**4. Improvement Over the Moment Estimator**

The moment estimator need not always remain in the support of the true parameter *α* (that is (0,2]). Hence, the moment estimators proposed above do not need to be proper estimators of *α*. A modified estimator free from this defect is given by

$$\begin{aligned} \hat{a}^\* &= \quad \pounds \quad \text{if } 0 < \mathbb{R} < 2\\ &= \quad \pounds \quad \text{if } \mathbb{R} \ge 2 \end{aligned}$$

(since support of *α* excludes non-positive values).

Thus, the density function of *α*ˆ∗ is given by

$$\begin{aligned} \text{g}(\hat{a}^{\star}) &= \frac{P[\hat{\mathbb{R}} < 2]}{P[\hat{\mathbb{R}} \ge 0]} \quad ; 0 < \hat{a}^{\star} < 2 \equiv -\infty < \mathbb{R} < 2 \\ &= P[\hat{a}^{\star} = 2] \quad ; \hat{a}^{\star} = 2 \equiv \mathbb{R} \ge 2 \\ &= \frac{P[\hat{\mathbb{R}} \ge 2]}{P[\hat{\mathbb{R}} \ge 0]} \quad ; \hat{a}^{\star} = 2 \equiv \mathbb{R} \ge 2 \end{aligned}$$

where *f*(*α*<sup>ˆ</sup>) is the density function of *α*ˆ ∼ *<sup>N</sup>*(*<sup>α</sup>*, *<sup>γ</sup>* <sup>Σ</sup>*γ*/*m*). Therefore,

$$\begin{split} \log(\hat{a}^{\star}) &= \frac{\Phi\left(\frac{2-\alpha}{\sqrt{2^{t}\Sigma\underline{\chi}/m}}\right)}{1-\Phi\left(\frac{-\alpha}{\sqrt{2^{t}\Sigma\underline{\chi}/m}}\right)} \quad \text{;} 0 < \hat{a}^{\star} < 2 \equiv -\infty < \hat{\kappa} < 2 \text{ } \hat{\kappa} \\ &= 1 - \frac{\Phi\left(\frac{2-\alpha}{\sqrt{2^{t}\Sigma\underline{\chi}/m}}\right)}{\Phi\left(\frac{\alpha}{\sqrt{2^{t}\Sigma\underline{\chi}/m}}\right)} \quad \text{;} \hat{a}^{\star} = 2 \equiv \hat{\kappa} \ge 2 \end{split}$$

Thus, we ge<sup>t</sup> *g*(*α*<sup>ˆ</sup>∗) as a mixed distribution of one atomic mass function and a continuous function.

$$\text{4.1. } \text{Special Case 1}: \mu = 0, \sigma = 1$$

Similar modifications can be made for the estimator *α*ˆ1. Let it be denoted by *α* ˆ ∗ 1.

 $4.2. \text{  $Special Case 2 : } \mu = 0, \sigma \text{ $ llukuown.}$ 

Similar modifications can be made for the estimator *α*ˆ2. Let it be denoted by *α* ˆ ∗ 2.

#### **5. Derivation of the Asymptotic Distribution of the Modified Truncated Estimators**

Now, using the asymptotic normal distribution of *α*ˆ, we can derive the same results for the modified truncated estimator of the index parameter *α* (given as below) as we have done for the method of moment estimator of *α*.

The mean of *α*ˆ∗ is given by

$$E(\hat{\alpha}^\*) = 0.P(\hbar < 0) + \int\_0^2 \hbar f(\hbar)d\hbar + 2.P(\hbar > 2).$$

where √*m*(*α*<sup>ˆ</sup> − *α*) →N(0,*γ* Σ*γ*) asymptotically (as noted above) and f(*α*ˆ) =probability density function of *α*ˆ.

The above is equivalent to *τ* = √ *α* ˆ−*α γ* Σ*γ*/*m* →N(0,1) asymptotically. Let *φ*(*τ*) and <sup>Φ</sup>(*τ*) denote the p.d.f. and c.d.f. of *τ*, respectively. Let *σ* = *γ* Σ*γ m* . Then, we get,

$$\begin{aligned} E(\hat{a}^\*) &= aP(\tau < a^\*) + \int\_{a^\*}^{b^\*} (\tau \sigma + a)\phi(\tau)d\tau + bP(\tau > b^\*) \\ \Rightarrow E(\hat{a}^\*) &= \sigma \left[ \{\phi(a^\*) - \phi(b^\*)\} \right] + a \left[ \Phi(b^\*) - \Phi(a^\*) \right] \end{aligned}$$

$$=\alpha$$

since [Φ(*b*∗) − <sup>Φ</sup>(*a*<sup>∗</sup>)] → 1, *b*[1 − Φ(*b*∗)] → 0 and *σ* → 0 as *m* → infinity where *<sup>a</sup>*<sup>∗</sup>= -<sup>−</sup>*αγ* <sup>Σ</sup>*γ m* and *b*<sup>∗</sup>= 2−*α*

