**3. Statistical Inference**

In this section, we introduce the double index divergence test statistic

$$\left(T^{a\_1}\_{\Phi\_1}\left(\hat{\boldsymbol{\theta}}^r\_{(\Phi\_2,a\_2)}\right)\right) = \frac{2N}{\Phi\_1^{\prime\prime}(1)}d^{a\_1}\_{\Phi\_1}\left(\boldsymbol{\mathfrak{p}},\boldsymbol{\mathfrak{p}}\left(\hat{\boldsymbol{\theta}}^r\_{(\Phi\_2,a\_2)}\right)\right) \tag{12}$$

with Φ1, Φ<sup>2</sup> ∈ *F*<sup>∗</sup> and *α*1, *α*<sup>2</sup> > 0 and make the additional assumptions by which we focus on the Csiszar's family of measures for testing purposes (the notation *ϕ* is used for clarity) and the equiprobable model:
