Various characterizations of the distribution of the ratio of two independent Rayleigh random variables are presented. These characterizations are based, on a truncated moment; on hazard function; and on certain functions of order statistics

Hamedani, Gholamhossein

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of10-1-2013Characterizations of Distribution of Ratio ofRayleigh Random VariablesGholamhossein HamedaniMarquette University, gholamhoss.hamedani@marquette.eduPublished version. Pakistan Journal of Statistics, Vol. 29, No. 4 (October 2013): 369-376. Permalink.© 2013 Pakistan Journal of Statistics. Used with permission.© 2013 Pakistan Journal of Statistics 369 Pak. J. Statist. 2013 Vol. 29(4), 369-376  CHARACTERIZATIONS OF DISTRIBUTION OF RATIO OF  RAYLEIGH RANDOM VARIABLES  G.G. Hamedani Department of Mathematics, Statistics and Computer Science Marquette University, Milwaukee, WI 53201-1881, USA Email: g.hamedani@mu.edu  ABSTRACT   Various characterizations of the distribution of the ratio of two independent Rayleigh random variables are presented. These characterizations are based, on a truncated moment; on hazard function; and on certain functions of order statistics.  1. INTRODUCTION   The problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions. As pointed out by Shakil and Ahsanullah (2011), the distribution of the ratio of independent random variables arises in many fields of studies such as biology, economics, engineering, genetics and order statistics, to name a few. For the detailed explanation of the importance of the distribution of the ratio of independent random variables, we refer the interested reader to Shakil and Ahsanullah (2011) where they consider the distributional properties of record values of the ratio of independent Rayleigh random variables. In Section 2, we present characterizations of the distribution of the ratio of two independent Rayleigh random variables in three different directions. Our results in subsections 2.1-2.3, will be based: on a truncated moment; on hazard function; and on truncated moments of certain functions of order statistics, respectively.   The pdf (probability density function) f , cdf (cumulative distribution function) F  and hazard function F  of the distribution of the ratio of two independent Rayleigh random variables are given, respectively, by         22; , 2 , 0f x f x x x x                (1.1) and      121 , 0,F x x x                  (1.2)       122 , 0F x x x x                   (1.3)  where 0   and 0   are parameters. We like to mention that pdf ( )1.1  is also the pdf of the square root of the Pareto random variable with the shape parameter 1.   Characterizations of Distribution of Ratio of Rayleigh Random Variables 370  Throughout this work we denote the ratio of two independent Rayleigh random variables with RR.  2. CHARACTERIZATION RESULTS   The distribution of the ratio of independent Rayleigh random variables has applications in many fields of study, in particular in life modeling. So, an investigator will be vitally interested to know if their model fits the requirements of RR distribution. To this end, the investigator relies on characterizations of RR distribution, which provide conditions under which the underlying distribution is indeed that of RR. In this section we will present several characterizations of this distribution.   Throughout this section we assume, where necessary, that the distribution function F  is twice differentiable on its support.  2.1 Characterization Based on a Truncated Moment  In this subsection we present characterizations of RR distribution in terms of a truncated moment. We like to mention here the works of Galambos and Kotz ( )1978 , Kotz and Shanbhag ( )1980 , Glänzel (1987, 1988, 1990), Glänzel et al. (1984, 1994), Glänzel and Hamedani ( )2001  and Hamedani (1993, 2002, 2006, 2011) in this direction. Our characterization results presented here will employ a special version of an interesting result due to Glänzel ( )1987  (Theorem 2.1.1 below).  Theorem 2.1.1   Let  , ,F P  be a given probability space and let  ,H a b  be an interval for some ( ,a b a b     might as well be allowed). Let :X H  be a continuous random variable with the distribution function F  and let g  be a real function defined on H such that        , ,E g X X x x x H          is defined with some real function  . Assume that    1 2,g C H C H   and F  is twice continuously differentiable and strictly monotone function on the set H . Finally, assume that the equation g  has no real solution in the interior of H . Then F  is uniquely determined by the functions g  and  , particularly           0 expx uF x C s u duu g u   ,   where the function s  is a solution of the differential equation sg  and C  is a constant, chosen to make 1HdF  .    