are first grouped to the points of an uniform grid and then the reconstruction is carried out from such a reduced data set. The data grouping is common for computer rounding errors and may also be viewed as a data compression process. The accuracy of the proposed grouping techniques is examined. These results are used to provide an understanding of the number of grid points required to achieve a given level of accuracy.

We consider the problem of recovering a band-limited signal f(t) from noisy data yk=f(k/spl tau/)+/spl Gt/epsilon//sub k/, where /spl tau/ is the sampling rate. Starting from the truncated Whittaker-Shannon cardinal expansion with or without sampling windows (both cases yield inconsistent estimates of f(t)) we propose estimators that are convergent to f(t) in the pointwise and uniform sense. The basic idea is to cut down high frequencies in the data and to use suitable oversampling /spl tau//spl les//spl pi///spl Omega/, /spl Omega/ being the bandwidth (maximum frequency) of f(t). The simplest estimator we propose is given by f/spl circ//sub n/(t)=/spl tau/ /spl Sigma//|t-k/spl tau/|/spl les/n/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/n/spl tau/. Generalizations of f/spl circ//sub n/ including sampling windows are also examined. The main aim is to examine the mean squared error (MSE) properties of such estimators in order to determine the optimal choice of the sampling rate /spl tau/ yielding the fastest possible rate of convergence. The best rate for the MSE we obtain is O(In(n)/n).