The skew-symmetric-Laplace-uniform distribution

Symmetry is something which we try to seek naturally in everything, but not everything in the world is symmetric. So expecting symmetry in everything is unrealistic. In statistics, most classical procedures assume some kind of symmetry. However, the absence of symmetry is much more common in many data sets. In particular, much interest has been shown recently in a family of distributions called “Skew-symmetric distributions”. Let $f$ be a density function symmetric about zero, and $K$ an absolutely continuous distribution function such that the corresponding density function $K\,^{\prime}$ is symmetric about zero. Then, Azzalini’s form of skew-symmetric density function for any real $\lambda$, as mentioned in Azzalini, (1985), is given as

When $f$ and $K$ are the density and distribution functions of the Laplace distribution in (1), respectively, it is called a skew-Laplace distribution. Aryal and Rao, (2005) studied some properties of truncated skew-Laplace distribution, and Kozubowski and Nolan, (2008) showed that a skew-Laplace distribution is infinitely divisible. Further, Nekoukhou and Alamatsaz, (2012) introduced a more general family of skew-Laplace distributions by considering $f$ as a standard Laplace pdf, $K$ as an arbitrary symmetric cdf, and $w$ as any odd continuous function in place of $\lambda x$ in (1). That is,

Recently much interest has been shown in the construction of flexible parametric classes of distributions that exhibit skewness and kurtosis, which is different from the normal distribution. While much of classical statistical analysis is based on Gaussian distributional assumptions, statistical modeling with the Laplace distribution has gained importance in many applied fields. The motivation originates from data sets, including environmental, financial, and biomedical ones, which often do not follow the normal law. Models based on the Laplace distributions are popular in economics and finance; see Zeckhauser and Thompson, (1970); Rachev and SenGupta, (1993); Rydén et al., (1998); Theodossiou, (1998); Kotz et al., (2001); Kozubowski and Podgórski, (2001). They are rapidly becoming distributions of the first choice whenever “something” with heavier than normal tail is observed in the data. The interesting characteristic has often bound on its support from either end along with skew nature. i.e., data is positively skewed but bounded below or negatively skewed but bounded above. For example, consider the scenario of family income, which is typically positively skewed and bounded below by a certain amount. In this paper, by considering interesting applications of Laplace distribution, the need for skewness and restriction on the support of variable of interest, skew-symmetric-Laplace-uniform distribution (SSLUD) is introduced. Here, we consider $f$ as the standard Laplace density function and $K$ as a distribution function of Uniform$(-\theta,\theta)$ in (1). It provides a more flexible model representing the data as adequately as possible. Thus, we can expect this to be useful in more practical situations. The standard Laplace pdf is

The skew-symmetric-Laplace-uniform distribution with parameter $\mu$, $SSLUD(\mu)$ appears not to have been introduced yet. We provide a comprehensive description of the mathematical properties of (6). This paper follows up on Nadarajah, (2009), where a comprehensive description of the mathematical properties for the skew-logistic distribution is provided. Here, we have derived formulas for moment generating function, characteristic function, and first four raw moments (Sect. 2), mode and median (Sect. 3), hazard rate function (Sect. 4), mean deviation about ‘$a$’ (Sect. 5), Rènyi entropy and Shannon entropy (Sect. 6), simulation and estimation by the methods of moments and maximum likelihood (Sect. 7). We also discuss these estimators’ finite sample and asymptotic properties (Sect. 7). Finally, the application of $SSLUD(\mu)$ to real-life data on the daily percentage change in the price of NIFTY 50, an Indian stock market index, is discussed. Comparison of fitting of $SSLUD(\mu)$ is done with fitting of normal distribution $N(\theta,\sigma^{2})$, Laplace distribution $L(\theta,\beta)$, and skew-Laplace distribution $SL(\lambda)$ for the above data (Sect. 8).

Here, we derive the moment generating function and the characteristic function of r. v. $X$ having pdf given in (6). The moment generating function (MGF) is $M_{X}(t)=E(e^{tX})$. By using (6), one obtains

The moments of a probability distribution are a collection of descriptive constants used for measuring its properties. Here, we derive the expression of the first four raw moments of $X$. They are as follows.

Figure 2 illustrates the behavior of the four measures E($X$), Var($X$), Skewness($X$) and Kurtosis($X$) for $\mu=-10,\ldots,10$. Mean and skewness are decreasing functions of $\mu$ over the range $(-\infty,0)$ and $(0,\infty)$, while variance and kurtosis are even functions of $\mu$. The variance strictly decreases as $\mu$ moves from $-\infty$ to 0 and increases as $\mu$ moves from 0 to $\infty$.

