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IJIRST –International Journal for Innovative Research in Science & Technology| Volume 3 | Issue 11 | April 2017 ISSN (online): 2349-6010

On Stereographic Semicircular New Weibull Pareto Model P.Srinivasa Subrahmanyam Research Scholar Department of Statistics Nagarjuna University, Guntur, & Joint Director, Treasuries & Accounts Department, Govt. of A.P, India

A.V.Dattatreya Rao Professor Department of Statistics Acharya Nagarjuna University Guntur, A.P, India

S.V.S.Girija Associate Professor Department of Mathematics Hindu College Guntur, A.P, India

Abstract A very good number of Circular and Semicircular models have been constructed for the important and widely used life testing linear distributions by employing techniques like wrapping, inverse stereographic projections, rising sun function etc. Taking cue from this work an attempt is made in this paper to develop a new Semicircular model called Stereographic Semicircular New Weibull Pareto Distribution using the method of inverse stereographic projection. The Probability density function, Cumulative density function, Characteristic function, Trigonometric moments and population characteristics for the new Stereographic Semicircular New Weibull Pareto Distribution are studied. Keywords: Circular Distribution, Stereographic projection, Semicircular Models, Trigonometric Moments, New Weibull Pareto distribution _______________________________________________________________________________________________________ I.

INTRODUCTION

In the literature of Directional Statistics we can find a significant number of Circular models defined on unit circle, developed from the existing linear distributions, both continuous and discrete by Fisher (1993), Jammalamadaka and Sengupta (2001), Dattatreya Rao et al (2007). A new method called Stereographic Projection was employed for developing the probability distributions of angular models. Minh and Farnum (2003) used a bilinear transformations to map points on the unit circle in the complex plane into points x on the real line and for a density function g(α) on the interval (−π, π), and demonstrated how a corresponding density function f(x) on (−∞, ∞) is induced. Toshihiro Abe et al (2010) developed symmetric circular models applying the Inverse Stereographic Projection. Dattatreya Rao et al (2011) developed Cauchy type models by inducing Stereographic Projection on circular Cardioid distribution. Phani et al (2012) developed circular model induced by Inverse Stereographic Projection on Extreme-Value distribution. Taking cue from these works, an attempt is made in this paper to develop a Semicircular model for the New Weibull Pareto distribution called Stereographic Semicircular New Weibull Pareto Distribution using the method of inverse stereographic projection. This paper has been organised in to four sections. Section 2 deals with the methodology of inverse stereographic projection on a linear probability distribution. In the Section 3 the proposed new Stereographic Semicircular New Weibull Pareto Distribution is presented with relevant graphs of probability density, distribution and characteristic functions at different values of parameters. Important population characteristics for the new Stereographic Semicircular New Weibull Pareto Distribution are also tabulated. Section 4 presents a brief summary on the findings of this study. For this paper software MATLAB is used for all the computations and for plotting of graphs. II. INVERSE STEREOGRAPHIC PROJECTION METHOD For analysing angular data new models can be constructed by applying inverse stereographic projection on the linear probability distributions. Inverse stereographic projection when applied on a linear model yields a circular model. Stereographic projection creates a one to one relationship between the points on the unit circle and those on the real line. Circular as well as linear Probability distributions can be obtained by applying stereographic projection. Inverse stereographic projection is defined by a one to one mapping given

  T     x  u  v tan   2 by

, where

x    ,   ,   [  ,  ), u 

, and v  0

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

If x is random variable defined on the interval

  ,  

with probability density functions

f  x

and cumulative distribution as

x  u 1 1 T  x     2 tan   F x v   is a random point on a unit circle. , then

G 



and

 , then G  



g   and

are respectively the cumulative distribution and the probability density functions of this random point

g  

can be derived in terms of

F x

and

f  x

using the following transformation

For v  0   G     F  u  v tan  2   2   sec  2 g    v  2   

    F  x  

     f  u  v tan    2  



…(1)

  

…(2)

Without loss of generality u is considered as zero in this paper. If Inverse Stereographic Projection is applied on a linear model whose probability density function is defined on the resultant distribution is a Stereographic Semicircular models automatically mapped onto demonstrates this fact.

