Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 R E S E A R C H Open Access Qusi-log concvity conjecture nd its pplictions in sttistics Ji Jin Wen 1,TinYongHn 1* nd Sui Sun Cheng 2 * Correspondence: hntin123_123@163.com 1 College of Mthemtics nd Informtion Science, Chengdu University, Sichun, 610106, Chin Full list of uthor informtion is vilble t the end of the rticle Abstrct This pper is motivted by severl interesting problems in sttistics. We first define the concept of qusi-log concvity, nd conjecture involving qusi-log concvity is proposed. By mens of nlysis nd inequlity theories, severl interesting results relted to the conjecture re obtined; in prticulr, we prove tht log concvity implies qusi-log concvity under proper hypotheses. As pplictions, we first prove tht the probbility density function ofk-norml distribution is qusi-log concve. Next, we point out the significnce of qusi-log concvity in the nlysis of vrince. Next, we prove tht the generlized hierrchicl teching model is usully better thn the generlized trditionl teching model. Finlly, we demonstrte the pplictions of our results in the reserch of the llownce function in the generlized trditionl teching model. MSC: 26D15; 62J10 Keywords: qusi-log concvity; qusi-log concvity conjecture; truncted rndom vrible; hierrchicl teching model; k-norml distribution 1 Introduction Convexity nd concvity re essentil ttributes of functions, their reserch nd pplictions re importnt topics in mthemtics (see [1 12]). There re mny types of convexity nd concvity, one of them is log concvity which hs mny pplictions in sttistics (see [2, 4, 7 12]). In [4], theuthorspply thelogconcvity to study the Roy model, nd some interesting results re obtined (see p.1128 in [4]), which include the following: If D is logconcve rndom vrible, then Vr[D D > d] d 0 nd Vr[D D d] d 0. (1.1) Recll the definitions of log-concve function (see [1 5]) nd β-log-concve function (see [13]): If the function p : I (0, ) stisfies the inequlity p [ θu +(1 θ)v ] e β p θ (u)p 1 θ (v), β [0, ), (u, v) I 2, θ [0, 1], (1.2) then we sy tht the function p : I (0, ) is β-log-concve function. 0-log-concve function is clled log-concve function. In other words, the function p : I (0, ) is log-concve function if nd only if the function log p is concve function. If log p is concve function, then we cll the function p : I (0, ) log-convex function. Here I is n intervl (or high dimension intervl). 2014 Wen et l.; licensee Springer. This is n Open Access rticle distributed under the terms of the Cretive Commons Attribution License (http://cretivecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 2 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 For the log-concve function, we hve the following results (see [5]). Let the function p : I (0, ) be differentible, where I is n intervl. Then the function p is log-concve function if nd only if the function (log p) is decresing, i.e., ifv 1, v 2 I, v 1 < v 2,thenwe hve [ log p(v1 ) ] [ log p(v2 ) ]. (1.3) Let the function p (0, ) be twice differentible. Then the function p is log-concve function if nd only if p(x)p (x) [ p (x) ] 2 0, x I. (1.4) Let for convenience tht p p(x), p p(x)dx, p (x) dp(x) dx, p (x) d3 p(x) dx 3, p (n) (x) dn p(x) dx n, n 4. p (x) d2 p(x) dx 2, It is well known tht there is wide rnge of pplictions of log concvity in probbility nd sttistics theories (see [2, 4, 7 12]). However, qusi-log concvity lso hs fscinting significnce in probbility nd sttistics theories, see Section 4 nd Section 5. The min object of this pper is to introduce the qusi-log concvity of function nd demonstrte its pplictions in the nlysis of vrince. Now we introduce the definition of qusi-log concvity nd qusi-log convexity s follows. Definition 1.1 A differentible function p : I (0, ) issidtobequsi-log concve if the following inequlity ( [p G p [, b] p) (b) p () ] [ p(b) p() ] 2 0,, b I (1.5) holds, here I is n intervl. If inequlity (1.5) is reversed, then the function p : I (0, ) is sid to be qusi-log convex. We remrk here if the function p : I (0, ) is twice continuously differentible, then inequlity (1.5) cn be rewritten s follows: ( ) 2 G p [, b] p p p 0,, b I. (1.6) Now we show tht for the twice continuously differentible function, qusi-log concvity implies log concvity, nd qusi-log convexity implies log convexity. Indeed, suppose tht p : I (0, ) is twice continuously differentible nd qusi-log concve. Then (1.6)holds.Hence p(x)p (x) [ p (x) ] { 2 1 b = lim p(t)dt b x (b x) 2 x x [ ] 2 } p (t)dt p (t)dt 0 x
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 3 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 for ll x I so tht d 2 log p(x) = p(x)p (x) [p (x)] 2 0, x I. dx 2 p 2 (x) Therefore, (1.4) holds nd p is log concve on I. Similrly, we cn prove tht qusi-log convexity implies log convexity. On the other hnd, we cn prove tht for the twice continuously differentible function log convexity implies qusi-log convexity. Indeed, suppose tht p : I (0, ) is twice continuously differentible nd log convex. Then (1.4)isreversed.Hence p(x)p (x) [ p (x) ] 2 0, x I p (x) 0, p (x) p(x)p (x), x I ( ) 2 ( p p 2 ( ) 2 ) pp p tht is, inequlity (1.6) is reversed, here we used the Cuchy inequlity ( 2 fg) f 2 g 2. Therefore, p is qusi-log convex on I. p,, b I, Unfortuntely, we hve not found the connection between qusi-log concvity nd β-log concvity, where β >0. Bsed on the bove nlysis, we hve reson to propose conjecture (bbrevited s qusi-log concvity conjecture) s follows. Conjecture 1.1 (Qusi-log concvity conjecture) Suppose tht the function p : I (0, ) is twice continuously differentible. If p is log concve, then p is qusi-log concve. Here I is n intervl. We hve done lot of experiments with mthemticl softwre to verify the correctness of Conjecture 1.1, but did not find counter-exmple. We remrk tht similr concepts my be defined for sequences {x n } n=1 (0, ). We first define x n x n+1 x n, n N {1,2,...