G.G. Hamedani 371 Remarks 2.1.2  ( )a  In Theorem 2.1.1, the interval H need not be closed since the restriction is on the interior of H . ( )b  The goal is to have the function   as simple as possible. For a more detailed discussion on the choice of  , we refer the reader to Glänzel and Hamedani ( )2001  and Hamedani (1993, 2002, 2006).  Proposition 2.1.3   Let  : 0,X    be a continuous random variable and let    122g x x    for  0,x  . The pdf of X  is ( )1.1  if and only if the function   defined in Theorem 2.1.1 has the form       12 , 0x x x     .  Proof:   Let X  have pdf ( )1.1 , then       12 , 0,E g X X x x x        and        12 0.x g x x            Conversely, if   is given as above, then          122 , 0,xs x x x xx g x          and hence       2ln .s x x       Now, in view of Theorem 2.1.1 (with C   ), X  has cdf ( )1.2  and pdf ( )1.1  .  Corollary 2.1.4   Let  : 0,X    be a continuous random variable. The pdf of X  is ( )1.1  if and only if there exist functions g  and   defined in Theorem 2.1.1 satisfying the differential equation         122 , 0.xx x xx g x          Characterizations of Distribution of Ratio of Rayleigh Random Variables 372 Remarks 2.1.5   ( )a  The general solution of the differential equation given in Corollary 2.1.4 is            22 22 ,x x g x x x dx D               for 0x  , where D  is a constant. One set of appropriate functions is given in Proposition 2.1.3 with 0D  .  ( )b  Clearly there are other pairs of functions  ,g   satisfying conditions of Theorem 2.1.1. We presented one such pair of functions in Proposition 2.1.3.  2.2 Characterization Based on Hazard Function  For the sake of completeness, we state the following simple fact.   Let F  be an absolutely continuous distribution with the corresponding pdf f . The hazard function corresponding to F  is        , ,1f xF x x Supp FF x             (2.2.1)  where Supp F  is the support of F .   It is obvious that the hazard function of a twice differentiable distribution function satisfies the first order differential equation          ,F FFxx k xx                (2.2.2)  where  k x  is an appropriate integrable function. Although this differential equation has an obvious form since         ,F FFf x xxf x x     for many univariate continuous distributions ( )2.2.2  seems to be the only differential equation in terms of the hazard function. The goal here is to establish a differential equation which has as simple form as possible and is not of the trivial form ( )2.2.2 . For some general families of distributions this may not be possible. Here is our characterization result for RR distribution.  Proposition 2.2.1   Let  : 0,X    be a continuous random variable. The pdf of X  is (1.1) if and only if its hazard function F , with boundary condition  0 0F  , satisfies the differential equation  G.G. Hamedani 373        1 2 0, 0.F F Fx x x x x                (2.2.3)  Proof:   If X  has pdf ( )1.1 , then obviously ( )2.2.3  holds. If F  satisfies ( )2.2.3 , then      1 ,Fdx x xdx    or       22,1Ff x xxF x x c      where c  is a constant.   Integrating both sides of the above equation with respect to x  from 0 to x  and after some computations, we arrive at ( )1.2  with c.  Remark 2.2.2   For characterizations of other well-known continuous distributions based on the hazard function, we refer the reader to Hamedani ( )2004  and Hamedani and Ahsanullah ( )2005 .  2.3 Characterization based on Truncated Moment of  Certain Functions of Order Statistics   Let 1: 2: :....n n n nX X X    be n  order statistics from a continuous cdf F . We present here characterization results base on some functions of these order statistics. We refer the reader to Ahsanullah and Hamedani (2007), Hamedani et al. ( )2008  and Hamedani (2010), among others, for characterizations of other well-known continuous distributions in this direction.  Proposition 2.3.1.   Let  : 0,X    be a continuous random variable with cdf F  such that       2lim 1 0, 1nx x F x n      , for some 0   and 0  . Then       2 21: 1: , 01n nnE X X t t tn                  (2.3.1)  if and only if X  has cdf ( )1.2 .  Proof:   If X  has cdf ( )1.2 , then clearly ( )2.3.1  is satisfied. Now, if ( )2.3.1  holds, then using integration by parts on the left hand side of (2.3.1), in view of the assumption     2lim 1 0,nx x F x      we have  Characterizations of Distribution of Ratio of Rayleigh Random Variables 374           212 1 1 , 01n ntx F x dx t F t tn             (2.3.2)   Differentiating both sides of ( )2.3.2  with respect to t, we arrive at       122 , 01f tt t tF t               (2.3.3)   Now, integrating both sides of ( )2.3.3  from 0 to x, we have        121 , 0F x x x     .  Proposition 2.3.2   Let  : 0,X    be a continuous random variable with cdf F . Then     22 1 2: : 1 , 0nn nn n n ntE X X t t t                      ,     (2.3.4)   for 1n   and some 0,  0   , if and only if X  has cdf (1.2).  Proof:   Is similar to that of Proposition 2.3.1.   