Mode is the value of the r. v. $X$ at which pdf $g(x)$ is maximum. When $\mu>0$, $g^{\prime}(x)$ is,

The median $M$ of (6) is the value of r. v. $X$ such that $G(M)=\frac{1}{2}$. Thus, for $\mu>0$ using (7b),

The reliability function $R(x)=1-G(x)$ for $\mu>0$ is obtained using (7b) as,

The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and the mean deviation about the median, respectively. Mean deviation about an arbitrary real number ‘$a$’ is defined by $\eta_{a}=E\lvert X-a\rvert$.

The entropy of a random variable $X$ measures the variation of uncertainty. The Rènyi entropy of order $\alpha$ is

Therefore,

The Shannon entropy function is the particular case of (17) for $\alpha\uparrow 1$, and it is $H=E[-\log_{2}g(X)]$, where $g(x)$ is pdf of random variable $X$. Using this definition, after some simplification we get,

Here, we first consider simulating values of a random variable $X$ with the pdf (6) using the inverse transformation technique. Let $r$ be a random number between zero and one. The generator to generate a random sample is

Now, we consider the estimation of $\mu$ by the method of moments and the method of maximum likelihood. To estimate unknown parameter $\mu$, we have to consider both the cases $\mu<0$ and $\mu>0$ together. Suppose $x_{1},...,x_{n}$ is an observed random sample of size ‘$n$’ from (6). For the method of moments estimation, after equating sample mean $\overline{x}$ to the first population raw moment of (6), one obtains the equation

We consider the estimation of $\mu$ by the method of maximum likelihood in the following. Let $x_{(1)},x_{(2)},\ldots,x_{(n)}$ be the order statistics of given sample. Suppose $\mu<0$ and ‘$r_{1}$’ denotes the number of observations less than $\mu$ such that $-\infty<x_{(1)}<x_{(2)}<\ldots<x_{(r_{1})}\leqslant\mu\leqslant x_{(r_{1}+1)}<\ldots<x_{(n)}<-\mu<\infty$, i.e. $-\infty<\mu<\min(0,\,-x_{(n)})$ where $r_{1}=0,1,2,\ldots,n$. Similarly, suppose $\mu>0$ and ‘$r_{2}$’ denotes the number of observations lying in the interval $[-\mu,\mu]$ such that $-\infty<-\mu<x_{(1)}<x_{(2)}<\ldots<x_{(r_{2})}\leqslant\mu\leqslant x_{(r_{2}+1)}<\ldots<x_{(n)}<\infty$, i.e. $\max(0,\,-x_{(1)})<\mu<\infty$ where $r_{2}=0,1,2,\ldots,n$. Hence, the log-likelihood function of $\mu$ is written as

Finite sample properties of $\tilde{\mu}$ and $\hat{\mu}$ are studied using simulation, and computations are done using R- software. Table 2 and Table 3 presents bias and MSE of $\tilde{\mu}$ and $\hat{\mu}$ for $n=100(100)1000$ and for $\mu=-1.5,-0.75,-0.25,0.25,0.75,1.5$. We see that bias and MSE decrease as sample size $n$ increases for both MLE $\hat{\mu}$ and moment estimator $\tilde{\mu}$, with few exceptions only for bias. Further, the MSE of $\hat{\mu}$ is always less than the corresponding MSE of $\tilde{\mu}$. Also, one can observe that sign of bias of MLE $\hat{\mu}$ is opposite to the sign of parameter $\mu$. As parameter $\mu$ approaches zero from any side, MSE and magnitude of bias of $\hat{\mu}$ decrease. But, no such observation in the case of the moment estimator $\tilde{\mu}$. To check the asymptotic nature of the distribution of $\hat{\mu}$ and $\tilde{\mu}$ using simulation, we plotted observed densities for various values of the sample size $n$. We observe that as $n$ increases, the distribution of both $\hat{\mu}$ and $\tilde{\mu}$ converges to the normal distribution, but the rate of convergence to normal distribution seems to be much higher for $\hat{\mu}$ than $\tilde{\mu}$. Thus, based on all the above results, we conclude that MLE is better than the moment estimator of $\mu$ for SSLUD($\mu$).