}, x n x n,, b N, < b, n<b ( ) (x G xn [, b] x n b x ) (xb x ) 2,, b N, < b, the sequence {x n } n=1 (0, ) is clled log-concve sequence if x x +2 x 2 +1 0, N, (1.7)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 4 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 nd is clled qusi-log concve if G xn [, b] 0,, b N, < b. (1.8) Set b = +1in(1.8). Then (1.8)cnberewrittens(1.7).Henceforthesequence{x n } n=1 (0, ), qusi-log concvity implies log concvity. Similrly, we cn define log-convex sequence nd qusi-log convexity of sequence. We expect inter-reltions between these concepts but they will be delt with elsewhere. In this pper, we re concerned with Conjecture 1.1 nd demonstrte the pplictions of our results in the nlysis of vrince nd the generlized hierrchicl teching model with generlized trditionl teching model. Our motivtion is to study severl interesting problems in sttistics. In Section 2, wetkeupconjecture1.1. InSection3, we give severl illustrtive exmples. In Section 4, we prove tht the probbility density function of the k-norml distribution is qusi-log concve. In Section 5, we demonstrte the pplictions of these results, we show tht the generlized hierrchicl teching model is normlly better thn the generlized trditionl teching model (see Remrk 5.3), nd we point out the significnce of qusi-log concvity in the nlysis of vrince nd the generlized trditionl teching model. 2 Study of Conjecture 1.1 For Conjecture 1.1,wehvethefollowingfivetheorems. Theorem 2.1 Let the function p : I (0, ) be twice continuously differentible, log concve nd monotone. If { 0<sup (log p) (log p) } { inf (log p) } + (log p), (2.1) t I t I then p is qusi-log concve. Proof Since the function p is log concve, we hve (log p) = [ log p(t) ] 0, t I, so inequlity (2.1)iswelldefined. Without loss of generlity, we my ssume tht, b I, < b. Note tht for ny positive rel number λ nd ny rel numbers x, y,wehvetheinequlity ( λ 2 ) x + y 2 xy, (2.2) 2λ the equlity holds if nd only if λ 2 x = y.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 5 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 According to inequlity (2.1), there exists positive rel number λ such tht 0<sup t I { (log p) (log p) } { λ inf (log p) } + (log p). (2.3) From (2.3)we know tht for the positive rel number λ,we hve t I λ 2 p(t) 2λ p (t) + p (t) 0, t I, (2.4) nd λ 2 p(t)+2λ p (t) + p (t) 0, t I. (2.5) Indeed, since (log p) = p p, (log p) = pp (p ) 2 p 2, p p = [ (log p) ] 2 +(log p), (2.6) inequlity (2.4) is equivlent to the inequlities (log p) (log p) λ (log p) + (log p), t I, (2.7) nd inequlity (2.5) is equivlent to the inequlities λ [ (log p) (log p) ], t I, (2.8) or λ [ (log p) + (log p) ], t I. (2.9) Hence if inequlities (2.3) hold, then both inequlity (2.4) nd inequlity (2.5)hold.Tht is to sy, inequlities (2.4)nd(2.5) re equivlent to inequlities (2.3). Since p : I (0, ) is monotonic, we obtin tht ( ) 2 ( p = 2 p ). (2.10) Combining with (2.2), (2.4), (2.5)nd(2.10), we get ( ) 2 p p p ( λ 2 b p + ) 2 ( p b ) 2 p 2λ ( λ 2 p + p ) 2 ( = p ) 2 2λ ( λ 2 p + p = + p )( λ 2 p + p 2λ 2λ p )
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 6 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 = 1 [ ( λ 2 p +2λ p + p )][ ( λ 2 p 2λ p + p )] 4λ 2 0. This mens tht inequlity (1.6)holds. The proof of Theorem 2.1 is completed. Corollry 2.1 Let the function p :[α, β] (0, ) be thrice continuously differentible nd log concve. If p 0, p >0, (log p) 2 ( (log p) ) 3, x [α, β], (2.11) then the function p :[α, β] (0, ) is qusi-log concve. Proof Let ϕ = (log p) (log p), ψ = (log p) + (log p). From (2.11), we hve nd ϕ =(log p) (log p) = p (p ) 2 pp >0, p ψ =(log p) + (log p) ϕ >0, dϕ dx =(log (log p) p) + 2 0, (log p) dψ dx =(log (log p) p) 2 0, (log p) 0<ϕ(x) ϕ(α) ψ(α) ψ(x), x [α, β], hence { 0< sup (log p) (log p) } = ϕ(α) x [α,β] { ψ(α)= inf (log p) } + (log p). x [α,β] By Theorem 2.1, the function p :[α, β] (0, ) is qusi-log concve. This ends the proof. Theorem 2.2 Let the function p :[α, β] (0, ) be twice continuously differentible nd log concve. If { (β α) 2 ( ) sup log p(x) 2} { ( ) 2 inf } log p(x) 0, (2.12) x [α,β] x [α,β] then p :[α, β] (0, ) is qusi-log concve.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 7 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Proof Now we prove tht (1.6) holds s follows. nd Without loss of generlity, we ssume tht, b [α, β]nd < b.notetht G p [, b] G p [, b] b 2 G p [, b] b p = p(b) ( ) 2 p p, p + p (b) p 2p (b) = p(b)p () p (b)p()+2p (b)p (). According to Lgrnge men vlue theorem, there re two rel numbers, b, p,, b I nd < < b < b, such tht nd hence G p [, b] b = G p[, b] G p [, ] = G p[, b ], b b G p [,b ] G p [,b ] b b = G p[b,b ] b = 2 G p [, b ], b b b G p [, b]=(b )(b ) [ p(b )p ( )+p (b )p( ) 2p (b )p ( ) ]. (2.13) From (2.6) nd the Lgrnge men vlue theorem, we get p(b )p ( )+p (b )p( ) 2p (b )p ( ) = p( )p(b ) {[( log p(b ) ) ( log p( ) ) ] 2 + ( log p( ) ) + ( log p(b ) ) } = p( )p(b ) {[ (b ) ( log p(ξ) ) ] 2 + ( log p( ) ) + ( log p(b ) ) } p( )p(b ) { (β α) 2[( log p(ξ) ) ] 2 + ( log