A New Family of F-Loss Distributions: Properties and Applications

Advances and Applications in Statistics © 2023 Pushpa Publishing House, Prayagraj, India http://www.pphmj.com http://dx.doi.org/10.17654/0972361723030 Volume 87, Number 1, 2023, Pages 81-117 P-ISSN: 0972-3617 A NEW FAMILY OF F-LOSS DISTRIBUTIONS: PROPERTIES AND APPLICATIONS Abstract In this paper, a new family of F-Loss distributions known as cosine F-Loss is proposed. This is an extension of the F-Loss family of Received: February 4, 2023; Accepted: April 14, 2023 2020 Mathematics Subject Classification: 62E15, 60E05. Keywords and phrases: trigonometric distributions, Monte Carlo simulation, heavy-tailed distributions, insurance loss, maximum likelihood estimation, F-Loss. Corresponding author How to cite this article: John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda, A new family of F-Loss distributions: properties and applications, Advances and Applications in Statistics 87(1) (2023), 81-117. http://dx.doi.org/10.17654/0972361723030 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Published Online: April 22, 2023 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 82 distributions by appropriate use of trigonometric functions. The proposed method ensures that no additional parameter(s) is/are introduced in the bit to make the F-Loss family of distributions flexible. The mathematical properties are derived and maximum likelihood estimates of the model parameters are obtained. Three special distributions are proposed; cosine Weibull loss, cosine Burr III loss and cosine Lomax loss. The densities exhibit different kinds of right-skewed, decreasing, reversed-J, and approximately symmetric shapes. The hazard rate functions show different kinds of increasing, decreasing, increasing-constant-increasing, increasing-constant- decreasing, reversed-J, and upside down bathtub shapes. Monte Carlo simulations are carried out to assess the behavior of the estimators. It is realized that the estimators are consistent. The applications of the proposed distributions are presented with insurance loss datasets. The results show that the cosine Burr III loss distribution provides the best parametric fit for both the business interruption claims and the automobile bodily injury claims datasets compared to the other classical heavy-tailed distributions considered. 1. Introduction Modeling insurance claims is vital in risk management problems. Most insurance loss datasets exhibit right skewness, unimodal shape, and a thick right tail [4]. A distribution showing these features is desirable in modeling financial data (insurance loss, assets returns, amongst others) and thus could be used for evaluating the business risk level. Insurance data are usually positive, and their distributions are typically unimodal humped shaped, with extreme values yielding tails heavier than conventional distributions [6]. Therefore, traditional distributions may not be flexible enough in modeling these heavy-tailed datasets [1]. In many applied fields, including engineering, medicine, and finance among others, right or left skewness, bi-modality or multi-modality are features of data sets that can be modeled using statistical distributions. The commonly used distributions include normal, Weibull, gamma, and Lindley because of their simple forms and identifiability properties. A New Family of F-Loss Distributions: Properties and Applications 83 Some methods of developing distributions like the beta generated, transformation of variables, compounding of distributions, and addition of parameter among others are mostly not flexible enough in modeling insurance loss data adequately and sometimes there are issues of overparameterization. That is, some statistical distributions proposed in the literature have many parameters in the bit to make them flexible. According to [3], estimates from such distributions may be challenging to obtain numerically. Hence, there is the need to develop distributions with fewer parameters yet with a greater flexibility for modeling insurance loss data. From the literature, we do not find sufficient study on heavy-tailed distributions based on trigonometric functions; most of them are based on algebraic functions [2]. Raab and Green in [8] introduced a cosine distribution to approximate the normal distribution. Some of the extensions are the Cos-G family of distributions proposed by [9] and [10] proposing the Cosine Weibull (CosW). Muhammad et al. [7] introduced the new extended cosine family of distributions with the extended cosine generalized halflogistic, extended cosine power, and extended cosine Weibull distributions as special cases. Moreover, an increased interest in data analysis requires further research so as to have more choices regarding distributions that dwell on trigonometric functions [7]. The trigonometric transformation provides flexibility because of the periodic function, which controls how the distribution curve behaves, and parameter(s) oscillate with value changes [11]. Ahmad et al. [2] proposed the F-Loss family of distributions by adding a shape parameter to the baseline distribution. They proposed the Weibull-Loss (W-Loss) distribution as a special distribution. The distributions from this family are mostly not flexible enough in fitting most data set adequately. We extend this family of distributions without adding any extra parameter(s) by using trigonometric functions. This is to increase the flexibility of the F-Loss family of distributions and by extension develop better parametric fit to a given dataset than some of the conventional heavy tailed distributions. 