**2. The Trigonometric Moment Estimator**

The regular symmetric stable distribution is defined through its characteristic function given by

$$\varphi(t) = \exp(it\mu - |\sigma t|^a)$$

where *μ* is the location parameter; *σ* is the scale parameter, which we take as 1; and *α* is the index or shape parameter of the distribution. Here, without loss of generality, we take *μ* = 0.

From the stable distribution, we can obtain the wrapped stable distribution (the process of wrapping explained in Jammalamadaka and SenGupta (2001)). Suppose *θ*1, *θ*2, ..., *θm* is a random sample of size *m* drawn from the wrapped stable (given in Jammalamadaka and SenGupta (2001)) distribution whose probability density function is given by

$$f(\theta, \rho, a, \mu) = \frac{1}{2\pi} [1 + 2 \sum\_{p=1}^{\infty} \rho^{p^u} \cos p(\theta - \mu)] \quad 0 < \rho \le 1, 0 < \mu \le 2, 0 < \mu \le 2\pi \tag{1}$$

It is known in general from Jammalamadaka and SenGupta (2001) that the characteristic function of *θ* at the integer *p* is defined as,

$$\psi\_{\theta}(p) = E[\exp(ip(\theta - \mu))] = \alpha\_p + i\beta\_p$$
 
$$\text{where} \quad \alpha\_p = E\cos p(\theta - \mu) \quad \text{and} \quad \beta\_p = E\sin p(\theta - \mu)$$

Furthermore, from Jammalamadaka and SenGupta (2001), it is known that for, the p.d.f given by Equation (1),

$$
\psi\_{\theta}(p) = \rho^{p^{\pm}}
$$

$$\text{Hence}, \quad E\cos p(\theta - \mu) = \rho^{p^a} \quad \text{and} \quad E\sin p(\theta - \mu) = 0 \tag{2}$$

We define

Then, we note

$$\begin{aligned} \mathcal{C}\_1 &= \frac{1}{m} \sum\_{i=1}^m \cos \theta\_{i\prime} & \mathcal{C}\_2 &= \frac{1}{m} \sum\_{i=1}^m \cos 2\theta\_{i\prime} & \mathcal{S}\_1 &= \frac{1}{m} \sum\_{i=1}^m \sin \theta\_i \\ \text{and} \quad \mathcal{S}\_2 &= \frac{1}{m} \sum\_{i=1}^m \sin 2\theta\_i \\ \text{that } \mathcal{R}\_1 &= \sqrt{\mathcal{C}\_1^2 + \mathcal{S}\_1^2} \text{ and } \mathcal{R}\_2 = \sqrt{\mathcal{C}\_2^2 + \mathcal{S}\_2^2} \end{aligned}$$

By the method of trigonometric moments estimation, equating *R* ¯ 1 and *R* ¯ 2 to the corresponding functions of the theoretical trigonometric moments, we ge<sup>t</sup> the estimator of index parameter *α* as (see SenGupta (1996)):

$$\mathfrak{A} = \frac{1}{\ln 2} \ln \frac{\ln \vec{\mathcal{R}}\_2}{\ln \vec{\mathcal{R}}\_1}$$

Then, we define *R* ¯ *j* = 1 *m* ∑*mi*=<sup>1</sup> cos *j*(*<sup>θ</sup>i* − ¯*<sup>θ</sup>*), *j* = 1, 2 and ¯*θ* is the mean direction given by ¯ *θ* = arctan *S*¯1 *C*¯1. Note that *R*¯1≡ *R*¯.

We consider two special cases.

#### *2.1. Special Case 1 : μ* = 0*, σ* = 1

We now consider the case as treated by Anderson and Arnold (1993), specifically *μ* = 0 and *σ* = 1, and hence the concentration parameter *ρ* = exp(−<sup>1</sup>) as both parameters are known. This case may arise when one has historical data or prior information on the scale parameter. In such a case, the probability density function reduces to

$$f(\theta, \kappa) = \frac{1}{2\pi} [1 + 2 \sum\_{p=1}^{\infty} \{ \exp(-1) \}^{p^a} \cos p\theta) ], \quad 0 < \kappa \le 2$$

In addition, by the method of trigonometric moments estimation, the estimator of index parameter *α* is given by

$$\pounds\_1 = -\frac{\ln C\_2}{\ln 2}$$

#### *2.2. Special Case 2 : μ* = 0*, σ Unknown*

Next, we consider a general case when *μ* = 0 and *σ*, and hence the estimator of the concentration parameter is *ρ* = *R* ¯ 1. This case is especially useful in many real life applications, for example, for price changes in financial data, *μ* = 0 is a standard assumption. In such a case, the probability density function reduces to

$$f(\theta, \alpha) = \frac{1}{2\pi} [1 + 2 \sum\_{p=1}^{\infty} \rho^{p^u} \cos(p\theta)], \quad 0 < \alpha \le 2\pi$$

In addition, by the method of trigonometric moments estimation, the estimator of index parameter *α* is given by

$$\hat{a}\_2 = \frac{1}{\ln 2} \ln \frac{\ln C\_2}{\ln C\_1}$$

As is also seen in Anderson and Arnold (1993), for financial data after using log-ratio transformation, the location parameter of the transformed variable becomes zero. Hence, the case of *μ* = 0 was not considered by Anderson and Arnold (1993) and accordingly by us also for the comparison made in this paper.