p( ) ) + ( log p(b ) ) } = p( )p(b ) { (β α) 2[( log p(ξ) ) ] 2 ( log p( ) ) ( log p(b ) ) } [ { p( )p(b ) (β α) 2 ( ) sup log p(x) 2} { ( ) 2 inf log p(x) } ] x [α,β] x [α,β] 0, i.e., p(b )p ( )+p (b )p( ) 2p (b )p ( ) 0, (2.14)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 8 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 where α < < ξ < b < b β. (2.15) Combining with (2.13), (2.14)nd(2.15), we get inequlity (1.6). This completes the proof of Theorem 2.2. Theorem 2.3 Let the function ϕ :(α, β) (, ) be thrice continuously differentible. If ϕ (x)>0, 2ϕ (x)ϕ (x) ϕ (x) 0, x (α, β), (2.16) then the function p :(α, β) (0, ), p(x) ce ϕ(x), c >0, is qusi-log concve. Proof Let G p [, b] p We just need to show tht ( ) 2 p p. G p [, b] 0,, b (α, β). (2.17) Since G p [, b] G p [b, ], G p [, ]=0,, b (α, β), without loss of generlity, we cn ssume tht α < b < < β, c =1, (2.18) nd is fixed constnt. Note tht log p(x)= ϕ, d log p(x) dx = p p = ϕ, nd Hence d 2 log p(x) dx 2 = pp (p ) 2 p 2 = ϕ. p (x)= ϕ p, x (α, β), (2.19)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 9 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 nd p (x)= [( ϕ ) 2 ϕ ] p, x (α, β). (2.20) From G p [, b] b = p(b) p + p (b) (2.19)nd(2.20), we my see tht Let Then i.e., G p [, b] b p 2p (b) { = p(b) p + [( ϕ (b) ) 2 ϕ (b) ] F(, b) 1 G p [, b] = p(b) b F(, b) b F(, b) b p, p +2ϕ (b) p + [( ϕ (b) ) 2 ϕ (b) ] p +2ϕ (b) p }. (2.21) = p (b)+ [( ϕ (b) ) 2 ϕ (b) ] p(b)+ [ 2ϕ (b)ϕ (b) ϕ (b) ] p [ +2 ϕ (b) p + ϕ (b)p ] =2 [( ϕ (b) ) 2 ϕ (b) ] p(b)+ [ 2ϕ (b)ϕ (b) ϕ (b) ] p +2 { ϕ (b) [ p(b) p() ] ( ϕ (b) ) 2 p(b) } = [ 2ϕ (b)ϕ (b) ϕ (b) ] p 2ϕ (b)p(), p. (2.22) = [ 2ϕ (b)ϕ (b) ϕ (b) ] p 2ϕ (b)p(). (2.23) Bsed on ssumption (2.16), p <0nd(2.23), we hve F(, b) b <0, b (α, ). (2.24) From (2.24), (2.18)nd(2.22), we hve F(, b)>f(, )=0, G p [, b] b >0. (2.25) By (2.25)nd(2.18), we get G p [, b]<g p [, ]=0. (2.26)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 10 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Tht is to sy, inequlity (2.17)holds. We remrk tht the equlity in (2.17) holds if nd only if = b. The proof of Theorem 2.3 is completed. Theorem 2.4 Let the function ϕ :(α, β) (, ) be four times continuously differentible. If nd ϕ (x)>0, 2 [ ϕ (x) ] 3 ϕ (4) (x)ϕ (x)+ [ ϕ (x) ] 2 0, x (α, β), (2.27) G p [α, β] then the function α p p :(α, β) (0, ), p(x) ce ϕ(x), c >0, is qusi-log concve, where c is constnt. α ( ) 2 p p 0, (2.28) α InordertoproveTheorem2.4, we need the following lemm. Lemm 2.1 Under the ssumptions of Theorem 2.4, if α < < b < β < β, (2.29) then we hve G p [, b] mx { 0, G p [, β ] }. (2.30) Proof Without loss of generlity, we cn ssume tht c =1nd is fixed constnt. We continue to use the proof of Theorem 2.3.Notethteqution(2.23)cnberewritten s where F(, b) b ( ) = ϕ (b) p F (, b), (2.31) F (, b) 2ϕ (b) ϕ (b) ϕ (b) 2p() p. (2.32) Bsed on ssumption (2.27), p >0nd(2.32), we hve F (, b) b = 2[ϕ (b)] 3 ϕ (4) (b)ϕ (b)+[ϕ (b)] 2 [ϕ (b)] 2 + 2p()p(b) ( p)2 >0, b (, β ), (2.33) which mens tht F (, b) is strictly incresing for the vrible b (, β ).
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 11 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 From (2.32), we my see tht lim b + F (, b)=f (, +) =. (2.34) We prove inequlity (2.30) in two cses (A) nd (B). (A) Assume tht F (, β )>0. (2.35) By (2.34), (2.35) nd the intermedite vlue theorem, there exists only one number b (, β )suchtht F (, b )=0. (2.36) From (2.33)nd(2.31), we get nd < b < b F (, b)<f (, b )=0 F(, b)<0, b b < b < β F (, b)>f (, b )=0 F(, b)>0, b hence F(, b) is strictly decresing if b (, b ] nd strictly incresing if b [b, β ). If F(, β ) 0, since F(, )=0,wehve nd < b b F(, b)<f(, )=0, b b < β F(, b) F(, β ) 0, F(, b)= 1 G p [, b] 0, b (, β ), p(b) b G p [, b] 0, b (, β ), b G p [, b] G p [, ]=0, b (, β ). This mens tht inequlity (2.30)holds. Now we ssume tht F(, β )>0. (2.37) Note tht F(, b) is strictly decresing if b (, b ], we hve b (, b ] F(, b )<F(, )=0. (2.38)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 12 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 By (2.37), (2.38), F(, b) is strictly incresing if b [b, β ) nd the continuity, we know tht there exists unique rel number b (b, β )suchtht F (, b ) =0. (2.39) Since nd < b b F(, b)<f(, )=0 b G p[, b]=p(b)f(, b)<0, b < b < b F(, b)<f (, b ) =0 b G p[, b]=p(b)f(, b)<0, < b < b b G p[, b]<0, b < b < β F(, b)>f (, b ) =0 b G p[, b]=p(b)f(, b)>0, we know tht G p [, b] is strictly decresing if b (, b ] nd strictly incresing if b [b, β ), so tht G p [, b] mx { G p [, ], G p [, β ] } = mx { 0, G p [, β ] }. (2.40) This mens tht inequlity (2.30)lsoholds. (B) Assume tht F (, β ) 0. (2.41) Since F (, b) is strictly incresing for the vrible b (, β ), we hve nd F (, b)<f (, β ) 0, b (α, β ), ( F(, b) b ) = ϕ (b) p F (, b) 0, b (α, β ), b F(, b) F(, )=0, b (α, β ), G p [, b] b = p(b)f(, b) 0, b (α, β ), G p [, b] G p [, ]=0 mx { 0, G p [, β ] }, b (α, β ). Tht is to sy, inequlity (2.30) still holds. The proof of Lemm 2.1 is completed.