84 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda The motivation for proposing an extension of the F-Loss family of distributions is to improve the flexibility of the F-Loss family of distributions without introducing any additional parameter(s); to develop heavy tailed distributions with fewer parameters giving better parametric fit to a given dataset than other existing distributions; to generate distributions which are capable of modeling monotonic and non-monotonic hazard rates. Therefore, there is the need to extend the F-Loss family of distributions. This method contributes to the literature on heavy-tailed distributions that are based on trigonometric functions. The rest of the paper is organized as follows: Section 2 presents the genesis of the proposed family of the F-Loss distributions. The mixture representation of the CFL family of distributions is presented in Section 3. Section 4 presents the mathematical properties of the CFL family of distributions. In Section 5, we present the parameter estimation of the CFL family using maximum likelihood. Section 6 presents three special distributions of the CFL family of distributions. The behavior of the parameters of the special distributions is ascertained in Section 7 using Monte Carlo simulations. In Section 8, the applications of the special distributions are illustrated using two insurance loss datasets and the conclusion is presented in Section 9. 2. Proposed Family of the F-Loss Distributions From Section 1, it is essential that researchers look for heavy-tailed distributions with fewer parameters and yet flexible in modeling insurance loss to improve the flexibility of existing distributions. In this section, we propose a new family of the F-Loss distributions as an extension of the F-Loss family of distributions without adding any extra parameter(s). The proposed method is flexible and avoids over-parameterization. The F-Loss family of distributions was introduced by [2]. A random variable X is said to follow the F-Loss family, if the cumulative distribution function (CDF) is given by H x; V, Z 1 VF x; Z , V  log F x; Z V ! 0, x  R, (1) A New Family of F-Loss Distributions: Properties and Applications 85 where F x; Z 1  F x; Z is the survival function of the baseline distribution, Z is a p u 1 vector of parameters and V is a shape parameter. Souza [9] proposed the cosine-G family of distributions. A random variable X is said to follow the cosine-G family of distributions, if the CDF is given by G x; V, Z S 1  cos§¨ H x; V, Z ·¸ , ©2 ¹ x  R. (2) From equations (1) and (2), we propose the cosine F-Loss (CFL) family of distributions. A random variable X is said to follow the CFL family of distributions if the CDF is designated as G x; V, Z VF x; Z ªS § ·º 1  cos « ¨1  ¸ , V  log F x; Z ¹»¼ ¬2 © where F x; Z V ! 0, x  R, (3) 1  F x; Z is the survival function (sf) of the baseline distribution which may depend on the vector parameter Z and V is a shape parameter. It is constructed from inserting the CDF in equation (1) into the CDF in equation (2). The PDF is given by g x; V, Z S ª Vf x; Z >1  V  log F x; Z @ º » 2 «¬ >V  log F x; Z @2 ¼ VF x; Z ªS § ·º u sin « ¨1  ¸», x  R. 2 log ; F x V  Z ¬ © ¹¼ (4) The hazard rate function of the CFL family of distributions is given by h x; V, Z S ª Vf x; Z >1  V  log F x; Z @ º » 2 «¬ >V  log F x; Z @2 ¼ VF x; Z ªS § ·º u tan « ¨1  ¸ , x  R. 2 V  log F x; Z ¹»¼ ¬ © (5) John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 86 3. Mixture Representation of the CFL Family of Distributions The mixture representation of the PDF of the CFL family of distributions is obtained in this section. Lemma 1. The PDF of the CFL family of distributions in equation (4) has a mixture representation of the form f 2 n 1 k ¦ ¦¦ ><nkm 1  V Am f g x; V, Z x; Z F x; Z m  s  w n, m 0 k 0 s 0  <nkm Am 1 f x; Z F x; Z m 1 s  w @, (6) where <nkm <nkm 1 s § k · ¨ ¸, Am Vm 1 © s ¹ m § w  m· ¸ ¨ w 0© w ¹ f ¦ 1 w  a a 0 ma ¦ w § w· ¨ ¸ Pa , w ©a¹ and Am 1 m 1 § w  m  1· ¸ ¨ w 0© m ¹ f ¦ 1 w  a § w · ¨ ¸P . a 0 m  1  a © a ¹ a, w ¦ w Proof. Using the power series expansion of sine function, sin x f ¦ n 0 1 n x 2n 1. 2n  1 ! (7) That is, from equation (4), VF x; Z ·º ªS § sin « ¨1  ¸ 2 V  log F x; Z ¹»¼ ¬ © f ¦ n 0 1 n ª S § VF x; Z 1 2n  1 ! «¬ 2 ¨© V  log F x; Z ·º ¸» ¹¼ 2 n 1 . (8) A New Family of F-Loss Distributions: Properties and Applications 87 From the generalized binomial expansion given by f ¦ 1 k §¨© k ·¸¹ z k , 1 z n n (9) z d 1, k 0 and f ¦ 1 m §¨© 1  z r m 0 r  m  1· m ¸z , m ¹ z d 1. (10) Substituting equation (8) in equation (4) and making use of equation (9) and the fact that 0  g x; V, Z VF x; Z  1, we have V  log F x; Z S Vf x; Z >1  V  log F x; Z @ 2 >V  log F x; Z @2 f u ¦ n 0 1 n ª S § VF x; Z 1 2n  1 ! «¬ 2 ¨© V  log F x; Z S Vf x; Z >1  V  log F x; Z @ 2 u f 2 n 1 ¦¦ n 0k 0 2 n 1 S 1 n  k §¨ ·¸ ©2¹ 2n  1 ! § 2 n  1· ¨ ¸ © k ¹ ª º V k F x; Z k u« 2 k » ¬ >V  log F x; Z @ ¼ f x; Z >1  V  log F x; Z @ u f 2 n 1 ¦¦ n 0k 0 2n  2 S 1 n  k §¨ ·¸ ©2¹ 2n  1 ! § 2 n  1· ¨ ¸ © k ¹ ·º ¸» ¹¼ 2 n 1 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 88 ª « F x; Z k u« « ª log F x; Z « V «1  V ¬ ¬ Using z  the binomial expansion º » ». 