Characterization of the Skew-Normal Distribution Via Order Statistics and Record Values

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1 International Journal of Statistics and Probabilit; Vol. 4, No. 1; 2015 ISSN E-ISSN Published b Canadian Center of Science and Education Characterization of the Sew-Normal Distribution Via Order Statistics and Record Values M. Gharib 1, M.M. Mohie El-Din 2 & A. Sharaw 3 1 Department of Mathematics, Facult of Science, Ain Shams Universit, Cairo, Egpt 2 Department of Mathematics, Facult of Science (men), Al-Azhar Universit, Cairo, Egpt 3 Department of Mathematics, Al-Safwa High Institute of Engineering, Cairo, Egpt Correspondence: A. Sharaw, Department of Mathematics, Al-Safwa High Institute of Engineering, Cairo, Egpt. mrali112@ahoo.com Received: November 12, 2014 Accepted: November 27, 2014 Online Published: December 22, 2014 doi: /ijsp.v4n1p59 URL: Abstract Characterization of a probabilit distribution is the investigation of those unique properties enjoed b that distribution. In this article the general condition moments technique is used to obtain some new characterization results for the sew-normal distribution based on the order statistics as well as the record values of a random sample drawn from this distribution. The achieved results can be used to improve fitting and modeling sew data using the sew-normal distribution. Further, the new results are specialized to the standard normal distribution. Kewords: Characterization, order statistics, record values, conditional moment, sew-normal distribution 1. Introduction Azzalini (1985, 1986) have introduced the sew-normal distribution which includes the normal distribution and has some properties lie the normal and et is sew. This class of distributions is useful in studing robustness and modelling sewness. Since the normal distribution is still the most commonl used distribution both in statistical theor and applications, then a famil of distributions lie the sew normal one that possesses the same properties of the normal famil have a great potential impact in the theoretical and applied probabilit and statistics. Despite of this potential impact there are relativel few statisticians who used this famil in their theoretical and applied wors. It is well nown that characterization theorems represent an indispensable tool for understanding the missing lins between the mathematical structure of statistical distributions and the actual behavior of real world random phenomena. It is reasonable to assert that although the characterization theor does not offer a well-defined method for solving real world problems, it guides the statistician as the case ma be toward a proper scientific solution. In fact, research on characterization for the sew-normal distribution is still in its earl stage. Characterizations concerning the sew normal distribution are given b Gupta and Wen-Jang ( 2002), Gupta and Jose (2004) and Gharib et al. (2014). In this article the general condition moment technique is used to obtain some new characterization results for the sew-normal distribution based on the order statistics as well as the record values of a random sample drawn from this distribution. The achieved results can be used to improve fitting and modeling sew data using the sew-normal distribution. Further, the new results are specialized to the standard normal distribution. A random variable (rv) Z has a sew-normal distribution with parameter, denoted b Z ~ SN ( ) if its probabilit densit function (pdf) is given b: f(z, ) = 2 (z) ( z) (1.1) where, and are, respectivel, the probabilit densit function and cumulative distribution function of the standard normal distribution, and z and are real numbers Azzalini (1985). The cumulative distribution function (cdf) corresponding to (1.1) is given b 59

