**Theorem 3.**

$$
\sqrt{m}(\pounds\_2 - \mathfrak{a}) \xrightarrow{L} N(0, \gamma\_1' \Sigma' \gamma\_1)
$$

*where*

$$\gamma\_1 = \frac{1}{\ln 2} \left( \frac{-1}{\rho \ln \rho}, \frac{1}{\rho^{2^\ast} \ln \rho^{2^\ast}} \right)' $$

*and*

$$\underline{\gamma\_1}\underline{\nu}\underline{\Sigma'}\underline{\gamma\_1} = \frac{1}{(\ln 2)^2} \left[ \frac{1 + \rho^{2^a} - 2\rho^2}{2(\rho \ln \rho)^2} + \frac{1 + \rho^{4^a} - 2(\rho^{2^a})^2}{2(\rho^{2^a} \ln \rho^{2^a})^2} + \frac{2\rho^{2^a + 1} - \rho - \rho^{3^a}}{\rho \ln \rho \rho^{2^a} \ln \rho^{2^a}} \right]$$

**Proof.** We know from Lemma 3 that √*m*(*<sup>T</sup> m* − *μ* ) *L*−→ *<sup>N</sup>*2(0, Σ )

Therefore, by delta method (given in Casella and Berger (2002)), we ge<sup>t</sup>

$$\begin{aligned} \sqrt{m}(\hat{a}\_2 - a) \stackrel{\mathcal{L}}{\rightarrow} N(0, \gamma\_1' \Sigma' \gamma\_1) \text{ where } \mathcal{g}(T\_{\mathfrak{m}}') = \frac{1}{\ln 2} \ln \frac{\ln \mathcal{L}\_2}{\ln \mathcal{C}\_1} \\\ \gamma\_1 = \begin{pmatrix} \frac{\partial \mathcal{g}}{\partial \mathcal{C}\_1} \\ \frac{\partial \mathcal{L}}{\partial \mathcal{L}\_2} \end{pmatrix} \text{ at } \mu^{\prime\prime} = \frac{1}{\ln 2} \begin{pmatrix} \frac{-1}{\mathcal{C}\_1 (\ln \mathcal{C}\_1)} \\ \frac{1}{\mathcal{C}\_2 (\ln \mathcal{C}\_2)} \end{pmatrix} \text{ at } \mu^{\prime\prime} = \frac{1}{\ln 2} \begin{pmatrix} \frac{-1}{\rho \ln \rho} \\ \frac{1}{\rho^{2\prime} \ln \rho^{2\prime}} \end{pmatrix} \end{aligned}$$

$$\frac{\gamma\_1 \mu \mathcal{L}^{\prime} \mathcal{V}\_1}{} = \frac{1}{(\ln 2)^2} \left[ \frac{1 + \rho^{2\prime} - 2\rho^2}{2(\rho \ln \rho)^2} + \frac{1 + \rho^{4\prime} - 2(\rho^{2\ast})^2}{2(\rho^{2\ast} \ln \rho^{2\ast})^2} + \frac{2\rho^{2\ast + 1} - \rho - \rho^{3\ast}}{\rho \ln \rho \rho^{2\ast} \ln \rho^{2\ast}} \right]$$

The above theorems imply the estimator to be consistent. Hence, in large samples, the performance of the estimator is reasonably good. Now, assuming the sample size is large, say 100, we calculate the asymptotic variance *γ* Σ*γ*/100 of *g*(*Tm*) = *α*ˆ for different values of *α* ranging from 0 to 2 and different values of *ρ* ranging from 0 to 1 in Table 1.

**Table 1.** Asymptotic Variances of the moment estimator *α*ˆ and modified truncated estimator *α* ∗ .



**Table 1.** *Cont.*
