**Assumption 2.**


The Theorem below provides the asymptotic distribution of (12) under Assumptions (*A*0)–(*A*7). Assumption (*A*7) will be later relaxed and a general asymptotic result will be presented in the next subsection. A discussion about Assumption *A*<sup>6</sup> will also be made in the sequel.

**Theorem 3.** *Under Assumptions* (*A*0)*–*(*A*7) *and for the hypothesis in (3) we have*

$$T^0\_{\boldsymbol{\uprho}}\left(\boldsymbol{\hat{\theta}}^{\boldsymbol{r}}\_{\left(\Phi\_2,\boldsymbol{a}\_2\right)}\right) = \frac{2N}{\boldsymbol{q}^{\prime\prime}(1)}d\_{\boldsymbol{\uprho}}\left(\boldsymbol{\uphat{\theta}}^{\boldsymbol{r}}\_{\left(\Phi\_2,\boldsymbol{a}\_2\right)}\right) \xrightarrow[N\to\infty]{L} \chi^2\_{\boldsymbol{m}-1-\boldsymbol{s}-\boldsymbol{\upmu}}$$

*with θ***ˆ** *r* (Φ2,*α*2) *given in (9).*

**Proof.** It is straightforward that

$$\mathbf{p}(\boldsymbol{\hat{\theta}}\_{(\Phi\_2,a\_2)}^r) = \mathbf{p}(\theta\_0) + \mathbf{J}(\theta\_0)(\boldsymbol{\hat{\theta}}\_{(\Phi\_2,a\_2)}^r - \theta\_0) + o(||\boldsymbol{\hat{\theta}}\_{(\Phi\_2,a\_2)}^r - \theta\_0||)$$

which by Theorem 2, expression (11), and for **M**(*θ*0) = **J**(*θ*0)**W**(*θ*0) reduces to

$$\mathbf{p}(\hat{\theta}\_{(\Phi\_2 a\_2)}^r) - \mathbf{p}(\theta\_0) = \mathbf{M}(\theta\_0)(\hat{\mathbf{p}} - \mathbf{p}(\theta\_0)) + o\_{\mathbb{P}}(N^{-1/2})$$

which implies that

$$\sqrt{N}(\mathbf{p}(\boldsymbol{\theta}\_{(\Phi\_2,a\_2)}^{\prime}) - \mathbf{p}(\boldsymbol{\theta}\_0)) \xrightarrow[N \to \infty]{L} \mathrm{N}(\mathbf{0}, \mathbf{M}(\boldsymbol{\theta}\_0) \Sigma\_{\mathbf{p}(\boldsymbol{\theta}\_0)} \mathbf{M}(\boldsymbol{\theta}\_0)^{\top}).\tag{13}$$

Combining the above we obtain

$$\sqrt{N}\begin{pmatrix}\mathfrak{p}-\mathfrak{p}(\mathfrak{e}\_{0})\\\mathfrak{p}(\mathfrak{d}^{\boldsymbol{r}}\_{\left(\mathfrak{d}\_{2},a\_{2}\right)})-\mathfrak{p}(\mathfrak{e}\_{0})\end{pmatrix}\xrightarrow[N\to\infty]{L}\mathrm{N}\left(\mathfrak{d},\begin{pmatrix}I\\\mathbf{M}(\mathfrak{e}\_{0})\end{pmatrix}\Sigma\_{\mathbf{p}(\mathfrak{e}\_{0})}\left(I,\mathbf{M}(\mathfrak{e}\_{0})^{\top}\right)\right)$$

and

$$(\sqrt{N}(\mathfrak{p} - \mathbf{p}(\boldsymbol{\theta}^{r}\_{(\mathfrak{d}\_{2}, \mathfrak{a}\_{2})})) \xrightarrow[N \to \infty]{L} \mathrm{N}(\mathbf{0}, \mathbf{L}(\mathfrak{e}\_{0})) $$

where

$$\mathbf{L}(\boldsymbol{\theta}\_{0}) = \boldsymbol{\Sigma}\_{\mathbf{p}(\boldsymbol{\theta}\_{0})} - \mathbf{M}(\boldsymbol{\theta}\_{0})\boldsymbol{\Sigma}\_{\mathbf{p}(\boldsymbol{\theta}\_{0})} - \boldsymbol{\Sigma}\_{\mathbf{p}(\boldsymbol{\theta}\_{0})}\mathbf{M}(\boldsymbol{\theta}\_{0})^{\top} + \mathbf{M}(\boldsymbol{\theta}\_{0})\boldsymbol{\Sigma}\_{\mathbf{p}(\boldsymbol{\theta}\_{0})}\mathbf{M}(\boldsymbol{\theta}\_{0})^{\top}.\tag{14}$$

The expansion of *dϕ*(**p**, **q**) around (**p**(*θ*0), **p**(*θ*0)) yields

$$T^0\_\varphi \left( \boldsymbol{\hat{\theta}}^r\_{(\Phi\_2, a\_2)} \right) = \sum\_{i=1}^m \frac{N}{p\_i(\boldsymbol{\theta}\_0)} \left( \boldsymbol{\hat{\rho}}\_i - p\_i(\boldsymbol{\hat{\theta}}^r\_{(\Phi\_2, a\_2)}) \right)^2 + o\_\mathcal{P}(1) = \mathbf{X}^\top \mathbf{X} + o\_\mathcal{P}(1)$$

where

$$\mathbf{X} = \sqrt{N}diag\left(\mathbf{p}\left(\theta\_0\right)^{-1/2}\right) \left(\boldsymbol{\mathfrak{p}} - \mathbf{p}\left(\hat{\theta}\_{\left(\Phi\_2,\mu\_2\right)}^{\boldsymbol{r}}\right)\right) \xrightarrow[N \to \infty]{L} \mathrm{N}\left(\mathbf{0}, \mathbf{T}\left(\theta\_0\right)\right).$$

Then, under *A*7, **T**(*θ*0) (see (14)) is a projection matrix of rank *m* − 1 − *s* + *ν* since the trace of the matrices **A**(*θ*0) **A**(*θ*0)**A**(*θ*0) <sup>−</sup><sup>1</sup> **A**(*θ*0) and **A**(*θ*0) **A**(*θ*0)**A**(*θ*0) <sup>−</sup><sup>1</sup> **Q**(*θ*0) **Q**(*θ*0)(**A**(*θ*0)**A**(*θ*0) <sup>−</sup><sup>1</sup> **Q**(*θ*0))−<sup>1</sup> **Q**(*θ*0) **A**(*θ*0)**A**(*θ*0) <sup>−</sup><sup>1</sup> **A**(*θ*0) is equal to *s* and *ν*, respectively.

Then, the result follows from the fact (see ([24], p. 57)) that **XX** has a chi-squared distribution with degrees of freedom equal to the rank of the variance-covariance matrix of the random vector **X** as long as it is a projection matrix.

**Remark 2.** *Relaxation of Assumption* (*A*6)*: Arguing as in [11], when the true model is not the equiprobable the result of Theorem 3 holds true as long as α*<sup>2</sup> = 0 *and approximately true when α*<sup>2</sup> → 0*.*