2 k » º » »¼ ¼ in equation (10) and letting log F x; Z , V g x; V, Z f x; Z >1  V  log F x; Z @ u f 2 n 1 ¦¦ n 0k 0 2n  2 S 1 n  k §¨ ·¸ ©2¹ 2n  1 ! § 2 n  1· F x ; Z k ¸ ¨ V © k ¹ f u m § 2  k  m  1· ª log F x; Z º 1 m ¨ ¸ « »¼ V m © ¹¬ m 0 ¦ 2n  2 ª nk m § S · ¨ ¸ « 1 § 2 n  1 · § k  m  1· ©2¹ « ¨ ¸¨ ¸  n 2 1 ! k © k ¹© ¹ n 0 k 0 m 0« «¬ f 2 n 1 f ¦¦¦ u 1  V f x; Z F x; Z k > log F x; Z @m Vm 1 S 2n  2 1 n  k  m §¨ ·¸ ©2¹  2n  1 ! u § 2 n  1· § k  m  1 · ¨ ¸¨ ¸ m © k ¹© ¹ m 1 f x; Z F x; Z >log F x; Z @ k Vm 1 º » ». » » ¼ A New Family of F-Loss Distributions: Properties and Applications 89 Now, using the expansion; f  log 1  x t t 1 w  a §w  t· ¨ ¸ ta w ¹ w 0© a 0 w ¦ ¦ § w· tw , ¨ ¸ Pa , w , x a © ¹ where a ! 0 is any real value. The constants Pa , w can be calculated, 1 w az  z  w Pa , w  z for w w ¦z 1 z  1 recursively, via Pa , w Pa , 0 1, 2, 3, ... and 1. S 2n  2 1 n  k  m §¨ ·¸ ©2¹ 2n  1 ! Let <nkm g x; V, Z f 2 n 1 f ¦ ¦ ¦ ª«¬ § 2 n  1 · § k  m  1· ¸, ¸¨ ¨ m ¹ © k ¹© <nkm 1  V Vm 1 n 0 k 0m 0  <nkm V m 1 f x; Z F x; Z k >log F x; Z @m º f x; Z F x; Z k >log F x; Z @k 1 » . ¼ Also, it implies that, g x; V, Z f 2 n 1 f ª < 1 V « nkm f x; Z F x; Z k m m 1  « V n 0 k 0 m 0¬ ¦¦¦ f u  1 w  a § w  m· ¸ ¨ ma w ¹ w 0© a 0 ¦ <nkm V m 1 ¦ f x ; Z F x; Z k § w· mw ¨ ¸ u Pa , w F x; Z a © ¹ m 1 f w  m  1· ¸ w ¹ w 0 ¦ §¨© º 1 w  a § w · m 1 w ». ¨ ¸ Pa , w F x; Z m 1 a ©a¹ »¼ a 0 w u w ¦ John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 90 Letting Am m § w  m· ¨ ¸ w 0© w ¹ f ¦ 1 w  a a 0 ma ¦ w § w· ¨ ¸ Pa , w , m z a ©a¹ and Am 1 m 1 § w  m  1· ¨ ¸u w 0© w ¹ f ¦ 1 w  a § w · ¨ ¸P a 0 m  1  a © a ¹ a, w ¦ w the fact that >1  F x; Z @k F x; Z k ¦s k §k · 1 s ¨ ¸ F x; Z s , 0 ©s¹ we get g x; V, Z f 2 n 1 f k ª <nkm 1  V 1 s Am § k · msw ¨ ¸ f x ; Z F x; Z « m 1 ©s¹ V n 0 k 0 m 0s 0 ¬ ¦ ¦ ¦¦  <nkm 1 s Am 1 § k · m 1 s  w º f x F x ; Z ; Z ¸ ¨ ». ©s¹ V m 1 ¼ Thus, we get the PDF of the CFL as g x; V, Z f 2 n 1 k ¦ ¦ ¦ ><nkm 1  V Am f n, m 0 k 0 s 0 x; Z F x; Z m  s  w  <nkm Am 1 f x; Z F x; Z m 1 s  w @. 4. Mathematical Properties of the CFL Family of Distributions In this section, some mathematical properties of the CFL family of A New Family of F-Loss Distributions: Properties and Applications 91 distributions including quantile function, moments, moment generating function, value at risk, tail value at risk, tail variance and tail variance premium are derived. 4.1. Quantile function The quantile function is vital in describing the random variable of a distribution. It helps in simulating random samples which are useful in simulations. It can also be used to compute measures of shape such as skewness and kurtosis. Lemma 2. The quantile function of the CFL family of distributions for u  0, 1 is given by ª1  § 2 arccos 1  u ·º >V  log 1  t @  V 1  t ¸» «¬ ¨© S ¹¼ 0, (11) where t is the solution of the equation ª1  § 2 arccos 1  u ·º >V  log 1  t @  V 1  t ¸» «¬ ¨© S ¹¼ 0, and u has the uniform distribution on the interval 0, 1 . Proof. By definition, the quantile function is given by xu Qu G 1 u . Thus, 2 1  §¨ arccos 1  u ·¸ ©S ¹ VF x; Z . V  log F x; Z (12) Solving equation (12) and replacing F x; Z by t, we get ª1  § 2 arccos 1  u ·º >V  log 1  t @  V 1  t ¸» «¬ ¨© S ¹¼ 0. 4.2. Moments The moments of a distribution are important in estimating measures of John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 92 variation like the variance, standard deviation, coefficient of variation, mean deviation, median deviation, kurtosis, skewness amongst others. Proposition 1. The rth non-central moment of the CFL family of distributions is given by f 2 n 1 k ¦ ¦¦ ><nkm 1  V AmMmsw x; Z Pcr n, m 0 k 0 s 0  <nkm Am 1Kmsw x; Z @, (13) where f ³0 Mmsw x; Z f ³0 Kmsw x; Z x r f x; Z F x; Z m  s  w dx, x r f x; Z F x; Z m 1 s  w dx and r 1, 2, ... . Proof. By definition, the rth non-central moment is given by Pcr f ³0 x r g x dx. This implies that, f 2 n 1 k f ¦ ¦¦ ª«¬<nkm 1  V Am ³ 0 Pcr x r f x; Z F x; Z m  s  w dx n, m 0 k 0 s 0  <nkm Am 1 f ³0 º x r f x; Z F x; Z m  s 1 w dx » . ¼ Thus, we get, Pcr f 2 n 1 k ¦ ¦¦ ><nkm 1  V AmMmsw x; Z n, m 0 k 0 s 0  <nkm Am 1Kmsw x; Z @. A New Family of F-Loss Distributions: Properties and Applications 93 4.3. Moment generating function The moment generating function (MGF) helps in determining the moments of a random variable. Proposition 2. The MGF of the CFL family of distributions is given by MX z f 2 1 k ª Z r <nkm 1  V Am Z r <nkm Am 1 Mmsw x; Z  Kmsw x; Z « r! r! « ¬ n , m, r 0 k 0 s 0 ¦ ¦¦ º ». »¼ (14) Proof. By definition, the MGF is given as E e zX MX z f ³0 e zx g x dx. Using series expansion, MX z ª f ZrX r º E« » r! » ¬« r 0 ¼ ¦ f MX z ¦ r 0 f ¦ r 0 Zr E Xr , r! Zr Pc . r! r This implies, f MX z 2 n 1 k ª Z r <nkm 1  V Am « r! n , m, r 0 k 0 s 0 ¬ ¦ ¦¦ f u ³0  º Z r <nkm Am 1 f r x f x; Z F x; Z m  s 1 w dx » . r! 0 ¼ x r f x; Z F x; Z m  s  w dx ³ John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 94 4.4. Value at risk Value at risk (VaR) is commonly used as a benchmark in measuring market risk. It is also called the quantile premium principle or quantile risk measure. It is usually expressed with a confidence level q (usually 90%, 95% or 99%), and represent the percentage of loss in portfolio value that will be equaled or exceeded only X percent of the time. The VaR of a random X is the qth quantile of its CDF [12]. Proposition 3. For V ! 