 0,  

then

 0,   . The graph depicted below

III. STEREOGRAPHIC SEMICIRCULAR NEW WEIBULL PARETO DISTRIBUTION A linear random variable X is said to follow a three parameter New Weibull Pareto (NWP) Distribution if the distribution function of X is given by

F(X )  1 -

 x  -    

Where c is the shape parameter and λ and δ are the scale parameters and 0 < x < ∞ and c >0, λ>0 and δ > 0 the pdf of New Weibull Pareto Distribution is

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

f (x) 

c .  x     

 x c    

c 1 e

Where 0 < x < ∞ and c >0, λ >0 and δ > 0 Here if δ = 1, this NWP distribution reduces to Weibull distribution and if δ = 1 and c = 1 this distribution reduces to Exponential distribution. Also when c=2 and δ =1 the New Weibull Pareto distribution becomes Rayleigh distribution. Probability density function for Stereographic Semicircular New Weibull Pareto (SSNWP) distribution: As discussed in the previous section, by applying the inverse stereographic projection on the New Weibull Pareto distribution, the one to one mapping

  T     x  v tan   ; where x   0,   ,   [0,  ) and v  0 2 will yield the Stereographic Semicircular New Weibull Pareto distribution with probability density function g(θ) and cumulative distribution function G(θ) as below:

g ( )



    v tan    2    c .  2  S ec   2   2      



c 1

      

     2     

v tan 

… (3)

and 

G ( )  1

where   ( 0 ,  ) , c ,  ,   0 , and v 

e

       

    2   

v ta n 



       

… (4)



The graph depicting the linear representation of the pdf of SSNWP distribution for different values of c keeping the value for the parameter λ and δ at 2.0 is as follows:

Fig. 1: PDF of SSNWP distribution (Linear Representation) The linear representation of pdf for different values of λ keeping the values for parameters c and δ at 2.0 is obtained as below

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

Fig. 2: PDF of SSNWP distribution (Linear Representation)

Also the linear representation of pdf for different values of δ keeping the values for parameters c and λ at 2.0 is obtained as below

Fig. 3: PDF of SSNWP distribution (Linear Representation)

The graph depicting the circular representation of the pdf of SSNWP Distribution for different values of c keeping the value for the parameter and λ and δ at 2.0 is shown below:

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

Fig. 4: PDF of SSNWP distribution (Circular Representation)

The circular representation for the pdf for the different values of λ keeping the values for parameters c and δ at 2.0 is obtained as below:

Fig. 5: PDF of SSNWP distribution (Circular Representation)

Also the circular representation for the pdf for different values of δ keeping the values for parameters c and λ at 2.0 is obtained as below

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

Fig. 6: PDF of SSNWP distribution (Circular Representation)

Cumulative Distribution Function for SSNWP distribution: The graph for the CDF, G (đ?&#x153;&#x192;) for SSNWP is obtained as below:

Fig. 7: CDF of SSNWP distribution

Characteristic function for SSNWP distribution: The characteristic function of a Stereographic Semicircular model If the cumulative distribution function and probability density function for a Stereographic Semicircular model are G ď&#x20AC;¨ ď ą ď&#x20AC;Š and

g ď&#x20AC;¨ď ą ď&#x20AC;Š respectively and if F ď&#x20AC;¨ x ď&#x20AC;Š and f ď&#x20AC;¨ x ď&#x20AC;Š are the corresponding cumulative distribution function and probability density function of the linear model, then for the Stereographic Semicircular model the characteristic function can obtained as ď °

ď ŞX

ď&#x20AC;¨ p ď&#x20AC;Š ď&#x20AC;˝ ď&#x192;˛ e ipď ą g ď&#x20AC;¨ď ą ď&#x20AC;Š d ď ą

, pď&#x192;&#x17D;

Without loss of generality, considering v =1, the characteristic function for the Stereographic Semicircular New Weibull Pareto distribution can be obtained from the following equation.

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

X

Taking



 p 

     ta n    2  u         



e 0

ip

    v tan    v 2    c .   2  S ec   2    2      



c 1

      

  

v tan     2



   

d

… (5)

c

and transforming (5) we get 

X

 p 



ip ( 2 tan



1 

 

1  



 .u  c    

 u

e

… (6)

Again considering k= u.λ and making necessary transformations for (6) the characteristic function for the Stereographic Semicircular New Weibull Pareto distribution can be obtained as below 

X

 p 



ip ( 2 tan



1   k c  .          

1 

k

…. (7)

It can be noticed that the integral obtained above cannot be evaluated analytically in its general form, for evaluating the characteristic function to obtain the trigonometric moments for the new Stereographic Semicircular New Weibull Pareto distribution, the n – point Gauss – Laguerre quadrature formula for numerical integration as given in Rao et al, (1975) is applied on the equation (7) above. The real and imaginary parts  p and  p respectively are obtained from the characteristic function of the SSNWP distribution. The following are the graphs for the characteristic function of the SSNWP distribution showing the real part and imaginary part separately for the parameters c = 2, λ=3 δ=4.0 in Fig 8, and for c = 5, λ=4 δ=3.0 in Fig 9.