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 13 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 The proofof Theorem 2.4 is now reltively esy. Proof of Theorem 2.4 We just need to show tht (2.17) holds. Without loss of generlity, we ssume tht α < < b < β, c =1. (2.42) Let α, β (α, β)suchtht α < α < < b < β < β. (2.43) By Lemm 2.1,inequlity(2.30)holds. We define the uxiliry function p s follows: p :( β, α) (0, ), p (x) ce ϕ( x), c >0. Then, by (2.27), we hve [ ϕ( x) ] = ϕ ( x)>0, x ( β, α), [ ϕ( x) ] = ϕ ( x), [ ϕ( x) ] (4) = ϕ (4) ( x), x ( β, α), nd 2 {[ ϕ( x) ] } 3 [ ϕ( x) ] (4) [ ϕ( x) ] + {[ ϕ( x) ] } 2 =2 [ ϕ ( x) ] 3 ϕ (4) ( x)ϕ ( x)+ [ ϕ ( x) ] 2 0, x ( β, α). According to Lemm 2.1 nd β < β < < α < α, we get G p [ β, ] mx { 0, G p [ β, α ] }. (2.44) Since we hve G p [ b, ]= b p( x)dx b [ 2 p ( x)dx p ( x)dx] G p [, b], b G p [ β, ]=G p [, β ], G p [ β, α ]=G p [α, β ], nd inequlity (2.44)cnberewrittens G p [, β ] mx { 0, G p [α, β ] }. (2.45)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 14 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Combining with inequlities (2.30)nd(2.45), we get G p [, b] mx { 0, G p [, β ] } mx { 0, G p [α, β ] }. (2.46) In (2.46), set α α, β β,weget G p [, b] mx { 0, G p [α, β] }. (2.47) According to conditions (2.28)nd(2.47), inequlity (2.17) holds. This completes the proof of Theorem 2.4. Theorem 2.5 Let the function ϕ :(α, β) (, ) be four times continuously differentible. If lim x β ϕ(x)=, lim ϕ (x) [ϕ (x)] 2 =0, lim x β e ϕ(x) x β ϕ (x)e ϕ(x) =0, α p <, (2.48) nd (2.27) holds, then the function p :(α, β) (0, ), p(x) ce ϕ(x), c >0, is qusi-log concve. Proof We just need to show tht (2.17) holds. Without loss of generlity, we ssume tht α < < b < β, c =1. (2.49) Set β β in Lemm 2.1,wehve G p [, b] mx { 0, G p [, β] }. (2.50) To complete the proof of inequlity (2.17), by (2.50), we just need to show tht G p [, β] 0, (α, β). (2.51) Now, webelieve thttherel number is vrible. By condition (2.48), we hve p (β) lim x β p (x)=0, p(β) lim x β p(x)=0 nd ( ) [p G p [, β]= p (β) p () ] [ p(β) p() ] 2 = p () [ = p() ϕ () p [ p() ] 2 ] p p(),
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 15 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 i.e., G p [, β]=p()φ(), (2.52) where φ() ϕ () p p(), (α, β). (2.53) If ϕ () 0, then φ()<0,(2.51)holdsby(2.52). Here we ssume tht ϕ ()>0. Note tht dφ() d = ϕ () = ϕ () >0. p ϕ ()p() p () p Since ϕ (x)>0, x (α, β), the limit lim β ϕ () exists. If 0<ϕ () lim β ϕ ()<, from α p <,wehve lim β ( p = lim p β α α ) p =0 (2.54) nd If lim β ϕ () p =0. (2.55) lim β ϕ ()=,
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 16 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 then, by (2.54), (2.48)ndL Hospitl srule,wehve lim β ϕ () p = lim β p [ϕ ()] 1 (d = lim p)/d β (d[ϕ ()] 1 )/d p() = lim β [ϕ ()] 2 ϕ () = lim x β [ϕ (x)] 2 ϕ (x)e ϕ(x) =0, tht is to sy, (2.55)lsoholds.Hence [ φ()<φ(β)=lim ϕ () β ] p p() = lim ϕ () β p =0. By (2.52), inequlity (2.51)holds. The proof of Theorem 2.5 is completed. 3 Four illustrtive exmples In order to illustrte the connottion of qusi-log concvity, we give four exmples s follows. Exmple 3.1 The function p :[0,π] (0, ), p(x) exp(sin x) is qusi-log concve. Proof Indeed, if [ x I 0, π ] 2 or [ ] π 2, π, then p(x) is twice continuous differentible nd log concve with monotonous function, nd (log p) { } (log p) = cos x sin x mx cos x, sin x 1, (log p) + (log p) = cos x + sin x cos x + sin x = 1+ sin 2x 1, hence 0<sup x I { (log p) (log p) } =1= inf x I { (log p) + (log p) }. By Theorem 2.1, the function p(x) is qusi-log concve. Tht is to sy, for ny, b I, inequlity (1.5)holds.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 17 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Let 0 < π 2 < b π. Since p (b)<0<p (), inequlity (1.5) still holds. The proofis completed. Exmple 3.2 The function p :(0, ) (0, β), p(x) exp(rctn x) is qusi-log concve, where β = 1 9( 7+4 3 10 + 3 100 ) =2.251036399304479... is the root of the eqution 16x 3 (1 + x 2 ) 2x[ 6 48x 3 (1+x 2 ) 4 24x (1+x 2 ) 3 ] 2 (1 + x 2 ) 2 + Proof Indeed, in Theorem 2.4,set ϕ(x)= rctn x,then ] [ 8x2 (1 + x 2 ) + 2 2 =0. (3.1) 3 (1 + x 2 ) 2 p(x)=exp [ ϕ(x) ], x (0, β). By mens of Mthemtic softwre, we get ϕ (x)= 2x 0, x (0, β), (1 + x 2 ) 2 2 [ ϕ (x) ] 3 ϕ (4) (x)ϕ (x)+ [ ϕ (x) ] 2 = 16x3 (1 + x 2 ) 2x[ 6 48x 3 (1+x 2 ) 4 24x (1+x 2 ) 3 ] 2 (1 + x 2 ) 2 + ] [ 8x2 (1 + x 2 ) + 2 2 3 (1 + x 2 ) 2 0, x (0, β], the eqution holds if nd only if x = β. nd Since 0< 0 p <, G p [0, β] 0 p 0 ( ) 2 p p = 7.095040628958467...<0, 0
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 18 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 so (2.27) nd(2.28) hold.bytheorem2.4, the function p(x) isqusi-logconcve.this ends the proof. Exmple 3.3 The function p :(0, ) (0, ), p(x) x α, is qusi-log concve, where α >0. Proof Note tht inequlity (1.6)cn be rewritten s α ( b α+1 α+1)( b α 1 α 1) ( b α α) 2 0,, b I. (3.2) α +1 If 0 < α 1, then α ( b α+1 α+1)( b α 1 α 1) 0, α +1 inequlity (3.2)holds.Letα >1.Then ( b α+1 α+1)( b α 1 α 1) 0. Since nd 0< α α +1 <1, ( b α+1 α+1)( b α 1 α 1) ( b α α) 2 = α 1 b α 1 ( b) 2 0, we hve α ( b α+1 α+1)( b α 1 α 1) ( b α α) 2 α +1 ( b α+1 α+1)( b α 1 α 1) ( b α α) 2 0, tht is to sy, inequlity (3.2) still holds. The proof is completed. Exmple 3.4 The function p :(0, ) (0, ), p(x) exp ( e x) is qusi-log concve. Proof Indeed, 0< 0 ϕ(x)=e x, p = 0.21938393439552026... <, lim ϕ(x)= lim x x ex =,
