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A GENERALIZED GAMMA DISTRIBUTION ASSOCIATED WITH pFq AND ITS APPLICATION IN RELIABILITY

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A GENERALIZED GAMMA DISTRIBUTION ASSOCIATED WITH pFq AND ITS APPLICATION IN RELIABILITY

International Journal of Engineering Research & Technology (IJERT)

A GENERALIZED GAMMA DISTRIBUTION ASSOCIATED ISSN: 2278-0181

ETRASCT' 14 Conference Proceedings

WITH pFq AND ITS APPLICATION IN RELIABILITY

Dr. Anjali Mathur1

Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan) E-mail :-

anjali24mathur@gmail.com

Dr. Sandeep Mathur2

Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan)

E-mail :- mathur.sandeep1979@gmail.com

ABSTRACT

In this paper a generalized gamma distribution involving pFq (generalized hypergeometric function), has been studied from which almost variance. Hazard function have been worked out for generalized gamma distribution having pFq.

  1. INTRODUCTION

    In recent years many generalizations of gamma and Weibull distributions are proposed notably by Bradley [1], Srivastava [2], Lee and Gross[4], Bondesson [5]. These generalized distributions are mainly introduced in order to extend the scope of ordering gamma and Weibull distributions and to develop a Model for failure to suit any given particular situation.

    Kobayashi [3] has introduced a new type of generalized gamma function as

    r (m , n) = xm1 (x+n)-re-x dx … (1.1)

    0

    For a positive integer r. Here m and n are parameters of the functions. This function occurs in many problems of diffraction theory [Kobayashi [3]]. However, this generalized gamma function has not yet drawn, the attention of statistician. Agarwal and Kalla [6] has introduced a slightly modified form of the generalized gamma function as

    xm1 (x n) ebx

    0

    dx bm

    m, bn

    … (1.2)

    In this paper we defined a new generalization of gamma distribution involving pFq by considering a modified from of the Agarwal and Kalla [6]. A few well known probability distributions are shown to be its particular cases.

    ISSN: 2278-0181

    FUNCTION OF MATRIX ARGUMENT IN THE COMIntPerLnaEtioXnal CJouArnSalEof E:ngineering Research & Technology (IJERT)

    ETRASCT' 14 C~onference Proceedings

    We consider real valued scalar function of a single matrix argument of the type Z =

    ~ ~ ~ ~

    X + i Y where X and Y are p x p matrices with real elements and i

    1 as well as

    scaler functions of many matrices

    ~

    Z j , j = 1, 2, .K where each

    ~ ~

    Z j is of the type Z

    above in the real case. We confined our discussion to the situation where the argument matrix was real symmetric positive definite. This was done so that the fractional power of matrices and functions of such matrices could be uniquely defined. Corresponding properties are of we restrict to the class of Hermitian positive definite matrices.

    Definition : Hermitian positive definite matrix due to Mathai [11], We will denote the

    ~ ~ ~ ~ ~

    conjugate of Z by Z if Z hermitian, then Z = Z *, that is

    ~ ~ ~ ~ ~ ~ ~ ~

    Z = Z * X + i Y = ( X + i )* = X + i Y

    ~ ~ ~ ~

    X = X and Y = Y

    ~ ~ ~

    Thus X is the symmetric and Y is skew symmetric. Further if Z is hermitian

    ~

    positive definite, then all the eigen values of gamma in the complex case is

    Z are real and positive. Further, matrix variate

    p

    ~ ()

    p ( p1) ( )( 1)……. ( 1)

    = 2

    ~

    We will use the notation

    Z 0

    to indicate that

    ~

    Z is hermitian positive definite.

    Constant matrices will be written without a tilde whether the elements are real or complex unless it has to be emphasized that the matrix involved has complex elements. Then in that case a constant matrix will also be written with a tilde.