0, the VaRq X of the CFL family of distributions is given by ª1  § 2 arccos 1  q «¬ ¨© S ·º >V  log 1  t @  V 1  t ¸» ¹¼ (15) 0, where t is the solution of the equation ª1  § 2 arccos 1  q «¬ ¨© S ·º >V  log 1  t @  V 1  t ¸» ¹¼ 0. Proof. By definition xq F 1 t . Thus, the VaR of CFL distribution is written as ª1  § 2 arccos 1  q ·º >V  log 1  t @  V 1  t ¸» «¬ ¨© S ¹¼ 0. 4.5. Tail value at risk The tail value at risk (TVaR) is also called the tail conditional expectation (TCE) or conditional tail expectation (CTE) and is for determining the average loss beyond a given probability level. Proposition 4. For V t 0, the TVaRq X for the CFL family of distributions is given by TVaRq X 1 1 q f 2 n 1 k ¦ ¦ ¦ ><nkm 1  V AmTmsw  <nkm Am 1Tmsw @, n, m 0 k 0 s 0 (16) A New Family of F-Loss Distributions: Properties and Applications 95 where f ³VaR Tmsw xf x; Z F x; Z m  s  w dx q and f ³VaRq xf Tmsw x; Z F x; Z m 1 s  w dx. Proof. By definition TVaRq X E X _ X ! VaRq f 1 xg x dx, 1  q VaRq ³ TVaRq X 1 1 q  f 2 n 1 k ª f x <nkm 1  V Am f x; Z F x; Z m  s  w dx « VaR q n, m 0 k 0 s 0 ¬ ¦ ¦¦ ³ º x <nkm Am 1 f x; Z F x; Z m 1 s  w dx » . VaRq ¼ ³ f Thus, TVaRq X 1 1 q f 2 n 1 k ¦ ¦¦ ><nkm 1  V AmTmsw  <nkm Am 1Tmsw @. n, m 0 k 0 s 0 4.6. Tail variance The tail variance (TV) is an important risk measure in insurance sciences. It is vital in determining the risk level at the tails. Proposition 5. For V t 0, the TVq X distributions is given by for the CFL family of 96 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda TVq X 1 1 q 2 n 1 k f ¦ ¦¦ ><nkm 1  V AmVmsw  <nkm Am 1Vmsw @ n, m 0 k 0 s 0 2 f 2 n 1 k ª º 1 « ^<nkm 1  V Am Tmsw  <nkm Am 1Tmsw `» ,  «1  q n, m 0 k 0 s 0 » ¬ ¼ ¦ ¦¦ (17) where f ³VaR Vmsw x 2 f x; Z F x; Z m  s  w dx q and f Vmsw ³VaR x 2 f x; Z F x; Z m 1 s  w dx. q Proof. From definition, TVq X E X 2 _ X ! xq  TVaRq 2 . (18) Using conditional moments, E Xn_X ! x 1 W x , s x n where Wn x f ³x y n g y dy and s x 1 F x . This implies that, E X 2 _ X ! xq 1 1 q f 2 n 1 k ­ f x2 f ®<nkm 1  V Am ³ ¦ ¦ ¦ VaRq n, m 0 k 0 s 0 ¯ x; Z F x; Z m  s  w dx A New Family of F-Loss Distributions: Properties and Applications 97  <nkm Am 1 f ³VaR q ½ x 2 f x; Z F x; Z m 1 s  w dx ¾ . ¿ Therefore, 1 1 q TVq X f 2 n 1 k ¦ ¦¦ ><nkm 1  V AmVmsw  <nkm Am 1Vmsw @ n, m 0 k 0 s 0 2 f 2 n 1 k º ª 1 « ^<nkm 1  V Am Tmsw  <nkm Am 1Tmsw `» .  » «1  q n, m 0 k 0 s 0 ¼ ¬ ¦ ¦¦ 4.7. Tail variance premium The tail variance premium (TVP) is another important measure that plays an important role in insurance sciences. It is vital in determining the premium for a risk. Proposition 6. For V ! 0 and the fact that 0  G  1 then, the TVPq X for the CFL distribution is given by TVPq 1 1 q f 2 n 1 k ¦ ¦¦ ><nkm 1  V AmTmsw  <nkm Am 1Tmsw @ n, m 0 k 0 s 0 ª f 2 n 1 k 1 « ><nkm 1  V AmVmsw  <nkm Am 1Vmsw @  G« 1 q n m k s , 0 0 0 «¬ ¦ ¦¦ f 2 n 1 k ª 1 º « ^<nkm 1  V Am T msw  <nkm Am 1T msw `» «¬1  q n, m 0 k 0 s 0 »¼ ¦ ¦¦ 2º ». » »¼ (19) John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 98 Proof. From definition, TVaRq  GTVq . TVPq X (20) Substituting equation (16) and equation (17) in equation (20) ends the proof. 4.8. Order statistics Let X1, X 2 , ..., X n be a sample of size n from the CFL family of distributions and X 1:n d X 2:n d " d X n:n denote the order statistics of the sample. The PDF of the jth order statistics g j:n x is defined as g j :n x n! >G x @ j 1>1  G x @n  j g x . j 1 ! n  j ! (21) Using binomial series expansion, we have n j >1  G x @ n j ¦ 1 r §¨© r 0 n  j· r ¸ >G x @ . r ¹ (22) That is, equation (21) becomes g j:n x n! g x j 1 ! n  j ! n j ¦ 1 r §¨© r 0 n  j· r  j 1 . ¸ >G x @ r ¹ (23) Substituting the CDF and PDF of the CFL in equation (23), we get the jth order statistics as g j:n x VF x; Z ªS § ·º SVf x; Z >1  V  log F x; Z @ sin « ¨1  ¸» n ! 2 Z V  F x log ; ¬ © ¹¼ 2>V  log F x; Z @2 j  1 ! n  j ! n j u r  j 1 VF x; Z §n  j· ª ªS § ·º º . 1 r ¨ ¸ «1  cos « ¨1  ¸ V  log F x; Z ¹»¼ »¼ ¬2 © © r ¹¬ r 0 ¦ (24) A New Family of F-Loss Distributions: Properties and Applications 99 The PDF of the first order statistics is defined as g1:n x n>1  G x @n 1 g x , (25) we get the PDF of the first order statistics as g1:n x VF x; Z ª ªS § ·ºº n «cos « ¨1  ¸ V  log F x; Z ¹»¼ »¼ ¬ ¬2 © u n 1 S ª Vf x; Z >1  V  log F x; Z @ º » 2 «¬ >V  log F x; Z @2 ¼ VF x; Z ªS § ·º u sin « ¨1  ¸ . V  log F x; Z ¹»¼ ¬2 © (26) Also, the PDF of the nth order statistics is defined as g n :n x n>G x @n 1 g x , (27) we get the PDF of the nth order statistics as g n :n x VF x; Z ·º º ªS § ª n «1  cos « ¨1  ¸ V  log F x; Z ¹»¼ »¼ ¬2 © ¬ u n 1 S ª Vf x; Z >1  V  log F x; Z @ º » 2 «¬ >V  log F x; Z @2 ¼ VF x; Z ªS § ·º u sin « ¨1  ¸ . V  log F x; Z ¹»¼ ¬2 © (28) 5. Parameter Estimation In this section, the unknown parameters of the CFL are estimated using the maximum likelihood estimation (MLE) technique. 