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 19 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 nd ϕ (x) lim = lim x e ϕ(x) x [ϕ (x)] 2 = lim ϕ (x)eϕ(x) x hence equtions in (2.48)hold.Since e x exp(e x ) =0, ϕ (x)=e x >0, x (0, ), nd 2 [ ϕ (x) ] 3 ϕ (4) (x)ϕ (x)+ [ ϕ (x) ] 2 =2e 3x >0, x (0, ), inequlities in (2.27)hold.ByTheorem2.5, the function p is qusi-log concve. This ends the proof. In the next section, we demonstrte the pplictions of Theorem 2.3 nd Theorem 2.5 in the theory of k-norml distribution. 4 Qusi-logconcvity ofpdfof k-norml distribution The norml distribution (see [14 16]) is considered s the most prominent probbility distribution in sttistics. Besides the importnt centrl limit theorem tht sys the men of lrge number of rndom vribles drwn from common distribution, under mild conditions, is distributed pproximtely normlly, the norml distribution is lso trctble in the sense tht lrge number of relted results cn be derived explicitly nd tht mny qulittive properties my be stted in terms of vrious inequlities. But perhps one of the min prcticl uses of the norml distribution is to model empiricl distributions of mny different rndom vribles encountered in prctice. In such cse, possible generliztion would be fmilies of distributions hving more thn two prmeters (nmely the men nd the stndrd vrition) which my be used to fit empiricl distributions more ccurtely. Exmples of such generliztions re the normlexponentil-gmm distribution which contins three prmeters nd the Person distribution which contins four prmeters for simulting different skewness nd kurtosis vlues. In this section, we first introduce nother generliztion of the norml distribution s follows: If the probbility density function of the rndom vrible X is p(x; μ, σ, k) ) ( k1 k 1 2Ɣ(k 1 )σ exp x μ k, (4.1) kσ k then we sy tht the rndom vrible X follows the k-norml distribution or generlized norml distribution (see [17]or[18]), denoted by X N k (μ, σ ), where x (, ), μ (, ), σ (0, ), k (1, ), nd Ɣ(s) is the well-known gmm function.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 20 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Figure 1 The grphs of the functions p(x;0,1,3/2),p(x;0,1,2)ndp(x;0,1,5/2),where 4 x 4. Figure 2 The grph of the function p(x;0,1,k), where 4 x 4, 1 < k 3. For the probbility density function p(x; μ, σ, k) ofk-norml distribution, the grphs of the functions p(x;0,1,3/2), p(x;0,1,2) nd p(x;0,1,5/2) re depicted in Figure 1 nd p(x;0,1,k) is depicted in Figure 2. Clerly, when k =2,p(x; μ, σ, k) is just the stndrd norml distribution N(μ, σ )with men μ nd stndrd devition σ,nditisesilycheckedthtp(x;0,1,k) is symmetric bout 0 nd tht p(x;0,1,k)=σ p(σ x + μ; μ, σ, k), x (, ). (4.2) According to (4.1), we get (see (2) in [17]) ( p x; μ, ) σ s, s = 1/s ( s 2σƔ(1/s) exp x μ σ s). (4.3) Accordingtotheresultsof[17], we my esily show tht (see [17], p.688) p(x; μ, σ, k)>0, p(x; μ, σ, k)dt = 1, (4.4) EX = μ, (4.5) E X EX k = σ k, (4.6)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 21 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 nd E(X EX) 2 = k2k 1 Ɣ(3k 1 ) Ɣ(k 1 ) σ 2 > σ 2, 1<k <2, = σ 2, k =2, < σ 2, k >2. (4.7) Here μ, σ k nd σ re the mthemticl expecttion, k-order bsolute centrl moment nd k-order men bsolute centrl moment of the rndom vrible X,respectively. We remrk here if then nd ] (x μ)k p 0 (x)=exp [, x (μ, ), kσ k ] w(x) p μ)k 1 (x μ)k 0 (x)=(x exp [ >0, σ k kσ k μ w(x)dx =1, where w(x) is the probbility density function of Weibull distribution. Therefore, there reclose reltionships between the k-norml distribution nd the Weibull distribution. Next, we study the qusi-log concvity of the probbility density function of k-norml distribution. Theorem 4.1 The probbility density function p(x; μ, σ, k) of the k-norml distribution is qusi-log concve on (, ) for ll μ (, ), σ (0, ) nd k (1, ). Proof In view of (4.2), we my ssume tht (μ, σ ) = (0, 1). Let for convenience tht p(x) p(x;0,1,k)=c exp [ ϕ(x) ], where c = k1 k 1 2Ɣ(k 1 ) >0, x k ϕ(x)= k. Then G p [, ]=0=G p [b, b], G p [, b]=g p [b, ] nd G p [, b]=g p [ b, ], (4.8) where the lst equlity holds becuse p is even.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 22 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Nowweshowtht G p [, b] 0,, b (, ) (4.9) in two steps (A) nd (B). (A) We first consider the cse where k 2. By (4.8) nd continuity, without loss of generlity, we my ssume tht either (i): 0 < < b, or (ii): <0<b. We first consider the cse (i): 0 < < b. By (4.4), we hve Since 0 p = 1 2 <. lim ϕ(x)= lim x x xk 1 =, ϕ (x) lim = lim x e ϕ(x) x x k 1 exp( xk k ) =0, nd lim x [ϕ (x)] 2 = lim ϕ (x)eϕ(x) x equtions in (2.48)hold.Since x 2k 2 (k 1)x k 2 exp( xk k ) =0, ϕ (x)=(k 1)x k 2 >0, x (0, ), nd 2 [ ϕ (x) ] 3 ϕ (4) (x)ϕ (x)+ [ ϕ (x) ] 2 =2(k 1) 2 x 3k 6 (k 1) 2 (k 2)(k 3)x 2k 6 +(k 1) 2 (k 2) 2 x 2k 6 =(k 1) 2 x 3k 6 +(k 1) 2 (k 2)x 2k 6 >0, x (0, ), inequlities in (2.27)hold.By Theorem 2.5,inequlity(4.9)holds. Next, we consider the cse (ii): <0<b. Since we hve p (b)<0<p (), ( [p G p [, b]= p) (b) p () ] ( ) 2 ( [p p p) (b) p () ] <0, tht is, inequlity (4.9)still holds.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 23 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 (B) Next we ssume tht 1 < k <2. Since ϕ (x)=(k 1)x k 2 >0, x (0, ), nd 2ϕ (x)ϕ (x) ϕ (x)=2(k 1)x 2k 3 +(k 1)(2 k)x k 3 0, x (0, ), inequlities in (2.16)hold.By Theorem 2.3,inequlity(4.9)holds. Bsed on the bove nlysis, inequlity (4.9)isproved. The proof of Theorem 4.1 is completed. In the next section, we demonstrte the pplictions of Theorem 4.1 in the generlized hierrchicl teching model nd the generlized trditionl teching model. 