A Generalization of Student s t-distribution from the Viewpoint of Special Functions

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1 A Generalizaion of Suden s -disribuion fro he Viewpoin of Special Funcions WOLFRAM KOEPF and MOHAMMAD MASJED-JAMEI Deparen of Maheaics, Universiy of Kassel, Heinrich-Ple-Sr. 4, D-343 Kassel, Gerany Deparen of Maheaics, K.N.Toosi Universiy of Technology, Sayed Khandan, Jolfa Av., Tehran, Iran (Received ) Suden s -disribuion has found various applicaions in aheaical saisics. One of he ain properies of he -disribuion is o converge o he noral disribuion as he nuber of saples ends o infiniy. In his paper, by using a Cauchy inegral we inroduce a generalizaion of he -disribuion funcion wih four free paraeers and show ha i converges o he noral disribuion again. We provide a coprehensive reaen of aheaical properies of his new disribuion. Moreover, since he Fisher F- disribuion has a close relaionship wih he -disribuion, we also inroduce a generalizaion of he F- disribuion and prove ha i converges o he chi-square disribuion as he nuber of saples ends o infiniy. Finally soe paricular sub-cases of hese disribuions are considered. Keywords: Probabiliy disribuions, Cauchy inegral, Doinaed convergence heore, Pearson disribuion faily, Suden s -disribuion, Fisher F-disribuion, Noral disribuion, Gaa disribuion, Chi- Square disribuion MSC : 6E5, 6E, 33C45. Inroducion Le us sar our discussion wih he Pearson differenial equaion dw dx dx + e W ( x), () ax + bx + c which is iniaely conneced wih classical orhogonal polynoials and defines heir weigh funcions ( ) W x, see e.g. [6], and is soluion d e dx + e W ( x) W x exp( dx), () a b c ax + bx + c where a, b, c, d, e are all real paraeers. There are several special sub-cases of (). One of he is he Bea disribuion, which is usually represened by he inegral [7] E-ail: oepf@aheai.uni-assel.de E-ail: jaei@yahoo.co

2 where L () and L () are linear funcions, b C a b ( L ( )) ( L ( )) d, (3) a, are coplex nubers and C is an appropriae conour. The Euler and Cauchy inegrals [] are wo iporan sub-classes of Bea ype inegrals which are ofen used in applied aheaics. The Euler inegral is given by b c d Γ( c) Γ( d) c+ d ( a) ( b) d ( a+ b) (Rec>, Red >, a>, b> ) a Γ ( c+ d), (4) while he Cauchy inegral is represened by he forula d Γ( c + d ) ( c+ d ) ( a + b) c d π ( a i) ( b i) Γ( c) Γ( d) +, (5) in which i, Re( c+ d) >, Rea> and Reb >. Noe ha in boh relaions (4) and (5) Γ a x ( a) x e dx denoes he Gaa funcion. The relaion (5) is a suiable ool o copue soe differen looing definie inegrals. For his purpose, we use he relaion iq a ib b exp(qarcan ) ( a, b, q R ) (6) a+ ib a which rewries he coplex lef hand side in ers of he real righ hand side. Consequenly p+ iq p iq p ( ) ( ) ( ) exp( arcan ) b i b+ i b + q. (7) b Now if (7) is subsiued, hen he inegral (5) changes owards p Γ p p+ ( b + ) exp(qarcan ) d ( b). (8) b p iq p iq Γ + Γ ( ) π ( ) ( ) The above inegral plays a ey role o inroduce a generalizaion of he -disribuion.. A generalizaion of he -disribuion The Suden -disribuion [8,9] having he probabiliy densiy funcion (pdf) + Γ (( + ) / ) T (, ) ( + ) ( < <, N ) (9) π Γ( /)