5.1. Maximum likelihood estimation Let X1, X 2 , ..., X n be n random sample from the CFL family of John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 100 distributions. Therefore, the log-likelihood function which is a p u 1 V, Z T is given by parameter vector 4 S n§¨ log§¨ ·¸ ·¸  n log V  © © 2 ¹¹ A n ¦ log f xi ; Z i 1 n  ¦ log>1  V  log 1  F xi ; Z @ i 1 n 2 ¦ log>V  log 1  F xi ; Z @ i 1 n  VF x ; Z ·º ¸ . xi ; Z ¹»¼ ¦ log sin ª«¬ S2 §¨©1  V  log Fi i 1 (29) The log-likelihood function in equation (29) is differentiated with respect to each parameter to obtain the score function, U 4 wA wV n  V  n ¦ S 2 i 1 1 2 >1  V  log F xi ; Z @ n T § wA , wA · , ¸ ¨ © wV wZ ¹ n ¦ >V  log 1F i 1 VF x ; Z F xi ; Z ·  ¸ V  log F xi ; Z ¹ xi ; Z ¦ §¨© V  log Fi i 1 VF xi ; Z ªS § ·º u cot « ¨1  ¸» 2 V  Z log ; F x ¬ © ¹¼ i (30) and wA wZ xi ; Z @ n ¦ i 1 f c xi ; Z  f xi ; Z n Fc x ; Z ¦ >1  V  logi F i 1 xi ; Z @ A New Family of F-Loss Distributions: Properties and Applications 101  S 2 n VF c xi ; Z VF c x ; Z ·  ¸ V  log F xi ; Z ¹ xi ; Z ¦ §¨© V  log Fi i 1 VF xi ; Z ªS § ·º u cot « ¨1  ¸ , 2 V  log F xi ; Z ¹»¼ ¬ © (31) where F c xi ; Z wf xi ; Z . wZ wF xi ; Z and f c xi ; Z wZ 6. Special Distributions In this section, three special distributions are presented. These are the cosine Weibull Loss (CWL) distribution, cosine Burr III Loss (CBIIIL) distribution and cosine Lomax Loss (CLoL) distribution. 6.1. Cosine Weibull loss distribution If we consider the Weibull distribution as the baseline distribution with CDF and PDF defined as F x 1  e  Dx E DExE 1e  Dx and f x E for x ! 0 and D, E ! 0, respectively, we obtain the CWL distribution. From equation (3), the CDF of the CWL distribution is given by G x; D, E, V E ªS § Ve  Dx ·¸º ¨ », 1  cos « 1  «¬ 2 ¨© V  DxE ¸¹»¼ x ! 0, D, E, V ! 0, (32) where E and V are shape parameters and D is a scale parameter. The related PDF is given by g x; D, E, V SDEV 2 ª xE 1e  DxE 1  V  DxE « «¬ V  DxE 2 E ªS § Ve  Dx ·¸º », u sin « ¨1  «¬ 2 ¨© V  DxE ¸¹»¼ º » »¼ x ! 0. (33) 102 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda The corresponding hazard rate function is given by E h x; D, E, V SDEVxE 1e  Dx 1  V  DxE 2 V  DxE 2 E ª ªS § Ve  Dx ·¸º º ¨ »» , u « tan « 1  «¬ «¬ 2 ¨© V  DxE ¸¹»¼ »¼ x ! 0. (34) The different plots of the density function of the CWL distribution are shown in Figure 1. The density function exhibits right-skewed, decreasing and approximately symmetric shapes. Figure 1. Different plots for the density function of the CWL distribution. Figure 2 shows the different plots of the hazard rate function of the CWL distribution. The hazard rate exhibits decreasing, increasing, increasingconstant-increasing, reversed-J and upside-down bathtub shapes. Figure 2. Different plots for the hazard rate function of the CWL distribution. A New Family of F-Loss Distributions: Properties and Applications 103 6.2. Cosine Burr III loss distribution If we consider the Burr III distribution as the baseline distribution with 1  x  c  k and CDF and PDF defined as F x f x ckx  c 1 1  x  c  k 1 for x ! 0 and c, k ! 0, respectively, we obtain the CBIIIL distribution. From equation (3), the CDF of the CBIIIL distribution is given by G x; c, k , V ªS § V 1  1  x c k 1  cos « ¨¨1  V  log 1  1  x  c  k ¬2 © ·º ¸» , ¸ ¹¼ x ! 0, c, k , V ! 0, (35) where c, k and V are shape parameters. The related PDF is given by g x; c, k , V SckV ª x  c 1 1  x  c  k 1  V  log 1  1  x  c  k º » 2 «¬ >V  log 1  1  x  c  k @2 ¼ ªS § V 1  1  x c k u sin « ¨¨1  V  log 1  1  x  c  k ¬2 © ·º ¸» , ¸ ¹¼ x ! 0. (36) The corresponding hazard rate function is given by h x; c, k , V SckV ª x  c 1 1  x  c  k 1  V  log 1  1  x  c  k º » 2 «¬ >V  log 1  1  x  c  k @2 ¼ ªS § V 1  1  x c k u tan « ¨¨1  V  log 1  1  x  c  k ¬2 © ·º ¸» , ¸ ¹¼ x ! 0. (37) Figure 3 shows the plots of the density function of the CBIIIL distribution. The density function exhibits right-skewed and reversed-J shapes. 104 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda Figure 3. Different plots for the density function of the CBIIIL distribution. From Figure 4, the plots of the hazard rate function of the CBIIIL distribution show decreasing, increasing-constant-decreasing, upside-down bathtub and reversed-J shapes. Figure 4. Different plots for the hazard rate function of the CBIIIL distribution. 6.3. Cosine Lomax loss distribution If we consider the Lomax distribution as the baseline distribution with CDF and PDF defined as F x x D 1  §¨1  ·¸ © O¹ and f x D§ x ·  D 1 ¨1  ¸ O© O¹ for x ! 0 and D, O ! 0, respectively, we obtain the CLoL distribution. From equation (3), the CDF of the CLoL distribution is given by A New Family of F-Loss Distributions: Properties and Applications 105 G x; D, O, V ª § x  D ·º ¸» V§¨1  ·¸ «S ¨ O¹ © ¸» , x ! 0, D, O, V ! 0, 1  cos « ¨1  x 2 ¨ ¸ · § « V  D log¨1  ¸ ¸» O ¹ ¹»¼ «¬ ¨© © (38) where D and V are shape parameters and O is a scale parameter. The related PDF is given by g x; D, O, V ª§ x ·  D 1 ª x º 1  V  D log§¨1  ·¸º » ¨1  ¸ « «¬ SDV © O ¹»¼ O¹ © » « 2 2O « » ªV  D log§1  x ·º ¨ ¸» »¼ «¬ «¬ O ¹¼ © D ·º ª § §1  x · ¨ ¸» V ¸ ¨ «S O¹ © ¸» , u sin « ¨1  x · ¸» § «2 ¨ V  D log¨1  ¸ ¸ O ¹ ¹»¼ «¬ ¨© © x ! 