5 Applictions in sttistics 5.1 Hierrchicl teching model nd truncted rndom vrible We first introduce the hierrchicl teching model s follows. The usul teching model ssumes tht the mth scores of ech student in clss re treted s continuous rndom vrible, written s ξ I, which tkes on some vlue in the rel intervl I =[ 0, m ], nd its probbility density function p I : I (0, ) is continuous. Suppose we now divide the students into m clsses, written s Clss[ 0, 1 ], Clss[ 1, 2 ],...,Clss[ m 1, m ], where 0 0 1 m, m 2, nd i, i+1, i =0,1,...,m 1, re the lowest nd the highest llowble scores of the students of Clss[ i, i+1 ], respectively. We introduce set HTM{ 0,..., m, p I } { Clss[ 0, 1 ], Clss[ 1, 2 ],...,Clss[ m 1, m ], p I } clled hierrchicl teching model (see [19 22]) such tht the trditionl teching model, denoted by HTM{ 0, m, p I },isjustspecilhtm{ 0,..., m, p I },wherem =1. If 0 = nd m =,thenhtm{,...,, p I } nd HTM{,, p I } re clled generlized hierrchicl teching model nd generlized trditionl teching model, respectively. In order to study the hierrchicl teching model nd the trditionl teching model from the ngle of the nlysis of vrince, we need to recll the definition of truncted rndom vrible.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 24 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Definition 5.1 Let ξ I I be continuous rndom vrible with continuous probbility density function p I : I (0, ). If ξ J J I is lso continuous rndom vrible nd its probbility density function is p J : J (0, ), p J (t) p I(t) J p, I then we cll the rndom vrible ξ J truncted rndom vrible of the rndom vrible ξ I,writtensξ J ξ I.Ifξ J ξ I nd J I, then we cll the rndom vrible ξ J proper truncted rndom vrible of the rndom vrible ξ I,writtensξ J ξ I.HereI nd J re high dimensionl intervls. We point out tht bsic property of the truncted rndom vrible is s follows: Let ξ I I be continuous rndom vrible with continuous probbility density function p I : I (0, ). If ξ I ξ I, ξ I ξ I nd I I, then ξ I ξ I.If ξ I ξ I, ξ I ξ I nd I I, then ξ I ξ I. Indeed, by Definition 5.1, the probbility density functions of the truncted rndom vribles ξ I, ξ I re p I : I (0, ), p I : I (0, ), p I (t)= p I(t) I p I, p I (t)= p I(t) I p I, respectively. Thus, the probbility density function of ξ I cn be rewritten s Hence p I : I (0, ), p I (t)= p I(t)/ I p I I (p I / I p I ) = p I (t) I p I I I ξ I ξ I nd I I ξ I ξ I. According to the definitions of the mthemticl expecttion Eϕ(ξ J ) nd the vrince Vr ϕ(ξ J ) with Definition 5.1,weesilyget J Eϕ(ξ J ) p J ϕ = p Iϕ J J p, (5.1) I. nd Vr ϕ(ξ J ) E [ ϕ(ξ J ) Eϕ(ξ J ) ] 2 J p Iϕ 2 ( J = J p p ) Iϕ 2 I J p, (5.2) I
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 25 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 where ξ J ξ I,ndthefunctionϕ : J (, ) oftherndomvribleξ J is continuous. In the generlized hierrchicl teching model HTM{,...,, p I }, the mth scores of ech student in Clss[ i, i+1 ] is lso rndom vrible, written s ξ [i, i+1 ].Since [ i, i+1 ] I, i =0,1,...,m 1, so ξ [i, i+1 ] is truncted rndom vrible of the rndom vrible ξ I. Assume tht the j i clsses, i.e., Clss[ i, i+1 ], Clss[ i+1, i+2 ],...,Clss[ j 1, j ] re merged into one, written s Clss[ i, j ]. Since [ i, j ] I, weknowthtξ [i, j ] is truncted rndom vrible of the rndom vrible ξ I,where0 i < j m. In generl, we hve ξ [i, j ] ξ [i, j ] ξ I, i, i, j, j :0 i i < j j m. (5.3) In the generlized hierrchicl teching model HTM{,...,, p I }, we re concerned with the reltionship between the vrince Vr ξ [i, j ] nd the vrince Vr ξ I,where0 i < j m, so s to decide the superiority nd inferiority of the hierrchicl nd the trditionl teching models. If Vr ξ [i, j ] Vr ξ I, i, j :0 i < j m, (5.4) then in view of the usul mening of the vrince, we tend to think tht this generlized hierrchicl teching model is better thn the generlized trditionl teching model. Otherwise, this generlized hierrchicl teching model is probbly not worth promoting, where I =(, ). In this section, one of our purposes is to study the generlized hierrchicl teching model nd the generlized trditionl teching model from the ngle of the nlysis of vrince so s to decide the superiority nd inferiority of the generlized hierrchicl nd the generlized trditionl teching models. In prticulr, we will study the conditions such tht inequlity (5.4) holds (see Theorem 5.2). In the generlized hierrchicl teching model HTM{,...,, p I },wecnchoosethe prmeters 1, 2,..., m 1 (, ) such tht the vrince of Vr(Vr ξ [0, 1 ],...,Vr ξ [m 1, m ]) ( 1 m 1 Vr ξ [j, m j+1 ] 1 m 1 2 Vr ξ [i, m i+1 ]) (5.5) j=0 i=0 Vr ξ [0, 1 ], Vr ξ [1, 2 ],...,Vr ξ [m 1, m ]
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 26 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 is the miniml by mens of mthemticl softwre, its purpose is to mke the scores of the clsses Clss[ 0, 1 ], Clss[ 1, 2 ],...,Clss[ m 1, m ] stble, where 0 = nd m =. Remrk 5.1 We remrk here if ξ I I is continuous rndom vrible with continuous probbility density function p I : I (0, ), then the integrtion I p I converges (see [23]), nd it stisfies the following conditions: I p I =1, P I (x) P(ξ I < x)= p I, x I. (5.6) (,x) I We cll the function P I : I [0, 1] probbility distribution function of the rndom vrible ξ I,whereP I (x) is the probbility of the rndom event ξ I < x, nd I is n intervl. 5.2 Applictions in the nlysis of vrince The nlysis of vrince is one of the centrl topics in sttistics. Recently, the uthors [24] hve expnded the connottion of nlysis of vrince nd obtined some interesting results. In this section, we point out the significnce of qusi-log concvity in the nlysis of vrince s follows. Theorem 5.1 Let ξ I be continuous rndom vrible nd its probbility density function p I : I (0, ) be twice continuously differentible. Then the function p I : I (0, ) is qusi-log concve if nd only if 0 Vr [ (log p I ) (ξ [,b] ) ] E [ (log p I ) (ξ [,b] ) ],, b I, < b, (5.7) where ξ [,b] [, b] is truncted rndom vrible of the rndom vrible ξ I. Proof By identities (2.6)nd(5.1)with(5.2), we get Vr [ (log p I ) (ξ [,b] ) ] +E [ (log p I ) (ξ [,b] ) ] ( p ) [ = Vr I pi p ] I +E (p I )2 p I p 2 I = = = ( p I p I ) 2 p I p I ( p I p I p I ( p I p I p I p I p I p I ) 2 p I ( p I )2 ( p, I) 2 ) 2 + p I p I (p I )2 p p 2 I I p I
