On Wrapping of Exponentiated Inverted Weibull Distribution

Transcription

1 IJIRST International Journal for Innovative Research in Science & Technology Volume 3 Issue 11 Aril 217 ISSN (online): On Wraing of Exonentiated Inverted Weibull Distribution P.Srinivasa Subrahmanyam Research Scholar Deartment of Statistics Nagarjuna University, Guntur, & Joint Director, Treasuries & Accounts Deartment, Govt. of A.P, India A.V.Dattatreya Rao Professor Deartment of Statistics Acharya Nagarjuna University, Guntur, A.P, India S.V.S.Girija Associate Professor Deartment of Mathematics Hindu College Guntur, A.P, India Abstract In many life testing exeriments directions are the observations. Directional data have many new and distinctive characteristics and challenges in terms of its modelling as well as in conducting statistical analysis. To draw more meaningful inferences, many circular models were develoed from the existing linear distributions using variety of techniques like wraing, inverse stereograhic rojections etc. In this article an attemt is made to construct a circular model for the Exonentiated Inverted Weibull Distribution by using the method of wraing. Exonentiated Inverted Weibull is considered to be the most frequently used robability distribution for analyzing the life time data with some monotone failure rates. In this aer the robability density function, distribution function and characteristic function are derived for this Wraed Exonentiated Inverted Weibull Distribution. The Trigonometric moments and imortant oulation characteristics for this wraed EIW Distribution are comuted. Keywords: Circular Statistics, Wraing, Exonentiated Inverted Weibull, Trigonometric Moments I. INTRODUCTION Dattatreya Rao et al (27) constructed Wraed Lognormal, Wraed Logistic, Wraed Weibull, and Wraed Extreme Value Distributions. Ramabhadra Sarma et al (29) derived characteristic function of Wraed Half Logistic and Wraed Binormal Distribution. Mardia and Ju (2) gave exressions for oulation characteristics such as variance, standard deviation, skewness, kurtosis etc. for circular distributions. Girija et al (21) introduced new construction rocedures of constructing Circular models calling Rising Sun Circular models and studied M L estimation arameters of Cardioid distribution from comlete samles. Contributing to this work, an attemt is made here to derive a new circular model for the Exonentiated Inverted Weibull distribution using the method of wraing. Wraing is a technique which reduces a linear variable to it s modulo 2π. The density, distribution function, characteristic function for the wraed Exonentiated Inverted Weibull distribution are derived and using the trigonometric moments, imortant oulation characteristics for the roosed wraed EIW distribution are also comuted. This aer is organised as follows. Section 2 describes the Circular robability distribution and the methodology of wraing a linear robability distribution. Section 3 defines the roosed wraed Exonentiated Inverted Weibull distribution, and resents the grahs of density, distribution and characteristic functions for various values of arameters. Imortant oulation characteristics for the wraed Exonentiated Inverted Weibull distribution are comuted. Section 4 summarises the findings of this study. For this aer software MATLAB is used for all the comutations and for lotting of grahs. II. CIRCULAR PROBABILITY DISTRIBUTION A circular random variable in a continuous circular distribution g :, 2 is said to be following a circular robability density function of g (θ) if and only if g has the following basic roerties g ( ), (1) All rights reserved by 18

2 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) 2 g( ) d 1 (2) Method of Wraing If X is a random variable defined on reduction. g ( ) g ( 2 k ) g is eriodic, for any integer k (Mardia,2) (3), then the corresonding circular random variable X w is defined by the modulo 2π mod 2 X W X If f (x) is the robability density function (df) of the linear random variable X then for the circular random variable X w, the corresonding df, g(θ) is defined as, It can be verified that ( ) g ( ) f ( 2 k ), where [, 2 ) k g with total robability concentrated on the unit circle (cos, sin ) / 2 lane and satisfies the roerties (1) to (3) above. Also the characteristic function for X w given its distribution function F ( ) is given by 2 i t t E e e df e t t it it It is clear from the above that whenever t, 2 it e 1 (Mardia 2). Imlies t integer values of t. Also the characteristic function for the wraed distribution is Trigonometric moments 2 i i in the can only be defined for and is defined as i E e e df e, Z also 1,,, the th trigonometric moment is value of the characteristic function at t =. The real art and the imaginary art of t are trigonometric moments and resectively and are denoted as cos, sin E E Where Z III. WRAPPED EXPONENTIATED INVERTED WEIBULL DISTRIBUTION(WEIW) The Exonentiated Inverted Weibull distribution is a generalization to the inverted Weibull distribution through adding a new shae arameter R by exonentiation to Inverted Weibull distribution function. A linear random variable X is said to follow a two arameter Exonentiated Inverted Weibull distribution, if the distribution function of X takes the following form x F ( x ) e Where c and λ are shae arameters and < x < and c >, λ> Hence the robability density function of Exonentiated Inverted Weibull distribution is c ( c 1) c x f ( x ). c. x e where < x < and c >, λ > Here if λ = 1, this EIW distribution becomes the standard Inverted Weibull distribution and if c = 1 this distribution reresents standard Inverted Exonential distribution. All rights reserved by 19

3 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) Probability density function for WEIW distribution Alying the method wraing the df for WEIW distribution g ( ) can be written as ( c1) ( 2 k) c g ( ). c ( 2 k ) e where k,2 and c >, λ > The grah deicting the linear reresentation of the df of WEIW distribution for different values of c keeing the value for the arameter λ at 2. is as follows: Fig. 1: PDF of WEIW distribution (Linear Reresentation) The same linear reresentation of df for different values of λ keeing the values for arameters c at 2. is obtained as below Fig. 2: PDF of WEIW distribution (Linear Reresentation) Now the grah deicting the circular reresentation of the df of WEIW Distribution for different values of c keeing the value for the arameter λ at 2. is shown below: All rights reserved by 2

