## Goodness-of-Fit Test for Long Range Dependent Processes

Citations: | 1 - 0 self |

### BibTeX

@MISC{Fay_goodness-of-fittest,

author = {Gilles Fay and Anne Philippe},

title = {Goodness-of-Fit Test for Long Range Dependent Processes},

year = {}

}

### OpenURL

### Abstract

In this paper, we make use of the information measure introduced by Mokkadem (1997) for building a goodness-of-fit test for long-range dependent processes. Our test statistic is in the frequency domain and writes as a non linear functional of the normalized periodogram. We establish the asymptotic distribution of our statistic under the null hypothesis. Under specific alternative hypotheses, we prove that the power converges to one. The performance of our test procedure is illustrated from different simulated series. In particular, we compare its size and its power with test of Chen and Deo.

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Citation Context ...r I n;k(1) f( k(N) ) ; : : : ; I n;k(N) f( k(N) ) converges weakly to a vector of i.i.d. exponential random variables as soon as X is stationary and short range dependent with nite variance (see Broc=-=kwell and Davis, 199-=-1, Theorem 10.3.2). The limit distribution is exactly the distribution of the periodogram of a Gaussian white noise. This fact translates with slight modications to the tapered and pooled periodogram,... |

115 |
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Citation Context ...s not derived. The empirical spectral measure (or integrated periodogram) R x I n ()d became a very popular alternative approach to goodness of t test for spectral densities (see Bartlett, 1955; Gren=-=ander and Rosenblatt, 1957-=-) since this quantity inherits of many of the asymptotic properties of the empirical distribution function of i.i.d observations. Consequently, usual statistics (KolmogorovSmirnov, Cramr-von Mises) ma... |

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Citation Context ...ge dependent (LRD) time series. This test is performed in the frequency domain, whereas many well known goodness-oft tests more likely check for the whiteness of the residuals for a tted model (see Bo=-=x and Pierce, 1970-=-). They include the so-called Portmanteau tests, which have recently been generalized in the spectral domain by Chen and Deo (2000). The authors show the validity of a GOF procedure in short-range dep... |

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Citation Context ...;k = a 1 p p X j=0 p j ( 1) j d n;k+j : (1.5) tapering is a major benet when dealing with possibly over-dierentiated (non-invertible) or nonstationary series (see Hurvich and Chen, 2000, see also Vel=-=asco, 199-=-9). Our GOF procedure may apply in those cases but we shall restrict our attention to invertible and stationary processes, which imply in particular 0 ds1=2. Pooling. After computing the periodogram ... |

44 |
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Citation Context ...or local ones) is not derived. The empirical spectral measure (or integrated periodogram) R x I n ()d became a very popular alternative approach to goodness of t test for spectral densities (see Bart=-=lett, 1955-=-; Grenander and Rosenblatt, 1957) since this quantity inherits of many of the asymptotic properties of the empirical distribution function of i.i.d observations. Consequently, usual statistics (Kolmog... |

17 |
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Citation Context ...Xn ) is a white noise (b) (Xn ) is an ARMA(P;Q) process (SRD) (c) (Xn ) is a FARIMA(0; d; 0) process (LRD) The FARIMA(0; d; 0) processes are simulated using the circulant matrix embedding method (see =-=Bardet et al., 200-=-2, for a review on the simulation of such processes). Figure 1, 2 and 3 give the size-power curves for these three null hypotheses. Whittle's estimator is used to estimate the dierent parameters of ea... |

12 | The FEXP estimator for potentially non-stationary linear time series - Hurvich, Moulines, et al. |

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Citation Context ...en, for any C, lim n!1 P( p KnSn ( I ; g(; ^ d; ^ ) > C) = 1: (3.4) Examples. The theory of parametric estimation and tests under misspecied models is not classical in long range dependence (see Hoso=-=ya, 199-=-7). We give below two practical examples for which the uniqueness of the minimizer h 2 F 0 of S(f; ) is established. Under the uniqueness hypothesis, the consistency (3.3) may be proved along the foll... |

