International Journal of Mathematical Archive-8(6), 2017, 1-7 Available online through ISSN

Transcription

1 nternational Journal o Mathematical Archive-8(6), 07, -7 Available online through SSN DETERMNATON OF ENTROPY FUNCTONAL FOR DHS DSTRBUTONS S. A. EL-SHEHAWY* Department o Mathematics, Faculty o Science, Menouia University, Shebin El-Kom, ZP-Code 35, Egypt. (Received On: ; Revised & Accepted On: ) ABSTRACT This paper presents a study o the entropy unctional or some continuous distributions. This study is concerned to determine the entropy unctional or some various amilies o deormed hyperbolic secant distributions "DHS distributions". Finally, we discuss some results based on the introduced parameters or suggested parametric unctions. Keywords: Deormed hyperbolic distributions, Entropy, Hyperbolic secant distribution. 00 Mathematics Subject Classiication: 37A35, 94A7, 37B40, 60E05, 60E0, 6E7, 6E0.. NTRODUCTON Nowadays, the deormation technique has been applied or the continuous hyperbolic secant distribution "HS- Distribution" [-7]. Some amilies o the continuous deormed hyperbolic secant distributions have been constructed and studied. Each DHS distribution has been obtained by introducing positive scalar deormation parameters as actors o the exponential growth and decay parts o the hyperbolic secant unction "HS unction" in the HS distribution. The probability density unction (pd) o the HS distribution is given as HS( x) = sech ( x / ) ; x R π, () see [-]. Based on [3, 5], some classes o deormed hyperbolic secant distributions have been presented by introducing two parametric unctions instead o the parameters. Some important properties o these classes o distributions have been discussed and some measures are derived in [3-7]. The main aim o this paper is to study and determine the entropy unctional or each class o the deormed hyperbolic secant distributions. Recently, the entropy unctional has been suggested or some dierent probability distributions [8-0]. The entropy is known as a measure o uncertainly and dispersion and it is deined or the continuous random variable X as ( ) H( ) = ( x) log ( x) () ( x ) is the pd, see [, ]. This paper is laid out as ollows: Section presents a amily o deormed hyperbolic secant distributions and its properties. Section 3 is devoted to the determined entropy unctional o the hyperbolic secant distribution based on using o the deormation technique. Section 4 is considered or power comparison or some dierent values o the introduced parameters or parametric unctions. Finally, the results o the paper have been concluded in Section 5. Corresponding Author: S. A. El-Shehawy*, Department o Mathematics, Faculty o Science, Menouia University, Shebin El-Kom, ZP-Code 35, Egypt. nternational Journal o Mathematical Archive- 8(6), June 07

2 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June-07.. FAMLY OF DEFORMED HYPERBOLC SECANT DSTRBUTONS n order to obtain the deormed hyperbolic secant distributions "DHS distributions" by means o the deormation or the hyperbolic unctions [4, 6,7], some properties o the deormed hyperbolic unctions (DHF) will irst be recalled, that is x x x x = and sech x =, x R and the deormation parameters p, q > 0. Based on [4], the pd o a -DHS distribution as: DHS ( x; p, q) = sech ( π x / ). (3) According to [6, 7], two special cases o this distribution are given as ollows: - at p =, q > 0, this gives a q-dhs distribution with pd q ( x; q) sech ( x / ) - at q =, p > 0, this gives a p-dhs distribution with pd q DHS = q π. (4) p ( x; p) sech ( x / ) p DHS = p π. (5) Similarly, according to [3, 5], i pw ( ) and qw ( ) are two arbitrary real positive deormation parametric dierentiable unctions o w, w R, we consider the ollowing pw ( ) qw ( )-deormed hyperbolic unctions: ϕ p( w ) e q( w ) e ϕ + pw ( ) qw ( ) ϕ = and sech pw ( ) qw ( ) ϕ =, pw ( ) qw ( ) ϕ with a real dierentiable unction ϕ = ϕ( xw, ) o x and w, and this unction is linear in x with positive partial derivative with respect to x, i.e. ϕ = Cw ( ) x+ Dw ( ), Cw ( ) (0, ) as a derivative o ϕ with respect to x, and Dw ( ) R. Moreover, a p(w)q(w)-dhs distribution has the ollowing pd: πϕ DHS ( ϕ; pw ( ), qw ( )) = sech, x, w R (6) Based on [5], two special cases o this distribution are also given as ollows: - a q(w)-dhs distribution with pd Cw ( ) qw ( ) πϕ q( w) DHS ( ϕ; qw ( )) = sech q( w) ; xw, R, (7) - a p(w)-dhs distribution with pd Cw ( ) pw ( ) πϕ p( w) DHS ( ϕ; pw ( )) = sech p( w) ; xw, R. (8) n [3-7], some properties, measures and unctions (expectation, variance, score unction, unmodallity, measures o tendency, moments generating unction, characteristic unction, cumulants generating unction, skewness, excess kurtosis and maximum likelihood estimators) o the amilies o DHS distribution are presented and discussed. n the next section we present a study o the entropy unctional or some various amilies o DHS distributions. 3. ENTROPY FUNCTONAL FOR SOME DHS DSTRBUTONS According to [8-] the dierentiable entropy H( ) is deined as a measure or a continuous random variable X by (). 07, JMA. All Rights Reserved