2 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 F(z, ) = z 2 (u) ( u)du. (1.2) For the structural and other properties of the sew-normal distribution see Brown (2001) and the references therein. The following lemma is needed in proving the results of the article. Lemma If X 1:n, X 2:n,..., X n:n, are the order statistics (os) of a random sample of size (n) drawn from a continuous distribution with cdf F(. ) then the conditional joint distribution of the os X 1:n,,X -1:n given X :n = is identical to the joint distribution of the os Y 1: 1, Y 2: 1,, Y 1: 1 from a random sample of size ( 1) with cdf H (u) having the form: F(u) F() u < H (u) = { 1 u, (1.3) and the conditional joint distribution of the os X +1:n, X +2:n,.,X n:n, given X :n =, is identical to the joint distribution of the os Z 1:n having the form:, Z 2:n,, Z n :n from a random sample of size (n ) with cdf R (u) [F(u) F()] F () u > R (u) = { 0 u, (1.4) where F () = 1 F(). This lemma is proved in Arnold et al. (2008). Using the above lemma, we have: and: E(X 1:n + + X 1:n X :n = ) = E(Y 1: 1 E(X 1:n + + X n:n X :n = ) = E(Z 1:n Further from (1.5) and (1.6), we readil obtain: 2. Main Results + + Y 1: 1 ) = ( 1)E(Y 1 ) = ( 1)[F()] 1 uf(u)du, (1.5) + + Z n :n ) = (n )E(Z 1 ) = (n )[F ()] 1 uf(u)du. (1.6) ( 1) E(X 1:n + + X n:n X :n = ) = + F() uf(u)du 2.1 Characterization via General Conditional Moments of Order Statistics + (n ) F () uf(u)du. (1.7) The problem of characterizing distributions through conditional moments of order statistics has been of increasing interest during the last few decades due to its several applications, for eample, the stud of (n r)-out-of-n sstems. A (n r)-out-of-n sstems consists of (n) independent and identicall distributed components, and it wors as long as at least (n r) components are woring. If X i represents the lifetime of the i th component, i = 1,2,.., n, the survival function of the (n r)-out-of-n sstem is the same as that of the (r + 1) th os X r+1,n from this sample of (n) random variables. Hence the results obtained for os hold for (n r)-out-of-n sstems, so characterizing distributions using os plas an important role in reliabilit theor. Theorems below characterizes the sew-normal distribution using conditional moments of os. Theorem 2.1 Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F ( ), failure rate function h(. ) and finite 60

3 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 mean. Let X 1:n X 2:n X n:n be the os of a random sample of size n drawn from X. Suppose that F(. ) has a continuous second order derivative on (, ). Then X has a sew normal distribution SN( ) with mean if and onl if, for = 1, 2,, n: E(X +1:n + + X n:n X :n = ) = (n ) {h(, ) + μ( ) ( ) [F (, )] 1 }, (2.1) where () = 1 ( ) and h(, ) = f(, )/F (, ). Theorem 2.2 Let X be a continuous rv with pdf f(. ), cdf F(. ) and finite mean and let X 1:n X 2:n X n:n be the os of a random sample of size n drawn from X. Let F(. ) has a continuous second order derivative on (, ). Then X has a sew-normal distribution SN( ) with finite mean if and onl if: Theorem 2.3 E(X 1:n + + X 1:n X :n = ) = ( 1) [ μ( ) ( ) f(, )] [F(, )] 1. (2.2) Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (. ) and finite mean and let X 1:n X n:n be the os of a random sample of size n drawn from X. Let F(. ) has a continuous second order derivative on (, ). Then X has a sew-normal distribution SN( ) with finite mean if and onl if: where Theorem 2.4 m E(S S X :n = ) = + [ μ( ) ( ) f(, ) μf(, )], (2.3) F(, )F (, ) S = X 1:n + + X :n, S = X +1:n + + X n:n, n = 2m + 1, = m + 1 and m = 1, 2, 3,. Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (.) and.nite mean and let X 1:n X n:n be the os of a random sample of size n drawn from X. Let F(. ) has a continuous second order derivative on (, ). Then X has a sew-normal distribution SN( ) with finite mean if and onl if: E(T X :n = ) = + μ( )m[f (, )] 1 + m[2f(, ) 1] F(, )F (, ) [ f(, ) μ( ) ( )], (2.4) where T = X 1:n + + X n:n, n = 2m + 1, = m + 1 and m = 1, 2, 3,. 2.2 Characterization via General Conditional Moments of Record Values Record values are found in man situations of dail life as well as in man situations applications suchas reliabilit theor (for comprehensive accounts of the theor and applications of record values see Arnold et al. (1998), Ahsanullah (1995), Ahsanullah (2004) and Ahsanullah et al. (2006). Characterizing the distributions via their record statistics has a long histor, for ecellent review one ma refer to Shaw et al. (2006) Shaw et al. (2008) and Shaw et al. (2008b) among others. Suppose that (X n ) n 1 is a sequence of independent and identicall distribution rvs. Let Y n = ma {X j 1 j n} for n 1. We sa X j is an upper record value of {X n n 1}, if Y j >Y j-1, j >1. B definition X 1 is an upper record value. The indices at which the upper record values occur are given b the record times {U(n); n >1}, where: U(n) = min {j j > U(n 1), X j > X U(n 1), n > 1} and U(1) = 1. Theorems 2.5 and 2.6 below characterize the sew-normal distribution using conditional moments of upper record values. Theorem 2.5 Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (.) and failure rate h(. ). Assume that F(. ) has a continuous second order derivative on (, ). and that E(X U(n+1) ) is finite. Then X has a sew-normal distribution SN( ) if and onl if, for = 1, 2, 2 E(X U(n+1) (2 1)!! X U(n) = ) = (2 1)!! + {[ 2i 1 h(, )] + μ( )α i (, ) F (, ) }, (2.5) (2i 1)!! 61