In this section, we present the application of skew-symmetric-Laplace-uniform distribution for modeling daily percentage change in the price of NIFTY 50, an Indian stock market index. Further, we have fitted and compared the proposed distribution $SSLUD(\mu)$ with normal distribution $N(\theta,\sigma^{2})$, Laplace distribution $L(\theta,\beta)$, and skew-Laplace distribution $SL(\lambda)$ for percentage change data. Here, $SL(\lambda)$ refers to a special case of skew-Laplace distribution using $f$ and $K$ as pdf and cdf of standard Laplace distribution in (1). The NIFTY 50 is a benchmark Indian stock market index representing the weighted average of 50 of the largest Indian companies listed on the National Stock Exchange (NSE). It is one of the two leading stock indices used in India. The daily price of NIFTY 50 quoted in the National Stock Exchange of India Ltd. is available at https://in.investing.com/indices/s-p-cnx-nifty-historical-data and is selected for the current study. We consider the daily percentage change $Y_{t}$ on day t given by $\displaystyle Y_{t}=\frac{X_{t}\,-\,X_{t-1}}{X_{t-1}}\times 100$, where $X_{t}$ denotes the price of NIFTY 50 on day t. This transformed data covering the period $16^{th}$ December 2021 to $13^{th}$ April 2022 (82 working days) is as follows :
0.16, - 1.53, - 2.18, 0.94, 1.10, 0.69, - 0.40, 0.49, 0.86, - 0.11, - 0.06, 0.87, 1.57, 1.02, 0.67, - 1.00, 0.38, 1.07, 0.29, 0.87, 0.25, - 0.01, 0.29, - 1.07, - 0.96, - 1.01, - 0.79, - 2.66, 0.75, - 0.97, - 0.05, 1.39, 1.37, 1.16, - 1.24, - 0.25, - 1.73, 0.31, 1.14, 0.81, - 1.31, - 3.06, 3.03, - 0.17, - 0.10, - 0.16, - 0.40, - 0.67, - 0.17, - 4.78, 2.53, 0.81, - 1.12, - 0.65, - 1.53, - 2.35, 0.95, 2.07, 1.53, 0.21, 1.45, - 1.23, 1.87, 1.84, - 0.98, 1.16, - 0.40, - 0.13, - 0.40, 0.40, 0.60, 1.00, - 0.19, 1.18, 2.17, - 0.53, - 0.83, - 0.94, 0.82, - 0.62, - 0.82, - 0.31.

Mean, variance, and skewness for the above data is 0.027, 1.671, and - 0.639 respectively. The Wald-Wolfowitz runs test for randomness of $Y_{t}$ yields a p-value of 0.076, justifying the assumption of independence of the $Y_{t}$ values. We consider fitting the proposed skew-symmetric-Laplace-uniform distribution $SSLUD(\mu)$ along with normal distribution $N(\theta,\sigma^{2})$, Laplace distribution $L(\theta,\beta)$, and skew-Laplace distribution $SL(\lambda)$ to the data on percentage change. Using R-software, the MLE of the parameters and hence, the estimated value of log-likelihood are obtained. Akaike’s Information Criteria (AIC) and Bayesian Information Criteria (BIC) are used for model comparison. Table 4 shows that the proposed $SSLUD(\mu)$ provides the best fit for the data set which is very close to $SL(\lambda)$ in terms of BIC. But in terms of AIC, N($\theta,\sigma^{2}$) seems to be better than $SSLUD(\mu)$ and the best among the four distributions.

For $SSLUD(\mu)$, MLE of $\mu$ is $\hat{\mu}$=62.38674 which is relatively high, and by definition of $g(x)$ in (6) for a large value of $\mu$, SSLUD approaches to Laplace distribution. But, from the histogram in Figure 4 and the value of skewness for $Y_{t}$, one can observe that data is negatively skewed. It might be due to a single parameter in the proposed distribution unable to give the best fit to the data. By changing the data location, significant change observed in SSLUD’s curve in terms of location, scale, and shape. So, by observation, one can choose an appropriate change in location such that the value of $\hat{\mu}$ is significantly small for possible better fitting of data using the proposed distribution. Through the trial and error method, here we consider a change as - 0.8 and define transformed daily percentage change in Nifty 50 index price, $Z_{t}=Y_{t}-0.8$ which gives $\hat{\mu}=-2.589259$, a significantly small value. A possible generalization of proposed distribution with additional location parameter to avoid hindrance to employ it is under consideration, in order to make it more flexible and apt to catch the features present in real data.

Table 5 shows the MLEs, estimated log-likelihood, AIC, and BIC by fitting the distributions mentioned above to $Z_{t}$. The graphical representation of the results is given in Figure 5. It is clear from Table 5 that the proposed $SSLUD(\mu)$ provides the best fit for the data set in terms of both AIC and BIC, but close to $SL(\lambda)$. The plot of observed and expected densities presented in Figure 5 also confirms our findings. Thus, $SSLUD(\mu)$ is better for modeling daily percentage change in the price of NIFTY 50 in comparison to normal distribution $N(\theta,\sigma^{2})$ and Laplace distribution $L(\theta,\beta)$, and one good alternative to skew-Laplace distribution $SL(\lambda)$.