3 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June-07. n this section, the suggested entropy unctional H( ) will be derived or the deormed hyperbolic secant variables ( X DHS p DHS q DHS pw ( ) DHS and X qw ( ) DHS ) which based on the constructed DHS distributions [3-7]. Theorem : Let X DHS and X be the -DHS variable and the p(w)q(w)-dhs variable with pd's DHS( x) = DHS( x; p, q) = sech ( π x / ) and πϕ DHS ( ϕ) = DHS ( ϕ; pw ( ), qw ( )) = sech, respectively. Then, (i) the entropy unctional or the -DHS variable is H( DHS ) = log ( ) + DHS, (9) π x / π x / log ( ) DHS = π x / π x / (ii) the entropy unctional or the p(w)q(w)-dhs variable is H( DHS) = [log Cw ( ) + log ( pwqw ( ) ( ))] + p ( w ) q ( w ) DHS (0) πϕ/ πϕ/ log( pw ( ) e + qw ( ) e ) p ( w ) q ( w ) DHS = πϕ/ πϕ/ pw ( ) e + qw ( ) e Proo: By using the orm () o H( ), the entropy unctional or the mentioned deormed hyperbolic secant variables in the two cases (i) and (ii) can be derived as ollows: (i) Similarly or the -DHS variable, we ind that H( DHS ) = ( ) DHS x log ( ( x) ) DHS = sech ( π x/ ) log sech ( π x/ ) / / / / log ( ) log ( π x π x = pe qe x x ) d x π π + pe qe = + π x/ π x/ log ( + ) log ( ) π x/ π x/ π x/ π x/ From the deinition o the pd o the -DHS variable, we ind that = π x / π x / and we can also obtain the required ormula (9) o the entropy unctional H( ) or the -DHS distribution. DHS (ii) Without loss o generality, we consider Dw ( ) = 0. n this case, or some dierent values o w and or each ixed pair o the parametric unctions pw ( ) and qw ( ) we can also derive the required ormula o the entropy unctional or the p(w)q(w)-dhs distribution as ollows: 07, JMA. All Rights Reserved 3