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Citation Context ...quantity inherits of many of the asymptotic properties of the empirical distribution function of i.i.d observations. Consequently, usual statistics (KolmogorovSmirnov, Cramr-von Mises) may apply (see =-=Anderson, 19-=-93). Moreover, some of those results still hold true for linear innite variance (stable) processes (Klueppelberg and Mikosch, 1996), where the shape of the linear lter is tested. Kokoszka and Mikosch ... |

5 |
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Citation Context ...k = 1 2n n X t=1 X t e it k 2 ; k = 2k n ; k = 1; : : : ; n: (1.3) Those quantities will be precisely dened below. The statistic Sn ( I ; 1) is a non-linear functional of the periodogram (see Tanigu=-=chi, 1980-=-; Janas and von Sachs, 1995; Fay et al., 1999, for the issues raised by those objects). Its weak convergence is derived under both whiteness and short range dependent linear hypotheses. We use the heu... |

4 | Consistency for non-linear functions of the periodogram of tapered data - Janas, Sachs - 1995 |

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Citation Context ...4). The hypotheses of Theorem 3.2 are then satised for this example. Note that the weak limit of ^ suitably normalized is also available (Gaussian if ds1=4, Rosenblatt if d > 1=4; for details, see Ya=-=jima, 1-=-993) . 4. Proofs 4.1. Proof of Theorem 3.1. To establish the weak convergence of Theorem 3.1, it suces to prove that i) p KnSn ( I ; f) has the asymptotic normal distribution of (3.2), ii) Sn ( I ; ... |

2 | A generalized portmanteau goodness-of-t test for time series models
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(Show Context)
Citation Context ...procedure introduced in Chen and Deo (2000) (called hereafter Chen and Deo's test). We restrict our comparison to their statistic associated to the Turkey kernel and the bandwidth pn = b3n 0:2 c (see =-=Chen and Deo, 2000-=-, for details). For each examples, we give the size-power curves based on 1000 independent replications. The curve is obtained as follows : for each replication, the test statistic is calculated and c... |

2 |
An e cient taper for potentially overdi erenced longmemory time series
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(Show Context)
Citation Context ...X 1 ; : : : ; Xn . Then d (m;p) n;k = a 1 p p X j=0 p j ( 1) j d n;k+j : (1.5) tapering is a major benet when dealing with possibly over-dierentiated (non-invertible) or nonstationary series (see Hur=-=vich and Chen, 200-=-0, see also Velasco, 1999). Our GOF procedure may apply in those cases but we shall restrict our attention to invertible and stationary processes, which imply in particular 0 ds1=2. Pooling. After co... |

2 |
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(Show Context)
Citation Context ...ns. Consequently, usual statistics (KolmogorovSmirnov, Cramr-von Mises) may apply (see Anderson, 1993). Moreover, some of those results still hold true for linear innite variance (stable) processes (K=-=lueppelberg and Mikosch, 1-=-996), where the shape of the linear lter is tested. Kokoszka and Mikosch (1997) achieved convergence in long range dependence under nite or innite variance hypotheses, using normalized and randomly ce... |

1 | Thormes limites pour les fonctionnelles du priodogramme - Fay - 2000 |

1 |
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(Show Context)
Citation Context ...n ; k = 1; : : : ; n: (1.3) Those quantities will be precisely dened below. The statistic Sn ( I ; 1) is a non-linear functional of the periodogram (see Taniguchi, 1980; Janas and von Sachs, 1995; Fa=-=y et al., 1999-=-, for the issues raised by those objects). Its weak convergence is derived under both whiteness and short range dependent linear hypotheses. We use the heuristic idea that the normalized periodogram (... |

1 |
Discrimination between monotonic trends and long-range dependence
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(Show Context)
Citation Context ...actly the distribution of the periodogram of a Gaussian white noise. This fact translates with slight modications to the tapered and pooled periodogram, but fails in the long memory context (see eg. K=-=nsc-=-h, 1986). Still, many statistical procedures translate into this framework. Thus, to test the hypothesis H 0 : f() 2 g(; d; ) for some 2 ; d; , we shall consider the following periodogram estimate ... |

1 | Une mesure d'information et son application des tests pour les processus arma. C.R. de l'Acad. Sci - Mokkadem - 1994 |

1 | A measure of information and its applications to test for randomness against ARMA alternatives and to goodness-of-t test - Mokkadem - 1997 |