    ZONAL POLYNOMIAL

    Let

    ~

    Vk be the vector space of homogeneous polynomial of degree k, then the Zonal

    polynomial ~ ~ is defined as the component of (tr ~ k ) in the subspace

    ~ . ~ ~ is

    Ck (X)

    X Vk Ck (X)

    also generalization of

    ~ k

    ~

    ~

    X . The exponential function has the following expansion

    ~

    etr( X)

    tr~

    C X

    (1.3)

    1

    X

    k

    K

    k 0 k! k0 K k!

    The binomial expansion is the following for I ~ ~ ~* 0and all eigen values

    X 0 that is X X

    X

    of ~ are between 0 and 1.

    ~

    ~ ~

    C

    International Journal of Engineering Research & Technology (IJERT)

    det I X

    where

    k 0 K

    K

    k! k

    (X),

    (1.4)ISSN: 2278-0181

    ETRASCT' 14 Conference Proceedings

    ()

    p

    j 1 ,

    2

    K

    j

    j1 k

    with

    K k ,…, k , k … k k

    1 p 1 p

    ~ ~ ~ ~

    ~ ~ ~ ~ ~ ~

    CK (X)CK (T)

    CK

    CK (H * XHT)dH

    ~ O ( P )

    ~ (I)

    (1.5)

    where I is the identity matrix, the integral is over the orthogonal group of p x p matrices and

    d ~

    H is the invariate Harr measure. For detailed study consult Mathai [10].

  2. STATISTICAL PROPERTIES OF GENERALIZED GAMMA DISTRIBUTION INVOLVING PFq

    In this section the expression of Mean, Variance, Moment generating function.

    Hazard function of generalized gamma distribution are discussed.

    Theorem 1: If the random variable x follows the generalized gamma distribution involving

    PFq with its respective mean and variance are

    p1

    m 1, bnp1 Fq , m 1;;k b

    Mean =

    bm, bn

    Fq , m ;; k b

    Mean =

    m

    b

    , for small bn (2.1)

    k=0 or either =0 or =0 and

    b2 m 2, bn m, bn

    F 2 m 1, bn,

    F

    Variance=

    p1 q

    p1 q

    m, bn F *

    p1 q

    m

    =

    b2

    , for small bn. (2.2)

    Here p1 Fq p1Fq , m 2;; k b

    q

    p1

    F

    p1

    Fq , m 1;; k b

    International Journal of Engineering Research & Technology (IJERT)

    ISSN: 2278-0181

    ETRASCT' 14 Conference Proceedings

    p1 Fq * p1Fq , m ;; k b

    Proof : By definition

    Mean = Ex xf xdx

    0

    p1

    b1 m 1, bn

    F , m 1; ; k b

    q

    = m, bn F , m ; ; k

    p1 q b

    Putting k = 0 or either 0 or 0

    b1 m 1, bn

    m, bn

    mn Um 1,2 m , bn b Um,1 m , bn

    m

    =

    b

    , for small bn

    and Variance = Ex2 Ex2

    Ex2 x2 f xdx

    m 2, bn F

    = p1 q

    b2 m, bn F *

    p1 q

    Putting k=0, 0 or 0

    b2 m 2, bn

    m, bn

    mm 1n2 Um 2,3 m , bn Um,1 m , bn

    1 m m d

    =

    b2

    , for small bn

    International Journal of Engineering Research & Technology (IJERT)

    ISSN: 2278-0181

    ETRASCT' 14 Conference Proceedings

    Variance =

    m

    b2

    Theorem 2 : The rth moment about origin and the moment generating function of random varable x following generalized gamma distribution are

    m r, bnp1 Fq , m r;; k

    ' b

    r br m, bn F , m ;; k

    p1 q b

    b

    1

    t m m, bn1

    t b

    p1

    F , m;; k

    q

    b t

    and M.G.F. =

    m, bn F , m ;; k

    p1 q b

    Theorem 3 : If x follows the generalized gamma distribution involving failure rate function is given by

    p Fq its Hazard or

    Hx

    e bx x m1 x n F ; ; kx

    F , m ;; k

    (2.3)

    q

    p

    bm m, bn, bx

    p1 q b

    where

    m, n, xis generalized in complete gamma function defined as

    x

    tm1e t

    m, n, x= t n dt

    REFERENCES

    International Journal of Engineering Research & Technology (IJERT)

