A NEW TRIGONOMETRIC DISTRIBUTION WITH BOUNDED SUPPORT AND AN APPLICATION

. In this paper, we introduce a new bounded distribution by using trigonometric functions, named the cosine-sine distribution. A comprehen- sive study of its statistical properties is presented along with an application to a unit-interval data set, namely ﬁrms risk management cost-eﬀectiveness data. The proposed distribution has increasing, bathtub and v hazard rate shapes. Further, we show that the distribution can be viewed as a truncated exponential sine distribution.


Introduction
In statistical literature, many extensions of the lifetime distributions have been studied by several researchers over the last two decades. Most of these extensions concern distributions with unbounded support, while there is a great scarcity of distributions with bounded support (see Marshall and Olkin [14]). In many practical situations, a common issue is dealing with uncertain phenomena observed in the bounded interval (0, 1). For example, in real-life situations we often encounter measures such as percentages, proportions or fractions. In reliability and survival analysis, failure of a device during the warranty period may not be counted. Moreover, test conditions, cost or other constraints may lead many reliability systems to be bounded above (Lai and Jones [12], Lai and Mukherjee [13], Mukherjee and Islam [15], Jiang [9]). Also, models defined on the unit interval are recommended when the reliability is measured as the ratio of the number of successful trials to the number of total trials (Genç [5]). Further, continuous distributions having a finite support are highly recommended to model and analyze reliability data because of physical reasons, such as the finite lifetime of a component system or the bounded signals occurring in industrial systems (see Jiang [9]). In insurance, a probability distribution with domain on (0, 1) can be used as a distortion function to define a premium principle (Gómez-Déniz et al. [6], Chowdhury and Nanda [3]). Further, Papke and Wooldridge [16] claim that variables whose values lie in (0, 1) arise naturally in many economic settings, such as the fraction of land area allocated to NEW TRIGONOMETRIC DISTRIBUTION WITH BOUNDED SUPPORT 461 X = arcsin π 2 Y , where Y has a truncated exponential distribution (0, 1) with parameter λ, i.e., Y has density Straightforward calculations, using the change of variable formula, lead to the following expression of the CS density function: (2.1) and the cumulative distribution function (cdf) of the CS model is which has a closed form. The corresponding hazard rate (hr) function is In Figures 1 and 2, we plot the pdf and hrf of the CS model for different values of the parameter λ. From Figure 1, we note that the pdf f (x; λ) decays rapidly to 0 as x goes to 1 with λ < 1 and slowly when λ ≥ 1. Figure 2 shows various shapes for the hrf, including increasing, bathtub and v-shapes. This variety of shapes may be due to trigonometric functions in expression (2.2), which makes the distribution capable of analyzing many practical data. Throughout the rest of the paper we will use the following definitions and results. For m > 0 and r = 1, 2, . . . , we define is the gamma function and Γ(., .) denotes the incomplete gamma function (see Abramowitz and Stegun [1]).
The entropy formulae that we will derive later will make use of the following functions.
(ii) The modified Struve function is defined by

Moments
In this section, we derive an expression for the marginal and conditional moments of the random variable X as a function of λ. Theorem 3.1. Let X be a random variable with CS density function (2.1). Then the r-th moment about the origin of X is given, for r = 1, 2, . . . , by Proof. Using the definition of the r-th moment of X around the origin, we have Using the expressions (see Gradshteyn and Ryzhik [7]) in the successive substitutions, Eq. (3.1) is attained. In particular, using (2.3), the mean of the CS distribution follows as In connection with lifetime distribution, it is important to determine the conditional moments E (X r | X > t), r = 1, 2, . . . , which are of interest in predictive inference.

Proposition 3.2. The conditional moments of the CS distribution are given by
Proof. The proof follows by applying the identity

Stochastic ordering and entropies
In this section we show that the CS model can be ordered in terms of the likelihood ratio order, which is a powerful tool in parametric models. We also provide expressions for the Shannon and Rényi entropies for this model.

Stochastic ordering.
The next theorem shows that the CS model can be ordered in terms of the likelihood ratio ordering.
respectively. If λ 2 < λ 1 , then X is said to be smaller than Y in the likelihood ratio order (denoted by X ≤ lr Y ).
Proof. The density ratio is given by Hence, if follows that Since That is, X ≤ lr Y , which completes the proof.

Entropies.
Entropy is a measure of uncertainty variation of a system modeled by a probability distribution f (x). Two popular entropy measures of f (x) are the Shannon entropy (Shannon [18]) and Rényi entropy (Rényi [17]), defined as and respectively. The next two theorems explore explicit expressions of such entropies for the CS model.