Fig. 8: Characteristic Function of SSNWP distribution

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

Fig. 9: Characteristic Function of SSNWP distribution

Population Characteristics: Mardia and Jupp (2000) had derived expressions for mean direction  o resultant length  1 , Circular variance V0, Central 



Trigonometric Moments  p ,  p , Skewness  1 and Kurtosis  2 from the values of the characteristic function  p at p=1 and o

2. The Population Characteristics for the Stereographic Semicircular New Weibull Pareto distribution for different values of the parameters c, λ and δ are computed and tabulated here under. Table – 1 Characteristics of Stereographic Semicircular New Weibull Pareto distribution at λ = 3.0 and δ=3.0 At different values of c c=2.0 c=3.0 c=4.0 c=5.0 c=6.0 Trigonometric Moments α1 -0.2318 -0.4640 -0.5740 -0.6328 -0.6683 α2 -0.4753 -0.4181 -0.2757 -0.1649 -0.0862 β1 0.8510 0.8311 0.7894 0.7561 0.7316 β2 -0.3288 -0.7000 -0.8630 -0.9303 -0.9603 Resultant Length ρ1 0.8820 0.9518 0.9760 0.9860 0.9910 ρ2 0.5780 0.8154 0.9060 0.9448 0.9641 Mean Direction µ0 1.8367 2.0799 2.1995 2.2676 2.3110 Circular Variance V0 0.1180 0.0482 0.0240 0.0140 0.0090 Circular Standard Deviation 0.5010 0.3142 0.2205 0.1678 0.1348 σ0 1.0471 0.6389 0.4445 0.3371 0.2703 Central Trigonometric Moments α1* 0.8820 0.9518 0.9760 0.9860 0.9910 α2* 0.5764 0.8153 0.9059 0.9448 0.9641 β1* 0.0000 0.0000 0.0000 0.0000 0.0000 β2* 0.0423 0.0114 0.0038 0.0016 0.0008 Skewness ϒ10 1.0444 1.0828 1.0184 0.9532 0.8989 Kurtosis ϒ20 -2.0747 -2.3482 -2.4262 -2.4460 -2.4476 Table – 2 Characteristics of Stereographic Semicircular New Weibull Pareto distribution at c = 3.0 and δ = 3.0

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

At different values of Îť Trigonometric Moments Îą1 Îą2 Î˛1 Î˛2 Resultant Length Ď 1 Ď 2 Mean Direction Âľ0 Circular Variance V0 Circular Standard Deviation Ď&#x192;0 Central Trigonometric Moments Îą1* Îą2* Î˛1* Î˛2* Skewness Ď&#x2019;10 Kurtosis Ď&#x2019;20