    ISSN: 2278-0181

    1. Bradeley M. (1988) : Generalized gamma parameter estimationETaRnAdSCTm' 14oCmonefnertence Proceedings

      evaluation common in Statist. Theory and Method, no. 17, p.p. 507-517.

    2. Srivastava H.M., Karlsson P.W. (1985) : Multiple Gaussian Hypergeometric series. Ellis Horwood Limited, Publishers, Chichester.

    3. Kobayashi K. (1991) : On generalized gamma function occurring in diffraction theory. J. Physical Society of Japan, 60, no. 5, p.p. 1501-1512.

    4. Lee M. and Gross A. (1991) : Lifetime distribution under unknown environment. J. Static. Plann. And Inference., no. 29, p.p. 137-143.

    5. Bondesson L. (1992) : Generalized gamma convolution and densities, lecture notes in Statistics, Springer-Verlag, New York.

    6. Agarwal S. K. and Kalla S. L. (1996) : A generalized gamma distribution and its application in reliability, Commun. Statist.-Theory Meth., 25(1), p.p. 201-210.

    7. Abramowitz M. and Stengun I. A. (1972) : Handbook of Mathematical Functions, Dover, New York.

    8. George A. and Mathai A. M. (1975): A generalized distribution for inter-live-birth intervals, Sankhya Vol. 37, Series B, p.p. 332-342.

    9. Saxena R. K. and Mishra K. N. (1991) : A generalized Probability Distribution, Mat. Acad. Math., India (Gorakhpur), Vol. 9.

    10. Mathai A.M. (1993) : Hypergeometric Functions of several Matrix Arguments, Centre of Mathematical Sciences, Trivandrum, India.

    11. Mathai A.M. (1995) : Jacobians of matrix transformation and functions of matrix argument . Word scientific Publishing Comp. New York.

    12. Mathai A.M. (1995) : Special Functions of Matrix Arguments-III ; Proceedings of the National Academy of Sciences, India ; LXV (IV) p.p. 367-393.

    13. Mathai A.M. (1995a) : Special function of matrix argument 1. Nat. Aca. Sci. India, Vol LXV Sec. A . part III. P.121 – 144 .

    14. Mathai A.M. (1995b) : Special function of matrix argument II. Nat. Aca . Sci. India

      , Vol LXV, SecA. Part IV, p. 227 – 246 .

    15. Mathai A.M. (1995c) : Special function of matrix argument III. Nat. Aca. Sci. India. Vol LXV. Sec A, part IV, p.367 – 393

    16. Mathai A.M., Pederzoli G. (1996) : Some Transformations for Functions of Matrix Arguments Indian J. Pure Appl. 27 (3), pp. 277-284.

    17. Saxena R.K., Sethi P.L. & Gupta O.P. (1997) : Appells Functions of Matrix Arguments. Indian J. Pure Appl. Math. ; 28, no. 3, p.p. 371-380.

    18. Srivastava H.M., Karlsson P.W. (1985) : Multiple Gaussian Hypergeometric series. Ellis Horwood Limited, Publishers, Chichester.

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    19. Upadhyaya Lalit Mohan, Dhami H.S. (2001) : MInteartnraitixonaGl JeonurenarlaolfizEnagtiinoeenrisngoRfesMearuchlt&ipTleechnology (IJERT) Hypergeometric Functions, # 1818 IMA Preprints Series, UniversityEoTRfAMSCiTn' 1n4eCsoontfear,ence Proceedings Minneapolis, U.S.A.

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