0. (39) The corresponding hazard rate function is given by h x; D, O, V ª§ x ·  D 1 ª x º  1 1  V  D log§¨1  ·¸º » ¨ ¸ « «¬ SDV © O¹ O ¹¼» © « » 2 2O « » ªV  D log§1  x ·º ¨ ¸» «¬ »¼ «¬ O © ¹¼ D ·º ª § §1  x · ¨ ¸» V ¸ ¨ «S O¹ © ¸» , u tan « ¨1  x 2 ¨ ¸ · § « V  D log¨1  ¸ ¸» ¨ O «¬ © ¹ ¹»¼ © x ! 0. (40) Figure 5 shows the plots of the density function of the CLoL distribution. The density function exhibits right-skewed and reversed-J shapes. John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 106 Figure 5. Different plots for the density function of the CLoL distribution. The plots of the hazard rate function of the CLoL distribution in Figure 6 show upside down bathtub and reversed-J shapes. Figure 6. Different plots for the hazard rate function of the CLoL distribution. 7. Monte Carlo Simulation In this section, the simulation results are presented in examining the properties of the maximum likelihood estimators for the parameters of the CLoL distribution. Five different combinations of the parameter values of this distribution were specified and the quantile function used in generating four different random samples of size, n simulations were replicated for N 50, 100, 150, 200. The 1000 times. The properties of the estimators were investigated by computing average bias (AB) and root mean square error (RMSE) for each of the parameters. The simulation steps are as follows: A New Family of F-Loss Distributions: Properties and Applications 107 (i) Specify the values of the parameters and the sample size n. (ii) Generate random samples of size n 50, 100, 150, 200 from CLoL distribution using the quantile. (iii) Find the maximum likelihood estimates for the parameters. (iv) Repeat steps (ii)-(iii) for 1000 times. (v) Calculate the average bias (AB) and root mean square error (RMSE) for the parameters of the distributions. Table 1 shows the simulation results for the CLoL distribution. It can be observed that, as the sample size increase, the AB and RMSE for the estimators of the parameters decrease. This shows that the estimators are consistent. Table 1. Monte Carlo simulation results: AB and RMSE for the parameters of the CLoL distribution Parameter value N D AB O V D RMSE O V D O V 50 1.62 0.12 0.02 0.420 0.232 0.224 0.197 0.184 0.313 100 1.62 0.12 0.02 0.419 0.201 0.182 0.191 0.166 0.236 150 1.62 0.12 0.02 0.412 0.200 0.164 0.190 0.159 0.215 200 1.62 0.12 0.02 0.402 0.176 0.107 0.186 0.117 0.118 50 1.8 3.4 0.08 0.456 2.048 0.582 0.317 4.730 0.912 100 1.8 3.4 0.08 0.433 1.845 0.359 0.271 3.839 0.512 150 1.8 3.4 0.08 0.419 1.835 0.335 0.244 3.786 0.482 200 1.8 3.4 0.08 0.417 1.816 0.229 0.237 3.713 0.418 50 3.64 0.13 1.11 1.654 0.076 0.783 2.740 0.006 0.656 100 3.64 0.13 1.11 1.645 0.075 0.780 2.706 0.006 0.651 150 3.64 0.13 1.11 1.642 0.072 0.774 2.697 0.006 0.647 200 3.64 0.13 1.11 1.641 0.072 0.774 2.692 0.005 0.638 50 1.5 2.02 0.6 0.342 0.405 0.867 0.153 0.352 1.067 100 1.5 2.02 0.6 0.286 0.377 0.761 0.113 0.324 0.907 150 1.5 2.02 0.6 0.239 0.372 0.714 0.084 0.310 0.842 200 1.5 2.02 0.6 0.216 0.355 0.665 0.070 0.290 0.774 50 4.4 0.52 0.2 2.441 0.383 0.857 5.976 0.154 1.323 100 4.4 0.52 0.2 2.438 0.382 0.745 5.960 0.153 1.078 150 4.4 0.52 0.2 2.430 0.380 0.637 5.915 0.151 0.860 200 4.4 0.52 0.2 2.421 0.376 0.555 5.866 0.150 0.670 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 108 8. Applications This section illustrates the usefulness and flexibility of the CFL family of distributions using two insurance loss datasets. The performance of the CWL, CBIIIL, and CLoL distributions were compared with other loss distributions. The performance of the distributions about providing proper parametric fit to the dataset was compared using the AIC, BIC, Cramér-Von Misses W , Anderson-Darling and K-S statistics. The distribution A with the least of these measures provides a reasonable fit to the dataset. The fit for the CWL, CBIIIL, and CLoL are compared with other heavy-tailed distributions, including the 2-parameter Weibull, 2-parameter Burr XII (BXII), Weibull-Loss (W-Loss), 2-parameter Burr III (BIII), Fréchet, WeibullLomax and Lomax. The distribution functions of the competitive models are: (1) Weibull F x D 1  e  Jx , x t 0, D, J ! 0. (2) B-XII F x 1  1  x c k , x t 0, c, k ! 0. (3) W-Loss F x 1 Ve  Jx D , x t 0, V, D ! 0. 1  x c k , x t 0, c, k ! 0. V  Jx J (4) BIII F x (5) Fréchet F x e  Dx E , x t 0, D, E ! 0. A New Family of F-Loss Distributions: Properties and Applications 109 (6) Weibull-Lomax F x 1 § x D ·  a ¨¨ §¨ 1 ·¸ 1¸¸ © E¹ ¹ e © b , x t 0, a , b, E, D ! 0. (7) Lomax x D 1  §¨1  ·¸ , O¹ © F x x t 0, D, O ! 0. 