Îť =2.0

Îť =3.0

Îť =4.0

Îť =5.0

Îť =6.0

-0.1395 -0.7376 0.9302 -0.2377

-0.4640 -0.4181 0.8311 -0.7000

-0.6443 -0.0837 0.7173 -0.8569

-0.7506 0.1751 0.6197 -0.8785

-0.8170 0.3631 0.5408 -0.8459

0.9406 0.7749

0.9518 0.8154

0.9641 0.8610

0.9733 0.8958

0.9798 0.9205

1.7196

2.0799

2.3027

2.4514

2.5568

0.0594

0.0482

0.0359

0.0267

0.0202

0.3500 0.7141

0.3142 0.6389

0.2702 0.5471

0.2325 0.4691

0.2021 0.4070

0.9406 0.7749 0.0000 0.0110

0.9518 0.8153 0.0000 0.0114

0.9641 0.8610 0.0000 0.0084

0.9733 0.8958 0.0000 0.0057

0.9798 0.9205 0.0000 0.0039

0.7564

1.0828

1.2329

1.3110

1.3561

-2.2216 -2.3482 -2.4269 -2.4786 -2.5131 Table â&#x20AC;&#x201C; 3 Characteristics of Stereographic Semicircular New Weibull Pareto distribution at c = 3.0 and Îť = 3.0 At different values of Î´ Î´=2.0 Î´=3.0 Î´=4.0 Î´=5.0 Î´=6.0 Trigonometric Moments Îą1 -0.5547 -0.4640 -0.3936 -0.3359 -0.2869 Îą2 -0.2653 -0.4181 -0.5166 -0.5849 -0.6344 Î˛1 0.7804 0.8311 0.8627 0.8841 0.8993 Î˛2 -0.7931 -0.7000 -0.6136 -0.5357 -0.4656 Resultant Length Ď 1 0.9575 0.9518 0.9482 0.9457 0.9439 Ď 2 0.8363 0.8154 0.8022 0.7932 0.7869 Mean Direction Âľ0 2.1887 2.0799 1.9988 1.9339 1.8797 Circular Variance V0 0.0425 0.0482 0.0518 0.0543 0.0561 Circular Standard Deviation 0.2948 0.3142 0.3262 0.3342 0.3397 Ď&#x192;0 0.5980 0.6389 0.6640 0.6808 0.6924 Central Trigonometric Moments Îą1* 0.9575 0.9518 0.9482 0.9457 0.9439 Îą2* 0.8362 0.8153 0.8021 0.7931 0.7868 Î˛1* 0.0000 0.0000 0.0000 0.0000 0.0000 Î˛2* 0.0102 0.0114 0.0120 0.0122 0.0121 Skewness Ď&#x2019;10 1.1611 1.0828 1.0181 0.9625 0.9137 Kurtosis Ď&#x2019;20 -2.3865 -2.3482 -2.3198 -2.2972 -2.2784

IV. CONCLUSION It can be observed that the Stereographic Semicircular New Weibull Pareto distribution becomes Stereographic Semicircular Exponential distribution when c = 1 and đ?&#x203A;ż =1 and when đ?&#x203A;ż =1, it reduces to Stereographic Semicircular Weibull distribution. Also when c=2 and đ?&#x203A;ż =1 the Stereographic Semicircular New Weibull Pareto distribution becomes Stereographic Semicircular Rayleigh distribution. From the population characteristics of Stereographic Semicircular New Weibull Pareto distribution tabulated in the last section, we can observe in all the different cases the distribution remained platykurtic. Circular Variance got reduced with incrementing the value of parameter c, when Îť and đ?&#x203A;ż kept constant at 3.0, and also with incrementing value of Îť when c and đ?&#x203A;ż are

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On Stereographic Semicircular New Weibull Pareto Model (IJIRST/ Volume 3 / Issue 11/ 048)

kept constant. The Skewness of the distribution also got reduced with higher value of c and δ when the other parameters kept constant. REFERENCES Dattatreya Rao, A.V., Ramabhadra Sarma, I., Girija S.V.S., (2007). “On wrapped version of some life testing models”. Communication in Statistics Theory and Methods, 36, 2027-2035 [2] Dattatreya Rao A.V, Girija S.V.S, Phani. Y. (2011) “Differential Approach to Cardioid Distribution”, Computer Engineering and Intelligent Systems, Vol/Issue: 2(8). Pp. 1-6, 2011. [3] Fisher, N. I. (1993). “Statistical Analysis of Circular Data”. Cambridge University Press, Cambridge. [4] Girija, S.V.S., 2010. “Construction of New Circular Models”. VDM - VERLAG, Germany. [5] Jammalamadaka S. Rao, Sen Gupta, A., (2001). “Topics in Circular Statistics”, World Scientific Press, Singapore. [6] Mardia, K.V. and Jupp, P.E. (2000), “Directional Statistics”, John Wiley, Chichester. [7] Minh, Do Le and Farnum, Nicholas R. (2003), “Using Bilinear Transformations to Induce Probability Distributions”, Communication in Statistics – Theory and Methods, 32, 1, pp. 1 – 9. [8] Phani. Y., Girija S.V.S and Dattatreya Rao A.V. (2012), “Circular Model Induced by Inverse Stereographic Projection On Extreme-Value Distribution”, IRACST – Engineering Science and Technology: An International Journal (ESTIJ), Vol.2, No. 5, 2250-3498 [9] Ramabhadra Sarma, I., Dattatreya Rao, A.V. and Girija S.V.S., (2009). “On Characteristic Functions of Wrapped Half Logistic and Binormal Distributions”, International Journal of Statistics and Systems, Vol 4(1), pp. 33–45. [10] Rao, C.R. & Mitra, S.K. (1975), “Formulae and Tables for Statistical Work”, Statistical Publishing Society. [11] Tahir M. H, Gauss M. Cordeiro, Ayman Alzaatreh, M. Mansoor, and M. Zubair, (2016), “A New Weibull-Pareto Distribution: Properties and Applications”, Communications in Statistics - Simulation and Computation Vol. 45, Iss. 10,2016 [12] Toshihiro Abe, Kunio Shimizu, Arthur Pewsey (2010). “Symmetric Unimodal Models for Directional Data Motivated by Inverse Stereographic Projection”, J. Japan Statist. Soc., Vol /Issue: 40(1). Pp 45-61. [1]

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