3 is perhaps one of he os iporan disribuions in he sapling probles of noral populaions. According o a heore in aheaical saisics, if X and S are respecively he ean value and variance of a sochasic saple wih he size of a noral populaion having he expeced value μ and variance σ, hen he rando variable T has he X μ S/ probabiliy densiy funcion (9) wih ( ) degrees of freedo [9]. This heore is used in he es of hypoheses and inerval esiaion heory when he size of he saple is sall, for insance less han 3. Now, by using (8) one can exend he pdf of he -disribuion. To ee his goal, we subsiue, b, p and q in (8). This yields + q + Γ ( ) π ( + ) exp( qarcan ) d + + iq + iq. () Γ( ) Γ( ) Since he righ hand side of () is an even funcion wih respec o he variable q, we ae a linear cobinaion and ge accordingly + ( + ) λexp( qarcan ) + λexp( qarcan ) d ( λ + λ ) Γ( ) π. () + + iq + iq Γ( ) Γ( ) Therefore, he above inegral can be used o generalize (9) by + + iq + iq Γ( ) Γ( ) Tq (,,, λλ, ) ( + ) λexp( qarcan ) + λexp( qarcan ) + Γ + ( λ λ) ( ) π () where < <, N, q is a coplex nuber and λ, λ. Noe ha λ, λ is a necessary condiion, because he probabiliy densiy funcion us always be posiive. Also noe ha he noralizaion consan + + iq + iq Γ( ) Γ( ) λ λ π ( + ) Γ( ) of () is real, because he corresponding inegrand is a real funcion on (, ). I is clear ha for q in () he usual -disribuion is derived. Moreover, for q he noralizaion consan of disribuion () is equal o he noralizaion consan of he -disribuion. This fac can be proved by applying Legendre s duplicaion forula [], i.e. 3

4 z z + π Γ( ) Γ( ) Γ( z). (3) z Bu, according o one of he basic heores in sapling heory, T (, ) converges o he pdf of he sandard noral disribuion N (,,) as [7,9], ha is li T (, ) N (,,). (4) Here we inend o show ha his aer is also valid for he generalized disribuion T,, q, λ, λ ). To prove his clai, we use he doinaed convergence heore (DCT) [] ( o he real sequence of funcions + ( ) () S (, q, λ, λ) ( + ) λexp( qarcan ) λexp( qarcan ) + (5) For every N i is no difficul o see ha () π S (, q, λ, λ) ( λ+ λ)exp( q ) ( R ). (6) On he oher hand, we have + li ( + ) ( λ exp( q arcan ) + λ exp( q arcan )) ( λ+ λ) exp( ). (7) Since he doinaed convergence heore saes ha if for a coninuous and inegrable funcion g (x) we have f ( x) g( x), hen considering he lii relaion (7) we obain li Tq (,,, λ, λ ) + + ( λ+ λ)exp( /) ( λ + λ )exp( / ) b b li f ( xdx ) li f ( xdx ), (8) a a ( λ λ ) ( λ λ ) li( + / ) exp( qarcan( / )) + exp( qarcan( / )) li( + / ) exp( qarcan( / )) + exp( qarcan( / )) d (9) exp( ) N(,,). π d, he following asypoic rela- Rear. Taing he lii on boh sides of () as ion is obained for he Gaa funcion 4

5 Γ ( x+ iy) Γ( x iy). () x x π li x (x ) Γ ( ) To copue he expeced value of he disribuion given by he pdf () i is sufficien o consider he definie inegral + Γ π q ( ) ( + ) exp( qarcan ) d ( ), () + + iq + iq Γ( ) Γ( ) which gives he expeced value of () as λ λ q E [ T] ( ) λ + λ. () On he oher hand, since E [ + T / ] can be easily copued, afer soe calculaions, we ge for he variance easure of () q ( + ) λ λ q Var [ T ] E [ T ] E [ T ] ( ). (3) ( )( ) λ λ + ( ) I is valuable o poin ou ha as expeced q in () and (3) gives he expeced value and variance of he usual -disribuion, respecively. I is nown ha he -disribuion has a close relaionship wih he Fisher F-disribuion [6], defined by is pdf / + Γ (( + ) / )( / ) F( x,, ) x ( + x) (, N, < x<), (4) Γ( /) Γ( /) where x and in (4). In oher words we have T (, ) F(,,). (5) By referring o he above relaion and he fac ha he -disribuion was generalized by relaion (), i is now naural o generalize he pdf of he F-disribuion (4) as follows + F( x,,, q, λ, λ ) Bx ( + x) ( λexp( qarcan x) + λ exp( qarcan x)), (6) where 5