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 27 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 i.e., Vr [ (log p I ) (ξ [,b] ) ] +E [ (log p I ) (ξ [,b] ) ] = p I p I ( p I )2 ( p. (5.8) I) 2 According to identity (5.8), we know tht inequlity (1.6)cnbe rewritten s(5.7). This completes the proof of Theorem 5.1. Remrk 5.2 According to Theorem 5.1,qusi-logconcvity is of gret significnce in the nlysis of vrince. 5.3 Applictions in the generlized hierrchicl teching model Now we demonstrte the ppliction of Theorem 4.1 in the generlized hierrchicl teching model. In the generlized hierrchicl teching model HTM{,...,, p I },themthscores of ech student re treted s rndom vrible ξ I,whereξ I I =(, ). By using the centrl limit theorem (see [25]), we my think tht the rndom vrible ξ I follows norml distribution, tht is, ξ I N 2 (μ, σ ), where μ isthevergescoreofthestudentsndσ is the men squre devition of the score. Hence p I (x)= ] 1 (x μ)2 exp [, x I. (5.9) 2πσ 2σ 2 We remrk here tht if the mth scores ξ I of ech student stisfies ξ I [0, 1] nd μ [0, 1], then, by (5.9), we hve P(ξ I <0) 0, P(ξ I >1) 0. Hence we cn use the generlized hierrchicl teching model insted of the hierrchicl teching model, pproximtely. Bsed on the bove nlysis nd Theorem 4.1,wehvethefollowingtheorem. Theorem 5.2 In the generlized hierrchicl teching model HTM{,...,, p I }, ssume tht ξ I N 2 (μ, σ ). Then we hve the following inequlity: Vr ξ [i, j ] Vr ξ I = σ 2, i, j :0 i < j m. (5.10) Proof Note tht p I (t)=p(t; μ, σ,2) is qusi-log concve by Theorem 4.1, henceinequlity(5.7) holdsbytheorem5.1, sowe obtin tht Vr [ (log p I ) (ξ [i, j ]) ] E [ (log p I ) (ξ [i, j ]) ]. (5.11)
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 28 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Note tht Vr(ϕ + C) Vr(ϕ), Vr(Cϕ) C 2 Vr(ϕ), E(Cϕ) CE(ϕ), (log p I ) (ξ [i, j ])= ξ [ i, j ] μ σ 2, (log p I ) (ξ [i, j ])= 1 σ 2, where C is constnt. By (5.11), we hve ( Vr ξ ) ( [ i, j ] μ E 1 ) σ 2 σ 2 1 σ 4 Vr ξ [ i, j ] 1 σ 2 E(1) Vr ξ [ i, j ] Vr ξ I = σ 2, E(1)=1, tht is to sy, inequlity (5.10)holds. This completes the proof of Theorem 5.2. Remrk 5.3 According to Theorem 5.2, we my conclude tht the generlized hierrchicl teching model is normlly better thn the generlized trditionl teching model. 5.4 Applictions in the generlized trditionl teching model Next, we demonstrte the pplictions of Theorem 4.1in the generlized trditionl teching model s follows. In the generlized trditionl teching model HTM{,, p I }, the mth scores of ech student re treted s rndom vrible ξ I,whereξ I I =(, ). By using the centrl limit theorem (see [25]), we my think tht the rndom vrible ξ follows norml distribution, tht is, ξ I N 2 (μ, σ ), where μ isthevergescoreofthestudentsndσ is the men squre devition of the score. If the top nd bottom students re insignificnt, tht is to sy, the vrince Vr ξ I of the rndom vrible ξ I is close to 0, ccording to Figure 1 nd Figure 2 with formul (4.7), we my think tht there is rel number k (2, ) such tht ξ I N k (μ, σ ). Otherwise, we my think tht there is rel number k (1, 2) such tht ξ I N k (μ, σ ). We cn estimte the number k by mens of smpling procedure. In the generlized trditionl teching model HTM{,, p I },wemyssumetht ξ J ξ I, ξ I N k (μ, σ ), k >1, where J =(μ, ), nd μ (0, ) is the verge mth score of the students nd σ is the k- order men bsolute centrl moment of the score. Then the probbility density function of ξ J is tht p(x; μ, σ, k) p J (t)= x J. J p(x; μ, σ, k), In the generlized trditionl teching model HTM{,, p I }, suppose tht the mth score of the student is ξ J. In order to stimulte the lerning enthusism of students, we my wnt to give ech student bonus pyment A(ξ J ). The function A : J (0, ) my be regrded s n llownce function.
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 29 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 In the generlized trditionl teching model HTM{,, p I }, we define the llownce function s follows: A : J (0, ), A(x) c(x μ) k 1, c >0,k >1. (5.12) Forthe bove llowncefunction (5.12), we hve the following theorem. Theorem 5.3 In the generlized trditionl teching model HTM{,, p I }, ssume tht ξ J ξ I, ξ I N k (μ, σ ), k >1. Then we hve the following inequlities: 0 Vr [ A(ξ [,b] ) ] cσ k E [ A (ξ [,b] ) ],, b J, < b. (5.13) Here the llownce function A is defined by (5.12). Proof By Theorem 4.1, the function p p J (t)isqusi-logconcveonj.henceinequlities (5.7) hold by Theorem 5.1.Notetht Vr(CA) C 2 Vr(A), E(CA) CE(A), (log p I ) (ξ [,b] )= (ξ [,b] μ) k 1 σ k = 1 cσ k A(ξ [,b]), nd (log p I ) (ξ [,b] )= 1 cσ k A (ξ [,b] ), where C is constnt. By inequlities (5.7), we get nd Vr [ (log p I ) (ξ [,b] ) ] E [ (log p I ) (ξ [,b] ) ], [ Vr 1 ] [ cσ A(ξ [,b]) E 1 ] k cσ k A (ξ [,b] ), 1 (cσ k ) Vr[ A(ξ 2 [,b] ) ] 1 cσ E[ A (ξ k [,b] ) ]. Tht is to sy, inequlities (5.13)hold. This completes the proof of Theorem 5.3. Remrk 5.4 A lrge number of inequlity nlysis nd sttisticl theories re used in this pper. Some theories in the proofs of our results re used in the references [5, 23, 24, 26 34].
