**Author details**

Example 2

*Asymptotic distribution of LRm.*

**Figure 4.**

I 5%

*Recent Advances in Numerical Simulations*

II 5%

**Table 6.**

10%

10%

*The rejection frequency computed on 10000 replications.*

**4. Conclusion**

**204**

**Table 6** shows the rejection frequency computed on 10000 replications of

**Model** *α* LR **test** ð Þ *n* ¼ **200** LR **test** ð Þ *n* ¼ **500**

0*:*0615 0*:*1225

0*:*9999 0*:*9999

Model II: Restricted *PEXPAR*4ð Þ1 with *φ* ¼ �ð Þ 0*:*8, 2, 0*:*5, �1*:*5, 0*:*9, 1*:*1, �0*:*4, 0*:*6 <sup>0</sup>

ð Þ� <sup>1</sup> � *<sup>α</sup>* quantile of the *<sup>χ</sup>*<sup>2</sup> distribution with 4 degrees of freedom. From **Figure 4**, we see that the asymptotic distribution of *LRm* (in full line) is close to the *χ*<sup>2</sup>

The periodic restricted *EXPAR* model is added to the family of nonlinear and periodic models. Interest is focused on estimation and testing problems. The

<sup>4</sup>ð Þ <sup>1</sup> � *<sup>α</sup>* , where *<sup>χ</sup>*<sup>2</sup>

.

0*:*0581 0*:*1075

> 1 1

<sup>4</sup>ð Þ 1 � *α* is the

<sup>4</sup> (in

Model I: *PAR*4ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*8, 0*:*5, 0*:*9, �0*:*4 <sup>0</sup>

and *γ* ¼ 1*:*, **Figure 4** shows the asymptotic distribution of *LRm* under the null hypothesis. The results show that the empirical levels are acceptable, for *n* ¼ 500, we have a relative rejection frequency of 5*:*81% (resp. 10*:*75%) which is very close to 5% (resp. 10%), the empirical power increases with the size *n* which means that the

simulations of length *n* ¼ 200 and 500 from the 2 models.

test is consistent. The rejection region is *LRm* >*χ*<sup>2</sup>

dashed lines), this confirm the above theoretical result.

Mouna Merzougui LaPS Laboratory, University Badji Mokhtar Annaba, Algeria

\*Address all correspondence to: merzouguimouna@yahoo.fr

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