4 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 where α i (, ) = i (2i 1)!! [ 2i 2 ( 1+ 2 )] j=1 (2j 1)!! [1+ 2 ] i j+1 2, i = 1, 2,, j = 1, 2, i, 1 if n = 1, n = 0, or n = 1 (n)!! = { n(n 2) if n 2, and (see Arfen (1985)). Special Case For = 1, (2.5) reduces to: Theorem E(X U(n+1) X U(n) = ) = 1 + h(, ) + μ( ) ( ) F (, ), (2.6) Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (.) and failure rate h(. ). Assume that F(. ) has a continuous second order derivative on (, ) and that E(X U(n+1) ) is finite. Then X has a sew-normal distribution SN( ) if and onl if, for = 0,1, 2, E(X 2+1 U(n+1) X U(n) = ) = (2 1)!! + (2)!! i=0 {[ 2i h(, )] + μ( )[F (, )] 1 [γ i (, ) + β i (, )]}, (2.7) (2i)!! where β i (, ) = (2i 1)!! (1+ 2 ) i [ ], i = 0,1,2,,, Special Case γ i (, ) = i j=1 For = 0, (2.7) reduces to: {[ 2i 1 ( )] [1 + 2 ] i j+1 2 } and j = 1, 2, i. (2j 1)!! 3. Proofs of the Theorems Proof of Theorem 2.1 Necessit: E(X U(n+1) X U(n) = ) = h(, ) + μ( )[F (, )] 1 ( ). (2.8) Suppose that X ~ SN( ) with pdf and cdf given b (1.1) and (1.2). Then using (1.6); we have On integrating b parts we get: Further: E(X +1:n + + X n:n X :n = ) = 2(n )[F (, )] 1 () ( )d. E(X +1:n + + X n:n X :n = ) = 2(n )[F (, )] 1 [ () ( ) + () ( )d]. (3.1) 2 () ( )d = μ( ) ( ). (3.2) Finall, recalling that h(, ) = f(, )/F (, ) and f(, ) = 2 () ( ) we get (2.1) from (3.1) and (3.2). Sufficienc: 62

5 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 Suppose that X is a continuous rv with pdf f(. ), cdf F(. ), finite mean and F(.) has a continuous second order derivative on (, ). Suppose, also, that (2.1) is true. Hence, from (1.6) and (2.1) one can write: f(, )d = f(, ) + μ( ) ( ). Differentiating both sides in this equation with respect to, we obtain: d f(, ) + f(, ) = μ( ) 2π () ( ). (3.3) d Which is a linear first order differential equation in the unnown function f(). Its solution is: Now, the normalizing condition: Hence: f(, ) = μ π [2 () ( )]. f(, )d = 1 gives μ( ) = 2 [ ]. π f(, ) = 2 () ( ). (3.4) Which is the pdf of the sew-normal distribution given b (1.1). This completes the proof of Theorem 2.1. Proof of Theorem 2.2 The proof is obtained b repeating word-to-word the steps of the proof of Theorem 2.1. Proof of Theorem 2.3 Necessit: Suppose that X ~ SN( ) with pdf and cdf given b (1.1) and (1.2) respectivel. Then using (1.5) and (1.6), we have: E(S S X :n = ) = + m F(, ) f(, )d m f(, )d F (, ) + m 1 E(X 1:n + + X 1:n X :n = ) m n E(X +1:n + + X n:n X :n = ). (3.5) The rest of the proof of necessit is obtained b appling (2.1) and (2.2). Sufficienc: Suppose that X is a continuous rv with pdf f(. ), cdf F(. ), finite mean and F(. ) has a continuous second order derivative on (, ). Suppose, also, that (2.3) is true. Hence, using (2.1), (2.2) and (2.3) one can get: Now, it is eas to see that: F (, ) Hence (3.6) reduces to: F (, ) f(, )d f(, )d F(, ) f(, )d μ( ) ( ) f(, ) μ( )F(, ). (3.6) F(, ) f(, )d = = f(, )d = μ( )F(, ). 63