$$\begin{aligned} &E(\frac{\sqrt{2\_2}}{\pi})\\ &E(\hat{a}^2) = 0^2.P(\hat{\mathfrak{n}} < 0) + \frac{2}{\pi}\hbar^2 f(\mathfrak{k})\mathrm{d}\mathfrak{k} + 4.P(\mathfrak{k} > 2) \\ &\Rightarrow \, E(\hat{a}^2) = \sigma^2 \Big[ \{a^\*\phi(a^\*) - b^\*\phi(b^\*) + \Phi(b^\*) - \Phi(a^\*)\} \Big] + a^2 \{\Phi(b^\*) - \Phi(a^\*)\} + 2a\sigma \{\phi(a^\*) - \Phi(b^\*)\} \\ &\phi(b^\*) \} \text{ since } b^2, [1 - \Phi(b^\*)] \to 0 \text{ as } m \to \text{infinity} \end{aligned}$$

The asymptotic variance of *α*ˆ∗ is given by

$$V(\hat{a^\*}) = E(\hat{a^\*}^2) - [E(\hat{a^\*})]^2$$

Similarly, the mean of *α* ˆ ∗ 1is given by

$$\begin{split} E(\hat{a}\_1^\circ) &= \frac{\frac{\partial q(\mathcal{T}\_m^\circ)}{\partial \rho^\circ}}{\sqrt{m}} \Big[ \{\Phi(a') - \Phi(b')\} \Big] + a \Big[ \Phi(b') - \Phi(a') \Big] \text{ since } b[1 - \Phi(b')] \to 0 \text{ as } m \to \text{infinity} \\ E(\hat{a}\_1^\circ) &= \frac{\sigma^2 \frac{(\partial q(\mathcal{T}\_m^\circ))^2}{\partial \rho^\circ}}{m} \Big[ \{a' \phi(a') - b' \phi(b') + \Phi(b') - \Phi(a')\} \Big] + a^2 \{\Phi(b') - \Phi(a')\} \Big] + \\ 2a \frac{\sigma \frac{(\partial q(\mathcal{T}\_m^\circ))}{\sqrt{m}}}{\sqrt{m}} \{\phi(a') - \phi(b')\} \text{ since } b^2. \left[1 - \Phi(b')\right] \to 0 \text{ as } m \to \text{infinity} \end{split}$$

The asymptotic variance of *α* ˆ ∗ 1is given by

$$V(\hat{\alpha}\_1^\*) = E(\hat{\alpha}\_1^{\*^2}) - [E(\hat{\alpha}\_1^\*)]^2$$

where *<sup>a</sup>* <sup>=</sup>−*<sup>α</sup> σ <sup>∂</sup>g*(*<sup>T</sup> m*) *∂μ* √*m* and *<sup>b</sup>* =(<sup>2</sup> − *α*) *σ <sup>∂</sup>g*(*<sup>T</sup> m*) *∂μ* √*m*

> The mean of *α* ˆ ∗ 2 is given by

$$\begin{aligned} E(\hat{a}\_2^\gamma) &= \frac{\sqrt{\gamma\_1 \mathcal{C} \mathcal{D}\_1}}{\sqrt{m}} \Big[ \left\{ \Phi(a^{\prime \prime}) - \Phi(b^{\prime \prime}) \right\} \Big] + a \Big[ \Phi(b^{\prime \prime}) - \Phi(a^{\prime \prime}) \Big] \text{ since } b[1 - \Phi(b^{\prime \prime})] \to 0 \text{ as } m \to \text{infinity} \\ E(\hat{a}\_2^{\gamma\_2}) &= \frac{\gamma\_1 \mathcal{C} \mathcal{D}\_1}{m} \Big[ \left\{ a^{\prime \prime} \Phi(a^{\prime \prime}) - b^{\prime \prime} \Phi(b^{\prime \prime}) + \Phi(b^{\prime \prime}) - \Phi(a^{\prime \prime}) \right\} \Big] + a^2 \{ \Phi(b^{\prime \prime}) - \Phi(a^{\prime \prime}) \} \, + \cdots \\ 2a \sqrt{\frac{\gamma\_1 \mathcal{C} \mathcal{D}\_1}{m}} \{ \phi(a^{\prime \prime}) - \phi(b^{\prime \prime}) \} \text{ since } b^2. [1 - \Phi(b^{\prime \prime})] \to 0 \text{ as } m \to \text{infinity} \end{aligned}$$

The asymptotic variance of *α* ˆ ∗ 2 is given by

$$V(\hat{a\_2^\*}) = E(\hat{a\_2^\*}^2) - [E(\hat{a\_2^\*})]^2$$

where *<sup>a</sup>* = −*α <sup>γ</sup>*1 <sup>Σ</sup> *<sup>γ</sup>*1 *m* and *b* = 2−*α <sup>γ</sup>*1 <sup>Σ</sup> *<sup>γ</sup>*1 *m*

> Thus, the following theorem is established
