Dependent SiZer: Goodness-of-Fit Tests for Time Series Models

In this paper, we extend SiZer (SIgnificant ZERo crossing of the derivatives) to dependent data for the purpose of goodness-of-fit tests for time series models. Dependent SiZer compares the observed data with a specific null model being tested by adjusting the statistical inference using an assumed autocovariance function. This new approach uses a SiZer type visualization to flag statistically significant differences between the data and a given null model. The power of this approach is demonstrated through some examples of time series of Internet traffic data. It is seen that such time series can have even more burstiness than is predicted by the popular, long- range dependent, Fractional Gaussian Noise model.

[1]  Sally Floyd,et al.  Wide-area traffic: the failure of Poisson modeling , 1994 .

[2]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[3]  Walter Willinger,et al.  On the Self-Similar Nature of Ethernet Traffic ( extended version ) , 1995 .

[4]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[5]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[6]  Patrice Abry,et al.  Wavelet Analysis of Long-Range-Dependent Traffic , 1998, IEEE Trans. Inf. Theory.

[7]  J. S. Marron,et al.  LASS: a tool for the local analysis of self-similarity , 2006, Comput. Stat. Data Anal..

[8]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[9]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[10]  James Stephen Marron,et al.  Advanced Distribution Theory for SiZer , 2006 .

[11]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[12]  Sujit K. Ghosh,et al.  Essential Wavelets for Statistical Applications and Data Analysis , 2001, Technometrics.

[13]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[14]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[15]  H. Müller,et al.  Local Polynomial Modeling and Its Applications , 1998 .

[16]  Vishal Misra,et al.  On the tails of web file size distributions , 2001 .

[17]  Bart M. ter Haar Romeny,et al.  Front-End Vision and Multi-Scale Image Analysis , 2003, Computational Imaging and Vision.

[18]  R. Ogden,et al.  Essential Wavelets for Statistical Applications and Data Analysis , 1996 .

[19]  Wolfgang Karl Härdle,et al.  An empirical likelihood goodness‐of‐fit test for time series , 2003 .