4 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June-07. H( ) = ( ϕ) log ( DHS( ϕ) ) DHS DHS = sech ( πϕ / ) Cw ( ) pwqw ( ) ( ) log sech ( πϕ/ ) = pwe ( ) + qwe ( ) {log Cw ( ) = {log Cw ( ) +C( w) pwqw ( ) ( ) πϕ/ πϕ/ + log ( pwqw ( ) ( )) + log ( ( ) ( ))} log ( pwe ( ) πϕ / + q w e π ϕ/ ( ) )} pwqw πϕ/ πϕ/ pwe ( ) + qwe ( ) πϕ/ πϕ/ log( pwe ( ) + qwe ( ) ) pwe ( ) + qwe ( ) πϕ/ πϕ/ From the deinition o the pd o the p(w)q(w)-dhs variable, we ind that Cw ( ) pw ( ) qw ( ) πϕ/ πϕ/ p( w ) e + q( w ) e = and we can also obtain the required ormula (0) o the entropy unctional H( ) or the p(w)q(w)-dhs distribution. Theorem : Let X p DHS q DHS pw ( ) DHS and X qw ( ) DHS be the p-dhs variable, the q-dhs variable, the p(w)-dhs variable and the q(w)-dhs variable with pd's p p DHS( x) = p DHS( x; p) = sech p( π x / ), q q DHS( x) = q DHS( x; q) = sech q( π x / ) ( ) ( ) ( ) ( ; ( )) pw πϕ p w ϕ = p w ϕ pw = sech p( w) and Cw ( ) qw ( ) πϕ q( w) DHS ( ϕ) = q( w) DHS ( ϕ; qw ( )) = sech q( w), respectively. Then, (i) The entropy unctional or the p-dhs variable is H( p DHS ) = log ( p) + p p DHS, () π x / π x / log ( pe + e ) p DHS = π x / π x / pe + e (ii) The entropy unctional or the q-dhs variable is H( q DHS ) = log ( q) + q q DHS, () π x / π x / log ( e + qe ) q DHS = π x / π x / e + qe 07, JMA. All Rights Reserved 4

5 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June-07. (iii) The entropy unctional or the p(w)-dhs variable is H( p( w) DHS ) = [ log Cw ( ) log pw ( ) ] πϕ/ πϕ/ log ( pw ( ) e + e ) pw ( ) DHS = πϕ/ πϕ/ pw ( ) e + e (iv) The entropy unctional or the q(w)-dhs variable is H( q( w) DHS ) = [ log Cw ( ) log qw ( ) ] πϕ/ πϕ/ log ( e + qw ( ) e ) qw ( ) DHS = πϕ/ πϕ/ e + qw ( ) e +, (3) Cw ( ) pw ( ) p w +. + (4) Cw ( ) qw ( ) q w + Proo: The previous results orms in (), (), (3) and (4) in this theorem can be directly derived by applying the orm () o the entropy and the rest o the proo is similar to the proo o theorem. Theorem 3: For the -DHS variable X DHS and the p(w)q(w)-dhs variable X the ollowing statements are veriied: H( DHS ) = H( q p DHS ) and H( pw ( ) qw ( ) DHS ) = H( qw ( ) pw ( ) DHS). Proo: These statements are satisied directly since and are veriied computationally. π x / π x / log ( ) DHS = = π x / π x / q p DHS π x/ π x/ log( pw ( ) e + qw ( ) e ) DHS = = π x/ π x/ q ( w ) p ( w ) DHS pw ( ) e + qw ( ) e Theorem 4: For p = q =, pw ( ) = qw ( ) =, p =, q =, pw ( ) = and qw ( ) = in (0), (), (), (3), (4) and (5) respectively the HS distribution with pd ( ) HS x given in () is recovered and in this case we ind that the entropy H( ()() DHS ) = H( HS ) o the HS distribution can be obtained computationally. Proo: The proo o this theorem can be directly achieved computationally. 4. CALCULATON OF ENTROPY FOR SOME SPECAL DHS DSTRBUTONS n this section we give the corresponding values or orms o some special cases o the -DHS distribution and the p(w)q(w)-dhs distribution according to the parameters p, q, p (w) and q (w). These obtained values or orms o the entropy can easily be worked out with the help o MAPLE or MATHEMATCA. We give the ollowing cases:. Case: or p = q = and pw ( ) = qw ( ) =, we ind that H( ) = H( ) = DHS p( w ) q( w ) DHS. Case: or p > q >, we ind that H( DHS ) = H( (9)() DHS ) , H( ) = H( ) DHS (9)(0) DHS 3. Case: or > p > q, we ind that H( DHS ) = H( (/ )(/9) DHS ) , H( ) = H( ) DHS (/0)(/9) DHS 07, JMA. All Rights Reserved 5