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 Pge 30 of 30 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 Competing interests The uthors declre tht they hve no competing interests. Authors contributions All uthors contributed eqully nd significntly in this pper. All uthors red nd pproved the finl mnuscript. Author detils 1 College of Mthemtics nd Informtion Science, Chengdu University, Sichun, 610106, Chin. 2 Deprtment of Mthemtics, Tsing Hu University, Hsinchu, Tiwn 30043, R.O. Chin. Acknowledgements This work ws supported in prt by the Nturl Science Foundtion of Chin (No. 61309015) nd the Nturl Science Foundtion of Sichun Province Eduction Deprtment (No. 07ZA207). The uthors re indebted to severl unknown referees for mny useful comments nd keen observtions which led to the present improved version of the pper s it stnds. Received: 10 Jnury 2014 Accepted: 12 August 2014 Published: 03 Sep 2014 References 1. Tong, Y: An dptive solution to rnking nd selection problems. Ann. Stt. 6(3),658-672 (1978) 2. Bgnoli, M, Bergstrom, T: Log-concve probbility nd its pplictions. Mimeo (1991) 3. Ptel, JK, Red, CB: Hndbook of the Norml Distribution, 2nd edn. CRC Press, Boc Rton (1996) 4. Heckmn, JJ, Honor, BE: The empiricl content of the Roy model. Econometric 58(5), 1121-1149 (1990) 5. Wng, WL: Approches to Prove Inequlities. Hrbin Institute of Technology Press, Hrbin (2011) (in Chinese) 6. Wen, JJ, Go, CB, Wng, WL: Inequlities of J-P-S-F type. J. Mth. Inequl. 7(2), 213-225 (2013) 7. Mohtshmi Borzdrn, GR, Mohtshmi Borzdrn, HA: Log concvity property for some well-known distributions. Surv. Mth. Appl. 6,203-219 (2011) 8. An, MY: Log-concve probbility distributions: theory nd sttisticl testing. Working pper, Duke University (1995) 9. An, MY: Log-concvity nd sttisticl inference of liner index modes. Mnuscript, Duke University (1997) 10. Finner, H, Roters, M: Log-concvity nd inequlities for Chi-squre, F nd Bet distributions with pplictions in multiple comprisons. Stt. Sin. 7,771-787 (1997) 11. Mirvete, EJ: Preserving log-concvity under convolution: Comment. Econometric 70(3),1253-1254 (2002) 12. Al-Zhrni, B, Stoynov, J: On some properties of life distributions with incresing elsticity nd log-concvity. Appl. Mth. Sci. 2(48), 2349-2361 (2008) 13. Crmnis, C, Mnnor, S: An inequlity for nerly log-concve distributions with pplictions to lerning. IEEE Trns. Inf. Theory 53(3), 1043-1057 (2007) 14. Wlodzimierz, B: The Norml Distribution: Chrcteriztions with Applictions. Springer, Berlin (1995) 15. Spiegel, MR: Theory nd Problems of Probbility nd Sttistics, pp. 109-111. McGrw-Hill, New York (1992) 16. Whittker, ET, Robinson, G: The Clculus of Observtions: A Tretise on Numericl Mthemtics, 4th edn., pp. 164-208. Dover, New York (1967) 17. Ndrjh, S: A generlized norml distribution. J. Appl. Stt. 32(7), 685-694 (2005) 18. Vrnsi, MK, Azhng, B: Prmetric generlized Gussin density estimtion. J. Acoust. Soc. Am. 86(4),1404-1415 (1989) 19. Hwkins, GE, Brown, SD, Steyvers, M, Wgenmkers, EJ: Context effects in multi-lterntive decision mking: empiricl dt nd Byesin model. Cogn. Sci. 36, 498-516 (2012) 20. Yng, CF, Pu, YJ: Byes nlysis of hierrchicl teching. Mth. Prct. Theory 34(9), 107-113 (2004) (in Chinese) 21. de Vlpine, P: Shred chllenges nd common ground for Byesin nd clssicl nlysis of hierrchicl models. Ecol. Appl. 19, 584-588 (2009) 22. Crlin, BP, Gelfnd, AE, Smith, AFM: Hierrchicl Byesin nlysis of chnge point problem. Appl. Stt. 41(2),389-405 (1992) 23. Wen, JJ, Hn, TY, Go, CB: Convergence tests on constnt Dirichlet series. Comput. Mth. Appl. 62(9),3472-3489 (2011) 24. Wen, JJ, Hn, TY, Cheng, SS: Inequlities involving Dresher vrince men. J. Inequl. Appl. 2013, 366 (2013) 25. Johnson, OT: Informtion Theory nd the Centrl Limit Theorem, p. 88. Imperil College Press, London (2004) 26. Wen, JJ, Zhng, ZH: Jensen type inequlities involving homogeneous polynomils. J. Inequl. Appl. 2010, Article ID 850215 (2010) 27. Wen, JJ, Cheng, SS: Optiml subliner inequlities involving geometric nd power mens. Mth. Bohem. 2009(2), 133-149 (2009) 28. Peĉrić, JE, Wen, JJ, Wng, WL, To, L: A generliztion of Mclurin s inequlities nd its pplictions. Mth. Inequl. Appl. 8(4),583-598 (2005) 29. Go, CB, Wen, JJ: Theory of surround system nd ssocited inequlities. Comput. Mth. Appl. 63, 1621-1640 (2012) 30. Wen, JJ, Wng, WL: Inequlities involving generlized interpoltion polynomil. Comput. Mth. Appl. 56(4), 1045-1058 (2008) 31. Wen, JJ, Wng, WL: Chebyshev type inequlities involving permnents nd their pplictions. Liner Algebr Appl. 422(1), 295-303 (2007) 32. Wen, JJ, Cheng, SS: Closed blls for interpolting qusi-polynomils. Comput. Appl. Mth. 30(3), 545-570 (2011) 33. Wen, JJ, Wu, SH, Go, CB: Shrp lower bounds involving circuit lyout system. J. Inequl. Appl. 2013, 592 (2013) 34. Wen, JJ, Wu, SH, Tin, YH: Minkowski-type inequlities involving Hrdy function nd symmetric functions. J. Inequl. Appl. 2014, 186 (2014) 10.1186/1029-242X-2014-339 Cite this rticle s: Wen et l.: Qusi-log concvity conjecture nd its pplictions in sttistics. Journl of Inequlities nd Applictions 2014, 2014:339