6 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 f(, )d = μ( ) ( ) f(, ). Differentiating both sides of this equation with respect to, we get: d d f(, ) + f(, ) = μ( ) 2π () ( ). Which is Equation (3.3) and thus having the solution given b (3.4). This completes the proof of Theorem 2.3. Proof of Theorem 2.4 The proof is obtained b repeating word-to-word the steps of the proof of Theorem 2.3. Proof of Theorem 2.5: Necessit: Suppose that X ~ SN( ) with pdf and cdf given b (1.1) and (1.2), respectivel. Then we have: 2 E(X U(n+1) X U(n) = ) = 2 F (, ) u2 Now, repeated routine integration b parts gives: + 2 u 2 1 (2 1)!! Again, repeated integration b parts gives: (u) ( u)du = 2 F (, ) u2 1 ( u)d (u). (3.7) ( u)d (u) = (2 1)!! {[ 2i 1 f(, )] + 2 u 2i 1 (u) ( u)du}. (3.8) 2 u 2i 1 2 π i (2i 2)!! (u) ( u)du = [ 2i 2 ( )] [1 + 2 i j+1 ] j=1 (2j 2)!! Finall from (3.7), (3.8) and (3.9) we get (2.5). Sufficienc: = μ( )α i (, ). (3.9) Suppose that X is a continuous rv with pdf f(. ) and cdf F(. ) and that F(. ) has a continuous second order derivative on (, ). Suppose, also, that E(X U(n+1) ) is finite and (2.5) is true. Then we have: 2 E(X U(n+1) = (2 1)!! + X U(n) = ) = [F (, )] 1 2 (2 1)!! f(, )d {[ 2i 1 h(, )] + μ( ) F (, ) α i(, )}. Multipling both sides of this equation b F () and then differentiating it with respect to, we obtain: 2i f(, ) = (2 1)!! f(, ) + (2 1)!! (2i 1)2i 2 f(, ) + + (2 1)!! [ 2i 1 d f(, )] d +μ( ) ( ) (2 1)!! i j=1 (2i 2)!! 64 (2j 2)!! [(2j 2) 2i 3 (1 + 2 ) 2i 1 ] [1 + 2 ] i j+1. (3.10)

7 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 Now, it is eas to see that: i j=1 (2i 2)!! [(2j 2) 2i 3 (1 + 2 ) 2i 1 ] (2j 2)!! [1 + 2 ] i j+1 = i 1. Hence (3.10) reduces, after simplification, to: Or, f(, ) (2 1)!! 2i + [ d (2 1)!! f(, )] 2i 1 = d μ( ) ( ) (2 1)!! 2i 1. [ d 2i 1 f(, ) + f(, )] = μ( ) ( ) 2i 1 d. (3.11) 2i 1 Finall, since is a polnomial in of degree (2-1) with rational coefficients, then (3.11) reduces: (2i 1)!! d d f(, ) + f(, ) = μ( ) 2π () ( ). It is eas to see that this first order differential equation in the unnown function f() has the solution given b (3.4). Therefore X ~ SN( ). This completes the proof of Theorem 2.5. Proof of Theorem 2.6: The proof of this theorem is similar to that of Theorem Specialization to the Standard Normal Distribution As the sew-normal distribution tends to the standard normal distribution for = 0, then putting = 0, theorems , will be valid for the standard normal distribution and can be restated as follows; eeping all the rotations used there: Theorem 4.1 Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (. ), failure rate h(. ) and let X 1:n. X n:n be the os of a random sample of size n drawn from X. Suppose that F(. ) has a continuous second order derivative on (, ). Then X has the standard normal distribution if and onl if: Theorem 4.2 E(X +1:n + + X n:n X :n = ) = (n )h 0 (). (4.1) Let X be a continuous rv with pdf f(. ), cdf F(. ) and let X 1:n X 2:n. X n:n be the os of a random sample of size n drawn from X. Suppose that F(. ) has a continuous second order derivative on (, ). Then X has the standard normal istribution if and onl if: Theorem 4.3 ( 1) E(X 1:n + + X 1:n X :n = ) = F 0 () f 0(). (4.2) Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (. ), failure rate h(. ) and let X 1:n X 2:n. X n:n be the os of a random sample of size n drawn from X. Suppose that F(.) has a continuous second order derivative on (, ). Then X has the standard normal distribution if and onl if: Theorem 4.4 E(S S X :n = ) = mh 0 ()[F 0 ()] 1. (4.3) Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (. ), failure rate h(. ) and finite mean 65