#### **3. Derivation of the Asymptotic Distribution of the Moment Estimator**

**Lemma 1.**

$$
\sqrt{m}(T\_m - \mu) \xrightarrow{L} N\_4(0, \Sigma)
$$

$$where \quad T\_m = (\mathcal{C}\_1, \mathcal{C}\_2, \mathcal{S}\_1, \mathcal{S}\_2)',$$

*μ is the mean vector given by*

$$\mu = (\rho \cos \mu\_{0\prime} \rho^{2^a} \cos 2\mu\_{0\prime} \rho \sin \mu\_{0\prime} \rho^{2^a} \sin 2\mu\_0)^{\prime\prime}$$

*and* Σ *is the dispersion matrix given by*

$$
\Sigma = \begin{pmatrix} A & B & C & D \\ B & E & F & G \\ C & F & H & I \\ D & G & I & J \end{pmatrix},
$$

*where*

*A* = *ρ*2*<sup>α</sup>* cos <sup>2</sup>*μ*0+1−2*ρ*<sup>2</sup> cos<sup>2</sup> *μ*0 2 *B* = *ρ* cos *<sup>μ</sup>*0+*ρ*3*<sup>α</sup>* cos <sup>3</sup>*μ*0−2*ρ*2*α*+<sup>1</sup> cos *μ*0 cos 2*μ*0 2 *C* = *ρ*2*<sup>α</sup>* sin <sup>2</sup>*μ*0−2*ρ*<sup>2</sup> cos *μ*0 sin *μ*0 2 *D* = *ρ*3*<sup>α</sup>* sin 3*μ*0+*ρ* sin *<sup>μ</sup>*0−2*ρ*2*α*+<sup>1</sup> cos *μ*0 sin 2*μ*0 2 *E* = *ρ*4*<sup>α</sup>* cos <sup>4</sup>*μ*0+1−<sup>2</sup>(*ρ*2*<sup>α</sup>* )2 cos<sup>2</sup> 2*μ*0 2 *F* = *ρ*3*<sup>α</sup>* sin 3*μ*0−*ρ* sin *<sup>μ</sup>*0−2*ρ*2*α*+<sup>1</sup> cos 2*μ*0 sin *μ*0 2 *G* = *ρ*4*<sup>α</sup>* sin <sup>4</sup>*μ*0−<sup>2</sup>(*ρ*2*<sup>α</sup>* )2 cos 2*μ*0 sin 2*μ*0 2 *H* = <sup>1</sup>−*ρ*2*<sup>α</sup>* cos <sup>2</sup>*μ*0−2*ρ*<sup>2</sup> sin<sup>2</sup> *μ*0 2 *I* = *ρ* cos *<sup>μ</sup>*0−*ρ*3*<sup>α</sup>* cos <sup>3</sup>*μ*0−2*ρ*2*α*+<sup>1</sup> sin *μ*0 sin 2*μ*0 2 *J* = <sup>1</sup>−*ρ*4*<sup>α</sup>* cos <sup>4</sup>*μ*0−<sup>2</sup>(*ρ*2*<sup>α</sup>* )2 sin<sup>2</sup> 2*μ*0 2

**Proof.** The derivations for the proof are given in Appendix A.

Hence, assuming large sample size, central limit theorem Feller (1971) gives (*C*¯1, *C*¯2, *S*¯1, *S*¯2) *L*−→ *<sup>N</sup>*4(*μ*, Σ*m* ) where *μ* is the mean vector given by

$$\mu = (\rho \cos \mu\_{0\prime} \rho^{2^a} \cos 2\mu\_{0\prime} \rho \sin \mu\_{0\prime} \rho^{2^a} \sin 2\mu\_0)^{\prime\prime}$$

and Σ is the dispersion matrix given by

$$
\Sigma = \begin{pmatrix} A & B & C & D \\ B & E & F & G \\ C & F & H & I \\ D & G & I & J \end{pmatrix},
$$

where *A*, *B*, *C*, *D*, *E*, *F*, *G*, *H*, *I* and *J* are as defined above.
