**8. Applications**

#### *8.1. Inference on the Gold Price Data (In US Dollars) (1980–2013)*

Gold price data, say *xt*, were collected per ounce in US dollars over the years 1980–2013. These were transformed as *zt* = <sup>100</sup>(ln(*xt*) − ln(*xt*−<sup>1</sup>)), which were then "wrapped" to obtain *θt* = *ztmod*2*π* and finally transformed to *θ* = (*θt* − ¯*θ*) mod 2*π*, where ¯*θ* denotes the mean direction of *θt* and *θ* denotes the variable thetamod as used in the graphs. The Durbin–Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. The test statistic of Watson's goodness of fit Jammalamadaka and SenGupta (2001) for wrapped stable distribution was obtained as 0.01632691 and the corresponding p-value was obtained as 0.9970284, which is greater than 0.05, indicating that the wrapped stable distribution fits the transformed gold price data (in US dollars). The modified truncated estimate *α* ˆ ∗ 1 is 0.3752206 while the estimate by characteristic function method is 0.401409. The value of the objective function using the characteristic function estimate is 2.218941 while that using our modified truncated estimate is 2.411018.

#### *8.2. Inference on the Silver Price Data (In US Dollars) (1980–2013)*

Data on the price of silver in US dollars collected per ounce over the same time period also underwent the same transformation. The Durbin–Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. Here, the Watson's goodness of fit test for wrapped stable distribution was also performed and the value of the statistic was obtained as 0.02530653 and the corresponding *p*-value is 0.9639666, which is greater than 0.05, indicating that the wrapped stable distribution also fits the transformed silver price data (in US dollars). The modified truncated estimate of the index parameter *α* is 0.4112475 while the estimate by characteristic function method is 0.644846. The value of the objective function using the characteristic function estimate is 2.234203 while that using our modified truncated estimate is 2.234432.

#### *8.3. Inference on the Silver Price Data (In INR) (1970–2011)*

Data on the price of silver in INR were also collected per 10 grams over the same time period. The *p*-value for the Durbin–Watson test performed on the log ratio transformed data is 0.3437, which indicates that the autocorrelation is zero. Here, the Watson's goodness of fit test was also performed on the transformed data and the value of the statistic was obtained as 0.03382334 and the corresponding *p*-value is 0.8919965, which is greater than 0.05, indicating that the wrapped stable distribution also fits the silver price data (in INR). The estimate *α* ˆ ∗ 1 is 1.142171, which is the same as the characteristic function estimate. The value of the objective function using the characteristic function estimate is 2.813234 while that using our modified truncated estimate is 2.665166. Since the estimate of *α* lies between 1 and 2, a mixture of normal and Cauchy distributions is used in Anderson and Arnold (1993) to estimate the respective parameters. The initial values of the scale parameter (*σ*1) for the normal distribution is taken as the sample standard deviation and that for the Cauchy distribution (*σ*2) is taken as the sample quartile deviation. In addition, different initial values of the mixing parameter *p* yield the same estimate of the parameters, viz. *p*ˆ = 0.165, *σ*ˆ1 = 14.38486, and *σ*ˆ2 = 0.077, and the value of the objective function was found to be 0.9308165. Then, the value of I.O. using modified truncated estimate (assuming stable distribution) is 4.665166 (2.665166 + 2), using the characteristic function estimate (assuming stable distribution) is 4.813234 (2.813234 + 2), and using the characteristic function estimate (assuming mixture of normal and Cauchy distribution) is 3.9308165 (0.9308165 + 3). Thus, it can be observed using the I.O. measure that a mixture of normal and Cauchy distribution gives the best fit to the data. The maximum likelihood estimate of *α* assuming wrapped stable distribution is 1.1421361. Akaike's information criterion (AIC) value assuming wrapped stable distribution is 153.5426 and that assuming a mixture of normal and Cauchy distribution is 201.4.

#### *8.4. Inference on the Box and Jenkins Stock Price Data*

Series B Box and Jenkins (IBM) common stock closing price data obtained from Box et al. (2016) were also transformed similarly as for the preceding one. The Durbin–Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. Watson's test statistic for the goodness of fit test was obtained as 0.0554223 and the corresponding *p*-value was obtained as 0.6442058, which is greater than 0.05, indicating that the wrapped stable distribution fits the stock price data. The estimates of the index parameter *α* and the concentration parameter *ρ* as obtained by modified truncation method are 1.102854 and 0.4335457, respectively.