6 + x ( x) ( λexp( qarcan x) λ exp( qarcan x)) dx. (6.) + + B For q, (6) is he usual F-disribuion defined in (4). According o he following heore, he generalized funcion (6) converges o a special case of he Gaa disribuion [9], defined by β x Gxα β x α β > < x< Γ( α) β α α (,, ) exp( ) (,, ). (7) Theore. If he Gaa disribuion is given by (7), hen we have li F( x,,, q, λ, λ ) G( x, α, β ) χ where χ denoes he pdf of he chi-square disribuion. Proof. Le us define he sequence + ( ) () S ( x,, q, λ, λ ) x ( + x) ( λexp( qarcan x) + λ exp( qarcan x)). I is easy o show ha () π S ( x,, q, λ, λ ) ( λ+ λ ) x exp( q ) ( x [, ), N ), (8) and () λ λ λ λ li S ( x,, q,, ) ( + ) x exp( x/). (9) Therefore, according o he DCT we have S x q x x li Fxq (,,,, λ, λ) Gx (,,). (3) S x q dx x x dx () li (,,, ( /), ) λ λ exp( / ) () ( /) li (,,, λ, λ) exp( / ) Moreover, i is no difficul o show ha F,,, q, λ, λ ) T(,, q, λ, λ ). (3) ( 6

7 3. Soe paricular sub-cases of he generalized (and F) disribuion In his secion, we are going o sudy soe syeric and asyeric sub-cases of he generalized disribuions () and (6). 3.. A syeric generalizaion of he -disribuion, he case q ib If he special case q ib and λ λ / is considered in (), hen and λ λ / + + b + b Γ( ) Γ( ) + ( ) Tib (,,,, ) TS (, b, ) ( + ) cos( barcan ) Γ( ) π (3) is a syeric generalizaion of he ordinary -disribuion in which b. The usual pdf of he -disribuion is obviously derived by b in (3). Noe ha according o he Legendre duplicaion forula we reach he noralizaion consan of he -disribuion if b is considered in (3). In oher words, we have b Γ (( + ) / ) Γ( ) π Γ(( + ) / ). (33) π Γ( / ) Also noe ha he paraeer b in he generalized disribuion (3) us belong o [-,], because he probabiliy densiy funcion us always be posiive and herefore we ough o have π π cos( barcan( / )). On he oher hand, since for θ we have cosθ, herefore o prove cos( barcan( / )) i is sufficien o prove ha π π b barcan [, ] ( R, N ). (34) For his purpose, le us consider he sequence U( ) arcan. We have π π U ( ) ( ) /( + ) > [in U( ), ax U( )] [ U( ), U( )] [, ]. (35) π π Now if we deand he sequence bu( ) barcan o belong o [, ], i is clear ha we us have b, which proves (34). The following figures clarify his aer for b [,] and b [, ] in he inerval (-,). 7

8 Figure : b /, 4 Figure : b 3, 4 Fig. shows he pdf T S (, 4,/ ) wih noralizaion consan 35 / 8 and Fig. shows 5/ he non-posiive funcion TS (,4,3) (4 / π )( + / 4) cos(3arcan( / )) in he inerval (,). As he above figures show, he generalized disribuion (3) is syeric, i.e. T( b,, ) T( b,, ) ( R ). (36) S S Moreover, according o () and (3) he expeced value and variance of disribuion (3) ae he fors ( b ) E [ ], Var[ ]. (37) ( )( ) Clearly b in hese relaions gives he expeced value and variance of he -disribuion. Theore. T S (,, q) converges o N (,,) as. + ( ) (3) Proof. If he sequence S (, b) cos( barcan )( + ) is considered, hen one can show ha + ( ) (3) S (, b) cos( barcan ) ( + ) ( R ). (38) Consequenly we have + cos( barcan( / ))( + / ) li TS (,, b) li + cos( barcan( / ))( + / ) d + li cos( barcan( / ))( + / ) exp( / ) + li cos( barcan( / ))( + / ) d exp( / ) d 8 N (,,).. (39)

9 By referring o (6), we can now define he generalized F-disribuion corresponding o he firs given sub-case as follows + F( xib,,,,, ) F ( xb,,, ) Bx ( + x) cos( barcan x) ( b ) (4) where + x ( x) cos( barcan x) dx B + π / / ( ) ( ) ( ) sin θ cos θ cos( bθ) dθ. (4.) Theore 3. ( x,,, b) converges o he chi-square disribuion as. F + ( ) (4) Proof. We define he sequence S ( x,, q) x ( + x) cos( barcan x) o ge (4) S ( x,, b) x ( x [, ), N, b < ). (4) Hence, according o DCT we find ou ha S ( x,, b) x exp( x/ ) li F ( x,,, b) li G( x,, ). (4) S ( x,, b) dx x exp( x /) dx (4) ( /) (4) ( /) I is no difficul o verify ha he generalized disribuions T S (,, b) and F ( x,,, b) are relaed wih each oher as follows F (,,, b) TS (,, b). (43) Rear : Here is a good posiion o enion ha in [4] a class of orhogonal polynoials is sudied, whose weigh funcion [3] corresponds o he ordinary -disribuion. The relaed polynoials are defined as n / ( p) p n n In ( x) n! ( ) ( x) n. (44) and saisfy he orhogonaliy relaion n ( p ) ( p) ( p) n! π Γ ( p) Γ(p n) ( + x ) I n ( x) I ( x) dx ( ) δ n, (44.) ( p n ) Γ( p n) Γ( p n + / ) Γ(p n ) if n where, n,,,..., N < p and δ n,. if n 9

