scholarly journals On the Bickel–Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes

2019 ◽  
Vol 23 ◽  
pp. 464-491
Author(s):  
Agnès Lagnoux ◽  
Thi Mong Ngoc Nguyen ◽  
Frédéric Proïa
Keyword(s):  
Unit Root ◽  
Goodness Of Fit ◽  
Density Estimator ◽  
Autoregressive Processes ◽  
True Density ◽  
Squared Error ◽  
Order Of Magnitude ◽  
Integrated Squared Error ◽  
Unstable Case ◽  
Non Gaussian

We investigate in this paper a Bickel–Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen–Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at − 1 whereas we give further results when the unit root is located at 1. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally, we build a goodness-of-fit Bickel–Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study.

scholarly journals Properties of Transmetric Density Estimation

2016 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Sigve Hovda
Keyword(s):  
Density Estimation ◽  
Kernel Density ◽  
Density Estimator ◽  
Unified Approach ◽  
Squared Error ◽  
Special Cases ◽  
Integrated Squared Error ◽  
The Common ◽  
The Mean ◽  
Convergence Orders

Transmetric density estimation is a generalization of kernel density estimation that is proposed in Hovda(2014) and Hovda (2016), This framework involves the possibility of making assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display.  In this paper we show that several state-of-the-art nonparametric, semiparametric and even parametric methods are special cases of this formulation, meaning that there is a unified approach. Moreover, it is shown that parameters can be trained using unbiased cross-validation.  When parameter estimation is included, the mean integrated squared error of the transmetric density estimator is lower than for the common kernel density estimator, when the number of dimensions is larger than two.

scholarly journals Optimizing Time Histograms for Non-Poissonian Spike Trains

2011 ◽  
Vol 23 (12) ◽  
pp. 3125-3144 ◽  
Author(s):  
Takahiro Omi ◽  
Shigeru Shinomoto
Keyword(s):  
Goodness Of Fit ◽  
Optimization Method ◽  
Spike Trains ◽  
Time Axis ◽  
Squared Error ◽  
Neurophysiological Studies ◽  
Inhomogeneous Density ◽  
Integrated Squared Error ◽  
The Mean ◽  
The Given

The time histogram is a fundamental tool for representing the inhomogeneous density of event occurrences such as neuronal firings. The shape of a histogram critically depends on the size of the bins that partition the time axis. In most neurophysiological studies, however, researchers have arbitrarily selected the bin size when analyzing fluctuations in neuronal activity. A rigorous method for selecting the appropriate bin size was recently derived so that the mean integrated squared error between the time histogram and the unknown underlying rate is minimized (Shimazaki & Shinomoto, 2007 ). This derivation assumes that spikes are independently drawn from a given rate. However, in practice, biological neurons express non-Poissonian features in their firing patterns, such that the spike occurrence depends on the preceding spikes, which inevitably deteriorate the optimization. In this letter, we revise the method for selecting the bin size by considering the possible non-Poissonian features. Improvement in the goodness of fit of the time histogram is assessed and confirmed by numerically simulated non-Poissonian spike trains derived from the given fluctuating rate. For some experimental data, the revised algorithm transforms the shape of the time histogram from the Poissonian optimization method.

NONLINEAR WAVELET DENSITY ESTIMATION FOR TRUNCATED AND DEPENDENT OBSERVATIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 587-609 ◽  
Author(s):  
JUAN-JUAN CAI ◽  
HAN-YING LIANG
Keyword(s):  
Asymptotic Expression ◽  
Kernel Estimator ◽  
Density Estimator ◽  
Mean Integrated Squared Error ◽  
Wavelet Estimator ◽  
Dependent Observations ◽  
Squared Error ◽  
Integrated Squared Error ◽  
Wavelet Density Estimation ◽  
Mixing Sequence

In this paper, we provide an asymptotic expression for mean integrated squared error (MISE) of nonlinear wavelet density estimator for a truncation model. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves. Also, we establish asymptotic normality of the nonlinear wavelet estimator.

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