8.1. Application 1: Business interruption losses dataset The first dataset consists of 2,387 French business interruption claims for losses above 100,000 French Francs. This data is available in CAS datasets package of R software. Table 2 shows the descriptive statistics of the business interruption claims dataset. It can be seen that, the losses are right skewed and leptokurtic, with a long right tail. Table 2. Descriptive statistics of the business interruption claims dataset No. of claims Mean Std. Skewness Kurtosis Min. Max. 2,387 1,096,223 4,751,003 24.444 719.424 100,289 152,449,017 Figure 7 shows the TTT-transform plot for the business interruption claims dataset. The data exhibits a decreasing hazard rate since it is convex below the 45 degree line. John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 110 Figure 7. TTT-transform plot for business interruption claims. Table 3 shows the maximum likelihood estimates for the parameters of the fitted distributions with their corresponding errors in brackets. The parameters of all the distributions fitted were significant at the 5% level with the exception of B-XII distribution which had all its parameters significant at 10% level and the CWL distribution also had D to be significant at the 10% level. Table 3. Maximum likelihood estimates of the parameters and standard errors for business interruption claims dataset Model CWL D̂ Ê Ô 0.001 0.375 0.076 (0.002) (0.015) (0.012) CBIIIL CLoL W-Loss Weibull V̂ Ĵ ĉ k̂ 0.002 0.257 1.483 (0.003) (0.002) (0.005) 0.331 0.003 9.536 (0.005) (0.007) (0.001) 0.444 0.003 3.510 (0.011) (0.002) (0.006) 0.403 0.001 (0.015) (0.003) A New Family of F-Loss Distributions: Properties and Applications 111 Lomax Fréchet 0.149 6.627 (0.003) (0.001) 7.171 0.150 (0.007) (0.003) BIII B-XII WeibullLomax â b̂ 4.663 7.550 0.002 0.117 (0.110) (0.001) (0.007) (0.003) 0.553 1.023 (0.002) (0.009) 0.898 0.085 (0.469) (0.045) Table 4 shows the goodness-of-fit and information criteria of the fitted distributions. It can be seen that, the CBIIIL distribution is the best distribution providing reasonable fit to the dataset among the other distributions fitted since it has the least AIC, BIC, K-S, A , W and –2l values compared with the rest of the competitive distributions. Table 4. Goodness-of-fit and information criteria of business interruption claims dataset Model –2l AIC BIC W* A* K-S CWL 72412.730 72418.730 72436.070 5.106 34.001 0.311 CBIIIL 69059.910 69065.920 69083.260 2.043 14.552 0.052 CLoL 72294.700 72300.700 72318.040 2.417 17.332 0.354 W-Loss 71571.410 71577.440 71594.770 10.508 51.244 0.364 Weibull 71233.400 71237.400 71248.960 11.283 22.083 0.334 Lomax 76037.090 76041.090 76052.650 5.841 35.701 0.855 Fréchet 75968.870 75972.870 75984.430 3.978 27.148 0.999 BIII 70624.600 70628.570 70640.130 2.420 17.340 0.197 B-XII 79313.320 79317.320 79328.870 4.281 29.006 0.586 Weibull-Lomax 70958.520 70966.520 70899.630 12.094 43.480 0.301 Figure 8 shows the plots of the empirical density, the fitted density, the empirical CDF and the CDF of the fitted distributions. It is evident that, the CWL, CBIIIL and CLoL distributions also provide reasonable fit to the data among the other distributions. John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 112 Figure 8. Empirical and fitted density (a) and CDF (b) plots of business interruption claims. 8.2. Application 2: Automobile bodily injury claim dataset The second dataset consists of 1,340 U.S automobile injury claims collected by the Insurance Research Council (part of AICPCU and IIA) in U.S. dollars. This dataset is reported in CAS datasets package of R software. Table 5 shows the descriptive statistics of the automobile bodily injury claim dataset. It can be seen that, the data is right-skewed and leptokurtic. Table 5. Descriptive statistics of automobile bodily injury claim dataset No. of claims Mean Std. Skewness Kurtosis Min. Max. 1,340 5.954 33.1362 25.688 794.666 0.005 1,067.697 Figure 9 shows the TTT-transform plot for the automobile bodily injury dataset. From the plot, there is evidence of a decreasing hazard rate function because the curve is convex below the 45 degree line. A New Family of F-Loss Distributions: Properties and Applications 113 Figure 9. TTT-transform plot for automobile bodily injury claims. Table 6 shows the maximum likelihood estimates for the parameters of the fitted distributions with their corresponding errors in brackets. The parameters of the BIII, B-XII, Fréchet, Lomax, Weibull and Weibull-Lomax distributions were all significant at the 5% level. The CWL distribution also had its parameters significant at the 5% level with the exception of D and V which were significant at 10%. CBILL had all its parameters significant at the 5% level with the exception of V which is significant at 10%. CLoL distribution had D to be significant at the 5% level whereas V and O are significant at the 10% level. Also, the W-Loss distribution had all its parameters significant at the 5% level with the exception of D which is significant at 10% level. Table 6. Maximum likelihood estimates of the parameters and standard errors for automobile bodily injury claims dataset Model CWL D̂ Ê Ô 0.092 0.755 0.093 (0.131) (0.002) (0.556) CBIIIL CLoL V̂ Ĵ ĉ k̂ 21.834 0.951 0.879 (0.407) (0.004) (0.008) 1.068 0.077 13.440 (0.001) (0.026) (0.039) John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda 114 W-Loss Weibull Lomax Fréchet 1.065 0.006 0.012 (0.308) (0.005) (0.175) 0.649 0.434 (0.009) (0.002) 1.911 4.359 (0.003) (0.007) 0.475 0.434 (0.008) (0.002) BIII B-XII WeibullLomax â b̂ 0.835 0.011 0.024 0.773 (0.004) (0.776) (0.663) (0.002) 1.041 1.484 (0.006) (0.001) 1.348 0.674 (0.005) (0.001) Table 7 shows the goodness-of-fit and information criteria of the fitted distributions. It can be seen that, the CBIIIL