6 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June Case: or q > p, we ind that H( DHS ) = H( ()(9) DHS ) , H( ) = H( ) DHS (0)(9) DHS 5. Case: or > q > p, we ind that H( DHS ) = H( (/9)(/ ) DHS ) , H( ) = H( ) DHS (/9)(/0) DHS 6. Case: Based on [3], or the pw ( ) qw ( )-DHS distribution we consider the case when pw ( ) =, qw ( ) = exp( w) and ϕ = ( w ) x + 3. n this case, the pd is given in the ollowing closed orm: exp( w/ ) π (( w) x+ 3) DHS ( ϕ;,exp( w)) = sech ; xw, R, sech( w) Thereore, we can ind that, H( pw ( ) qw ( ) DHS ) = H( ()(exp( w)) DHS ) w w = ( w ) e (log (( w )) + ) π ( x ( w ) + 3)/ w π ( x ( w ) + 3)/ log ( e + e ) ()(exp( w )) DHS = π ( x ( w ) + 3)/ w π ( x ( w ) + 3)/ e + e ()(exp( w )) DHS For a ixed real value o the parameter w we can obtain on ()(exp( w )) DHS and so we can determine the entropy unctional in this case. 5. CONCLUSON n this paper, we concerned to derive some closed orms o the entropy unctional or the deormed hyperbolic secant variables or various cases. These closed orms are studied based on some constructed amilies o the deormed hyperbolic secant distributions. t is ound that these orms o entropy unctional depend on the introduced parameters or parametric unctions. Samples o the entropy unctional o the -DHS distribution and the p(w)q(w)-dhs distribution or some special cases o p and q and some orms o p(w) and q(w) have been computed and discussed. REFERENCES. M. Fischer, The Skew Generalized Secant Hyperbolic Family, Austrian Journal o Statistics, 35 (006), No. 4, M. Fischer and D. Vaughan, The Beta-Hyperbolic Secant Distribution, Austrian Journal o Statistics, 39 (00), No. 3, S.A. El-Shehawy, A Class o Deormed Hyperbolic Secant Distributions Using Two Parametric Functions, Lie Science Journal 0 () (03) S.A. El-Shehawy, Study o the Balance or the DHS Distribution, nternational Mathematical Forum 7 (3) (0) S.A. El-Shehawy, S.A. El-Serai, Using a parametric unction to study the deormation o the hyperbolic secant distribution, Journal o Natural Sciences and Mathematics, Qassim university 8 () (05) S.A. El-Shehawy, E. A-B. Abdel-Salam, The q-deormed Hyperbolic Secant Family, nternational Journal o Applied Mathematics and Statistics 9 (5) (0) S.A. El-Shehawy, E. A-B. Abdel-Salam, On Deormation Technique o the Hyperbolic Secant Distribution, Far East Journal o Mathematical Science 64 () (0) A.. Al-Omari, Estimation o entropy using random sampling, Journal o Computational and Applied Mathematics 6 (04) A.. Al-Omari, A. Hag, Entropy estimation and goodness-o-it tests or the inverse Gaussian and Laplace distributions using paired ranked set sampling, Journal o Statistical Computation and Simulation 86 () (06) H. A. Noughabi, R.A. Noughabi, R.A., On the entropy estimators. Journal o Statistical Computation and Simulation 83 (4) (03) , JMA. All Rights Reserved 6

7 S. A. El-Shehawy* / Determination o entropy unctional or DHS distributions / JMA- 8(6), June-07.. C.E. Shannon, A mathematical theory o communications, Bell System Technical Journal 7 (3, 4) (948) , O. Vasicek, A test or normality based on sample entropy, Journal o Royal Statistical Society, Series B 38 (976) Source o support: Nil, Conlict o interest: None Declared. [Copy right 07. This is an Open Access article distributed under the terms o the nternational Journal o Mathematical Archive (JMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.] 07, JMA. All Rights Reserved 7