Let , 1,2,...,jX j n  be n  i.i.d. random variables with cdf F  and corresponding pdf f  and let 1: 2: :...n n n nX X X    be their corresponding order statistics. Let *1: 1n iX    be the 1st order statistic from a sample of size 1n i   of random variables with cdf      ,1tF x F tF x x tF t  ( t  is fixed) and corresponding pdf    ,1tf xf x x tF t . Then      *: 1: 1: 1di n i n n iX X t X     (d means equal in distribution),   that is               *: 1: 1: 11 1 ,1i n i n n in itX X Xf xf x t f x n i F x x tF t       .   Now we can state the following characterization of the RR distribution in yet somewhat different direction.  Proposition 2.3.3  Let  : 0,X    be a continuous random variable with cdf F. Then G.G. Hamedani 375     2 2: 1:1, 0i n i nn iE X X t t tn i                  (2.3.5)  for n i  and some 0,  0    if and only if X  has cdf ( )1.2 .  Proof:   If X  has cdf ( )1.2. , then clearly (2.3.5) is satisfied. Now, if (2.3.5) holds, then the left hand side of (2.3.5) can, in view of the previous explanation, be written as             2111 11n itn ix n i F x f x dxF t      .   Now, the rest of the proof is similar to that of Proposition 2.3.1.  Remark 2.3.4  For characterizations of some other continuous distributions in the direction of Proposition 2.3.3, we refer the reader to Ahsanullah ( )2009 .  ACKNOWLEDGEMENTS   The author is grateful to the Editor and two referees for their comments and suggestions improving the presentation of the content of this work.  REFERENCES  1. Ahsanullah, M. (2009). On some characterizations of univariate distributions based on truncated moments of order statistics. Pak. J. Statist., 25(2) 83-91. 2. Ahsanullah, M. and Hamedani, G.G. (2007). Certain characterizations of the power function and beta distributions based on order statistics. JSTA, 6, 220-225. 3. Galambos, J. and Kotz, S. (1978). Characterizations of probability distributions. A unified approach with an emphasis on exponential and related models. Lecture Notes in Mathematics, 675, Springer, Berlin. 4. Glänzel, W. (1987). A characterization theorem based on truncated moments and its application to some distribution families. Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, 1987, 75-84. 5. Glänzel, W. (1988). A characterization of the normal distribution. Studia Sci. Math. Hungar., 23, 89-91. 6. Glänzel, W. (1990). Some consequences of a characterization theorem based on truncated moments. Statistics, 21, 613-618. 7. Glänzel, W. and Hamedani, G.G. (2001). Characterizations of univariate continuous distributions. Studia Sci. Math. Hungar., 37, 83-118. 8. Glänzel, W., IrWin (1994). A characterization tool for discrete distributions under Window(R), Proc. COMPSTAT’ 94 (Vienna, 1994), Short Communications in Computational Statistics, ed. by R. Dutter and W. Grossman, Vienna, 199-200. 9. Glänzel, W., Telcs, A. and Schubert, A. (1984). Characterization by truncated moments and its application to Pearson-type distributions. Z. Wahrsch. Verw. Gebiete 66, 173-183. Characterizations of Distribution of Ratio of Rayleigh Random Variables 376 10. Hamedani, G.G. (1993). Characterizations of Cauchy, normal and uniform distributions. Studia Sci. Math. Hungar., 28, 243-247. 11. Hamedani, G.G. (2002). Characterizations of univariate continuous distributions. II, Studia Sci. Math. Hungar., 39, 407-424. 12. Hamedani, G.G. (2004). Characterizations of univariate continuous distributions based on hazard function. J. of Applied Statistical Science, 13, 169-183. 13. Hamedani, G.G. (2006). Characterizations of univariate continuous distributions. III, Studia Sci. Math. Hungar., 43, 361-385.  14. Hamedani, G.G. (2010). Characterizations of continuous univariate distributions based on the truncated moments of functions of order statistics. Studia Sci. Math. Hungar., 47, 462-484. 15. Hamedani, G.G. (2011). Characterizations of the Shakil-Kibria-Singh distribution. Austrian J. of Statistics, 40, 201-207. 16. Hamedani, G.G. and Ahsanullah, M. (2005). Characterizations of univariate continuous distributions based on hazard function II, J. of Statist. Theo. and Applica., 4, 218-238. 17. Hamedani, G.G., Ahsanullah, M. and Sheng, R. (2008). Characterizations of certain continuous univariate distributions based on truncated moment of the first order statistic. Aligrah J. of Statistics, 28, 75-81. 18. Kotz, S. and Shanbhag, D.N. (1980). Some new approaches to probability distributions. Adv. in Appl. Probab. 12, 903-921. 19. Shakil, M. and Ahsanullah, M. (2011). Record values of the ratio of Rayleigh random variables. Pak. J. Statist. 27(3), 307-325.  

Characterizations of Distribution of Ratio of Rayleigh Random Variables

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Characterizations of Distribution of Ratio of Rayleigh Random Variables

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