4 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) Fig. 3: PDF of WEIW distribution (Circular Reresentation) Same circular reresentation for the df now for the different values of λ keeing the values for arameters c at 2. is obtained as below: Fig. 4: PDF of WEIW distribution (Circular Reresentation) Cumulative Distribution Function for WEIW distribution: The Distribution function of the WEIW distribution can be derived as Taking m 2 k ( c 1) ( 2 k) c G ( ). c ( 2 k ) e d k c and solving the integral we get.( 2 ).( 2 ) c k c k G ( ) ( e e ) k The grah for the CDF, G (θ) for WEIW distribution is obtained as below where,2 and c >, λ > All rights reserved by 21

5 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) Fig. 5: CDF of WEIW distribution Characteristic function for WEIW distribution: As discussed in the revious section the characteristic function of EIW distribution is taking considering v c itx ( c 1) x ( t ) e. cx. ( e ) dx (4) c x U we get u it ( u ) ( 1/ c ) () t e e du (5) u then (5) can be reduced to v it Now Equation (6) can be written as ( 1/ c ) v dv () t e e (6) () t k ( 1/ c ) v it v e dv k!. (7) k k k 1/ c k it v k / c k! 1/ c it 1 k / c k! c > and λ> (8) k The convergence of the series in (8) fails at least for some values of c for examle when c takes values between and 1 and k. To solve this for obtaining the trigonometric moments, the n oint Gauss Laguerre quadrature formula for numerical e v dv All rights reserved by 22

6 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) integration as given in Rao et al, (1975) is alied for equation (6). For the characteristic function of the wraed Exonentiated Inverted Weibull distribution is hence given by ( ) 2 The real and imaginary arts and resectively are obtained from the characteristic function of the WEIW distribution. The following are the grahs for the characteristic function of the WEIW distribution showing the real art and imaginary art searately for different values of c and λ e ix g d Fig. 6: Characteristic Function of WEIW distribution (1) Poulation Characteristics: Fig. 7: Characteristic Function of WEIW distribution (2) Given a Circular distribution, Mardia (2) had derived exressions for mean direction o resultant length 1, Circular variance V, Central Trigonometric Moments, o, Skewness and Kurtosis. Using these exressions the Poulation 1o 2 Characteristics for the Exonentiated Inverted Weibull distribution for different values of the arameters c and λ are comuted and tabulated here under. All rights reserved by 23

7 Table 1 Characteristics of Exonentiated Inverted Weibull Distribution at λ= 2. c c=.5 c=1.5 c=2. c=2.5 c=3. Trigonometric Moments α α β β Resultant Length ρ ρ Mean Direction µ Variance V Circular Standard Deviation σ Circular Trigonometric Moments α1 * α2 * β1 *..... β2 * Skewness ϒ Kurtosis ϒ Table 2 Characteristics of Exonentiated Inverted Weibull Distribution at c= 2. λ λ=.5 λ=1.5 λ=2. λ=2.5 λ=3. Trigonometric Moments α α β β Resultant Length ρ ρ Mean Direction µ Variance V Circular Standard Deviation σ Circular Trigonometric Moments α1 * α2 * β1 *..... β2 * Skewness ϒ Kurtosis ϒ On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) IV. CONCLUSION It can be observed that the Wraed Exonentiated Inverted Weibull distribution becomes Wraed model for standard Inverted Weibull distribution when λ = 1 and when c =1, Wraed Exonentiated Inverted Weibull distribution becomes wraed model for Exonentiated Inverted Exonential distribution. From the oulation characteristics for the Wraed Exonentiated Inverted Weibull distribution tabulated above in the last section, we can observe that with increasing value of shae arameter c, keeing other shae arameters λ = 2., the Circular variance gradually decreased, the distribution is negatively skewed and remained latykurtic. With increasing value of the scale All rights reserved by 24

8 On Wraing of Exonentiated Inverted Weibull Distribution (IJIRST/ Volume 3 / Issue 11/ 4) arameter λ keeing other scale arameter at c = 2. the Circular variance gradually increased, the distribution started shifting from negatively skewed to near symmetric and from latykurtic to mesokurtic. REFERENCES [1] Dattatreya Rao, A.V., Ramabhadra Sarma, I., Girija S.V.S., (27). On wraed version of some life testing models. Communication in Statistics - Theory and Methods, 36, [2] Girija, S.V.S., 21. Construction of New Circular Models. VDM - VERLAG, Germany. [3] A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova (212), The Exonentiated Inverted Weibull Distribution, Al. Math. Inf. Sci. 6, No. 2, [4] Jammalamadaka S. Rao, Sen Guta, A., 21. Toics in Circular Statistics, World Scientific Press, Singaore. [5] Mardia, K.V. and Ju, P.E. (2), Directional Statistics, John Wiley, Chichester. [6] Ramabhadra Sarma, I., Dattatreya Rao, A.V. and Girija S.V.S., (29). On Characteristic Functions of Wraed Half Logistic and Binormal Distributions, International Journal of Statistics and Systems, Vol 4(1), [7] Ramabhadra Sarma, I., Dattatreya Rao, A.V. and Girija, S.V.S., (211). On Characteristic Functions of Wraed Lognormal and Weibull Distributions, Journal of Statistical Comutation and Simulation Vol. 81(5), [8] Rao, C.R. & Mitra, S.K. (1975), Formulae and Tables for Statistical Work, Statistical Publishing Society. All rights reserved by 25