8 International Journal of Statistics and Probabilit Vol. 4, No. 1; 2015 and let X 1:n X 2:n. X n:n be the os of a random sample of size n drawn from X. Let F(. ) has a continuous second order derivative on (, ). Then X has the standard normal distribution if and onl if: where n = 2m + 1, = m + 1 and m = 1, 2, 3,. Theorem 4.5 E(T X :n = ) = + m F 0 () h 0()[2F 0 () 1], (4.4) Let X be a continuous rv with pdf f(. ), cdf F(. ), survival function F (. ) and failure rate h(. ). Assume that E(X U(n+1) ) is finite. Then X has the standard normal distribution if and onl if: Theorem E(X U(n+1) X U(n) = ) = (2 1)!! + h 0 () (2 1)!! 2i 1, = 1,2,3,. (4.5) Let X be a continuous rv with pdf pdf f(. ), cdf F(.), survival function F (. ) and failure rate h(. ). Assume that E(X U(n+1) ) is finite. Then X has the standard normal distribution if and onl if: 5. Conclusion 2+1 X U(n) = ) = h 0 () (2)!! 2i, = 0, 1, 2,. (4.6) (2i)!! E(X U(n+1) In this stud, some characterization results for the sew-normal distribution are proved based on the conditional moments of order statistics as well as the conditional moments of record values of a random sample drawn from this distribution. Special cases regarding the standard normal distribution are also given. The achieved results are of direct relevance to fitting and modeling sew data using the sew-normal distribution. Moreover, the new results are more appropriate for practical applications comparing with the results of Gupta et al. (2004) which is based on distribution properties. Also, our results motivate the researchers toward a serious consideration of other characterization problems for the sew-normal distribution such as characterization using regression properties and residual life time. Further, the future research challenge is how to etend the present results to the case of the multivariate sew-normal distribution. The limitations is summarized in that it is applied onl for the sew-normal distribution and its specialized distribution. Acnowledgement The authors are greatl indebted to the referees and the editors for their constructive comments. References Ahsanullah, M. (1995). Record statistics. Nova Science Publishers, New Yor. Ahsanullah, M. (2004). Record values. Theor and Applications. Universit press of America, New Yor. Ahsanullah, M. & Ragab, M. Z. (2006). Bounds and characterization of Record statistics. Nova Science Publishers, Hauppauge, New Yor. Arfen, G. (1985). Mathematical methods for phsicists, 3rd ed. Orlando, FL: Academic Press. Arnold, B. C., Balarishnan, N. & Nagarajua, H. N. (1998). Records. John Wiel and Sons, New Yor. Arnold, B. C., Balarishnan, N. & Nagarajua, H. N. (2008). A First Course in Order Statistics. 1st Edn.,Wiel, New Yor. Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand J. Stat., 12, Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistical Papers, 46, Brown, N.D. (2001). Reliabilit studies of the sew-normal distribution. Thesis, Graduate School, Maine Universit. USA. Gharib, M., Salehi, S. & Sharaw, A. (2014). Characterizations of the sew-normal distribution Via characteristic function and conditional moments. J. Advanced Research in Applied Math., 6, i=0 66

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