distribution is the best distribution providing reasonable fit to the dataset among the ten distributions fitted since it has the least AIC, BIC, K-S, A , W and –2l values compared with all the competitive distributions. Table 7. Goodness-of-fit and information criteria of automobile bodily injury claims dataset Model –2l AIC BIC W* A* K-S CWL 6384.347 6390.347 6405.948 4.390 21.741 0.113 CBIIIL 6384.234 6390.234 6405.835 2.938 15.045 0.083 CLoL 6401.219 6407.219 6422.820 4.068 20.074 0.104 W-Loss 6390.452 6395.279 6408.035 2.992 16.319 0.098 Weibull 6588.228 6592.228 6602.629 3.152 18.409 0.114 Lomax 6491.842 6495.842 6506.243 13.082 47.349 0.410 Fréchet 6406.624 6410.624 6421.025 9.869 51.085 0.515 BIII 6365.081 6369.081 6379.482 3.9985 19.753 0.110 B-XII 6392.537 6396.537 6406.938 4.422 22.028 0.128 Weibull-Lomax 6587.243 6595.244 6616.045 3.137 19.074 0.113 A New Family of F-Loss Distributions: Properties and Applications 115 Figure 10 shows the plots of the empirical density, the fitted density, the empirical CDF and the CDF of the fitted distributions. It is evident that, the CWL, CBIIIL and CLoL distributions also provide reasonable fit to the data among the other distributions. Figure 10. Empirical and fitted density (a) and CDF (b) plots of automobile bodily injury claims. 9. Conclusion In this article, we have proposed a new family of F-Loss distributions known as cosine F-Loss family of distributions; an extension of the F-Loss family of distributions. The purpose of the paper is to develop heavy-tailed distributions with fewer parameters and yet flexible in providing better parametric fit to a given dataset than some classical distributions and to generate distributions which are capable of modeling monotonic and nonmonotonic hazard rates. The mathematical properties and maximum likelihood estimators of the family are studied. Three special distributions, namely the cosine Weibull loss, cosine Burr III loss, and cosine Lomax loss distributions, are proposed. Simulations are carried out to examine the behavior of the parameters of the proposed distributions. It is realized that the estimators are consistent. The densities exhibit different kinds of rightskewed, decreasing, reversed-J, and approximately symmetric shapes. The 116 John Abonongo, Ivivi J. Mwaniki and Jane A. Aduda hazard rate functions show different kinds of increasing, decreasing, increasing-constant-increasing, increasing-constant-decreasing, reversed-J, and upside down bathtub shapes. The usefulness of the proposed distributions is analyzed with two insurance loss datasets. From the applications, the cosine Burr III loss distribution provides the best parametric fit for both the business interruption claims and the automobile bodily injury claims datasets. The proposed model is reasonably good compared with the competitors. Insurance practitioners can employ the proposed models in modeling insurance loss since they are flexible. We hope the proposed model will attract broader application in the actuarial sciences and other related fields. Future extensions of this work will be to consider inverse trigonometric and hyperbolic functions. References [1] Z. Ahmad, E. Mahmoudi and S. Dey, A new family of heavy tailed distributions with an application to the heavy tailed insurance loss data, Commun. Stat. Simul. Comput. (2020), 1-24. doi: 10.1080/03610918.2020.1741623 [2] Z. Ahmad, E. Mahmoudi and G. G. Hamedani, A family of loss distributions with an application to the vehicle insurance loss data, Pakistan Journal of Statistics and Operations Research 15(3) (2019), 731-744. [3] D. Bhati, E. Calderín-Ojeda and M. A. Meenakshi, A new heavy tailed class of distributions which includes the Pareto, Risks 7(4) (2019), 99. [4] D. Bhati and S. Ravi, On generalized log-moyal distribution: a new heavy tailed size distribution, Insurance: Mathematics and Economics 79 (2018), 247-259. [5] C. Chesneau, H. S. Bakouch and T. Hussain, A new class of probability distributions via cosine and sine functions with applications, Commun. Stat. Simul. Comput. 48 (2019), 2287-2300. [6] S. A. Klugman, H. H. Panjer and G. E. Willmot, Loss Models: from data to decisions (Vol. 715), John Wiley & Sons, 2012. [7] M. Muhammad, H. M. Alshanbari, A. R. A. Alanzi, L. Liu, W. Sami, C. Chesneau and F. Jamal, A new generator of probability models: the exponentiated sine-G family for lifetime studies, Entropy 23(11) (2021), 1394. A New Family of F-Loss Distributions: Properties and Applications 117 [8] D. Raab and Edward Green, A cosine approximation to the normal distribution, Psychometrika 26(4) (1961), 447-450. [9] L. Souza, New Trigonometric Classes of Probabilistic Distributions, Thesis, Universidade Federal Rural de Pernambuco, 2015. [10] L. Souza, W. R. O. Junior, C. C. R. de Brito, C. Chesneau, T. A. E. Ferreira and L. Soares, General properties for the Cos-G class of distributions with applications, Eurasian Bull. Math. 2 (2019), 63-79. [11] L. Tomy and G. Satish, A review study on trigonometric transformations of statistical distributions, Biom. Biostat. Int. J. 10(4) (2021), 130-136. [12] A. Alzaatreh, C. Lee and F. Famoye, A new method for generating families of continuous distributions, Metron 71 (2013), 63-79.