Theorem 4.2. The Shannon entropy for the CS model is given by
Proof. Using the definition of Shannon entropy (4.1), we have The expression E (sin (πX/2)) is immediate from the fact that X = 2 π arcsin (Y ), where Y follows a truncated exponential distribution. Hence, Also, the quantity E (log [cos (πX/2)]) can be computed as Applying all the above results in the expression for Λ, we have the desired proof.

Theorem 4.3. The Rényi entropy for the CS distribution is
Proof. Using Eq. (2.1), we get Using (4.2), it follows that which completes the proof. Table 1 gives some numerical values for the Shannon entropy and the Rényi entropy. It can be observed that the Shannon entropy decreases with increasing λ and takes negative values, which may mean loss of system information. Also, it can be noted that the Rényi entropy takes negative values and decreases with increasing λ.

Some reliability measures
Residual life and reversed residual life random variables are used extensively in reliability analysis and risk theory. In this section, we get some of their related statistical functions, such as survival function (SF), mean and variance of residual life functions for the CS model.

Additionally, the mean of residual lifetime (MRL) of ψ (t) for the CS distribution is
where E(x) is obtained by (3.2) and υ (t; 1, λ) is given by (2.3). Finally, the variance of residual lifetime of ψ (t) for the CS model is given by where S(.) is the SF of the CS distribution, µ 2 can be obtained using (3.1), and υ(t; 2, λ) is given by (2.3). Some numerical values of the MRL are displayed in Table 2 for different values of λ at the time points t = 0.1, 0.3, 0.5, 0.7, 0.9. From this table, the MRL decreases with increasing the time points t and decreases with increasing λ.

Estimation with data analysis
In this section, we obtain the maximum likelihood estimator (MLE) of the unknown parameter of the CS model. Using this estimator we check the capability of this distribution to model a practical data set.
Let X 1 , X 2 , . . . , X n be a random sample of size n from the CS distribution with observed values x 1 , x 2 , . . . , x n . Then, the corresponding log-likelihood function is Differentiating with respect to λ, we have By setting d dλ = 0, the MLEλ of the parameter λ can be obtained by solving the non-linear equation above numerically for λ using some statistical software package, such as Mathematica.
6.1. Data analysis. By making use of a practical data set, we illustrate the applicability of the CS model among a set of classical and recent models -containing beta, Kumaraswamy and Log-Lindley-which also have support on (0, 1). We estimate the model parameters by using the maximum likelihood method. We compare goodness-of-fit of the models with the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Also, we apply the following goodness-offit tests: Cramér-von Mises (W * ), Anderson-Darling (A * ), the P-values based on Cramér-von Mises statistic and the Anderson-Darling statistic. In general, the smaller the values of these statistics and the largest P-values, the better the model's fit.
Description of the data is as follows.
The data set is given in Table 4. This data set is extracted from firms risk management cost-effectiveness, available at the personal web page of Professor E. Frees (Wisconsin School of Business). The data is defined as the total property and casualty premiums and uninsured losses as a percentage of the total assets leading to a bounded set on (0, 1). This data set was used by Gómez-Déniz et al. [6] to illustrate the applicability of the Log-Lindley distribution. Table 5 gives some descriptive statistics for the data set and using it we note that the data are under-dispersed (variance < mean) and positively skewed. Further, the data set possesses a negative kurtosis. Description of such data adapts features of the CS distribution and this proves the suitability of this distribution for analyzing such data. From Tables 5 and 8, we find that the descriptive and the corresponding theoretical estimated measures (under the CS model) are closed approximately for the data set. Also, Table 8 shows negative values for the considered entropies that adapt the theoretical results given in Table 1.
On the other hand, comparing the CS model with other bounded distributions is investigated as follows. For the data set, we estimate the unknown parameters of each distribution by the maximum-likelihood method, and using those estimates, we obtain the statistics AIC, BIC, W * , A * , and the P-values based on Cramérvon Mises statistic and the Anderson-Darling statistic. The results obtained are reported in Tables 6 and 7. From these tables, the smallest values of the AIC, BIC, W * , A * , and the largest P-values based on Cramér-von Mises statistic and the Anderson-Darling statistic are obtained for the CS model. Hence, we conclude that the CS model provides the best fit among the compared distributions.

Concluding remarks
A new trigonometric distribution with bounded support, named cosine-sine distribution, is introduced. The probability density function of this model decays slowly and rapidly to 0 at the end of the distribution support based on the distribution parameter values, which may be a desired feature for some real-life models. Also, this model can be displayed as a truncated exponential sine distribution. Additionally, the new model has a closed form expression for the cumulative distribution function and various shapes of its hazard rate function. Finally, the cosine-sine model and other distributions are fitted to a practical data set and the usefulness of the model is illustrated.