**Assumption 2.**


$$S := \begin{pmatrix} S\_{11} & S\_{12} \\ S\_{21} & S\_{22} \end{pmatrix} \\ \tag{52}$$

*with <sup>S</sup>*<sup>11</sup> := ( *<sup>∂</sup> <sup>∂</sup>tψ*(*y*, *<sup>θ</sup>*0, *<sup>t</sup>θ*<sup>0</sup> )*dP*0(*y*))*, <sup>S</sup>*<sup>12</sup> := ( *<sup>∂</sup> ∂θ ψ*(*y*, *θ*0, *tθ*<sup>0</sup> )*dP*0(*y*))*, <sup>S</sup>*<sup>21</sup> := ( *<sup>∂</sup>*<sup>2</sup> *∂θ∂tm*(*y*, *<sup>θ</sup>*0, *<sup>t</sup>θ*<sup>0</sup> )*dP*0(*y*)) *and <sup>S</sup>*<sup>22</sup> :<sup>=</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>*2*<sup>θ</sup> <sup>m</sup>*(*y*, *<sup>θ</sup>*0, *<sup>t</sup>θ*<sup>0</sup> )*dP*0(*y*)*, exists and is invertible.*

**Proposition 2.** *Let <sup>P</sup>*<sup>0</sup> *belong to the model* <sup>M</sup>1*, and suppose that Assumption 2 holds. Then, both* <sup>√</sup>*n*(*<sup>θ</sup>* 4*c <sup>ϕ</sup>* <sup>−</sup> *<sup>θ</sup>*0) *and* <sup>√</sup>*n*(4*<sup>t</sup> c θ* 4*c ϕ* − *tθ*<sup>0</sup> ) *converge in distribution to a centred multivariate normal variable with covariance matrices given by*

$$\left[ [\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}]^{-1}\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1} \right] \times \left[ [\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}]^{-1}\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1} \right]^{\top},\tag{53}$$

*and*

$$\left[\mathbf{S}\_{11}^{-1} - \mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}[\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}]^{-1}\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\right] \times \left[\mathbf{S}\_{11}^{-1} - \mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}[\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\mathbf{S}\_{12}]^{-1}\mathbf{S}\_{21}\mathbf{S}\_{11}^{-1}\right]^{\top}.\tag{54}$$

The condition of Type (a) from Assumption 1 is usually considered to apply the uniform law of large numbers. For many choices of divergence (for example, those from the Cressie–Read family), the function Ψ is continuous in (*θ*, *t*), and consequently, this condition is verified. The second condition from Assumption 1 is imposed for the uniqueness of (*θ*0, *tθ*<sup>0</sup> ) as a solution of the equation and is verified, for example, whenever Ψ is continuous and the parameter space is compact ([28], p. 46). Furthermore, the conditions of Type (b)–(d), included in Assumption 2, are often imposed in order to apply the law of large numbers or the central limit theorem and can be verified for the functions appearing in the definitions of estimators proposed in the present paper.