10 For n in (44.) an inegral is derived ha corresponds o he -disribuion. Furherore, he enioned coen holds for he F-disribuion. In [4], a sequence of orhogonal polynoials is sudied which is defined by n ( p, q) n p ( n+ ) q+ n M n ( x) ( ) n! ( x) n, (45) and saisfies he orhogonaliy relaion x ( + x) p where, n,,,..., N <, q >. n!( p ( n + ))!( q + n)! ( p (n ))( p q ( n ))! δ q ( p, q) ( p, q) M p q n ( x) M ( x) dx + n, (46) Clearly he weigh funcion of inegral (46) corresponds o he usual F-disribuion in he case n. 3.. An asyeric generalizaion of he -disribuion, he case λ Fro he orhogonaliy relaions (44.) and (46) i can be concluded ha he caegory of Pearson disribuions should have a relaed class of orhogonal polynoials. In [5], a class of orhogonal polynoials is sudied, whose weigh funcion is a specific case of () and is represened by ( pq, ) p ax + b Wn ( xabcd ;,,, ) (( ax+ b) + ( cx+ d) ) exp( qarcan ) ( < x<), (47) cx + d where a, b, c, d, p, q are all real paraeers. This funcion is a sub-case of he Pearson disribuion (), because he logarihic derivaive of (47) is a raional funcion. Hence, (47) is a special case of he Pearson disribuion faily. For convenience, we chose a paricular subcase of (47) in [5] o generalize he usual -disribuion. If a, b, c, d and + p ( N ) is seleced in (47) one ges + + (, q) W ( ;,,,) ( + ) exp( qarcan ) ( N, q R ). (48) Since + π / qθ ( ) ( + ) exp( qarcan ) d e cos θ dθ, (49) π / we have

11 π / π / e qθ cos ( ) θ dθ + ( ) q q e e qπ qπ ( )!( ( )) ( + ( ) ) ( )/ ( + ( ) ) q, (5) hence an asyeric generalizaion of he -disribuion ay be defined as + TA(,, q) K ( + ) exp( qarcan ) ( < <, N, q R ) (5) such ha K ( )/ ( q + ( ) ) ( )!( qπ qπ + ( ) ( )( )) ( ( ) ) q q e + e. (5.) The disribuion (5) wih noralizaion consan given by (5.) was defined in [5] based on his paricular approach. Bu here we can odify and siplify i. For his as, we se λ in (), and ge + + iq + iq Γ( ) Γ( ) + ( ) TA(,, q) ( + ) exp( qarcan ). (5) Γ( ) π This is in fac an explici represenaion of he asyeric generalizaion of he -disribuion. For his disribuion, we clearly have T (,, q) T (,, q). (5.) The asyery is also shown by Fig. 3 and 4. A A Figure 3: q, 4 Figure 4: q, 3 According o (5) and (5.), he explici definiions of he wo enioned figures have respecively he fors

12 Fig. 3: Fig. 4: 5 TA(,4,) ( + ) e 6 cosh( π / ) TA(,3,) ( + ) sinh( π / ) 3 5 arcan arcan 3 e The following saeens (A o A5) collec he properies of he asyeric disribuion (5). A) The expeced value and variance of (5) are respecively represened by q ( q + ( ) ) E [ ], Var[ ], (53) ( )( ) q in hese relaions gives he expeced value and variance of he -disribuion. A) T A (,, q) converges o N (,,) as. The proof is siilar o he firs case if one chooses λ and λ in he defined sequence () S, q, λ, λ ). ( A3) By he definiion (6) and considering he case λ we can define + F( x,,, q, λ,) F( x,,, q) D x ( + x) exp( qarcan x) ( q R,, N, < x<), where (54) + π / / ( ) ( ) qθ x ( + x) exp( qarcan x) dx ( ) sin θ cos θ e dθ D. (54.) A4) F ( x,,, ) converges o he chi-square disribuion as. q The proof is siilar o he proof of Theore when λ and λ. A5) The disribuions F ( x,,, ) and (,, q) are relaed o each oher by q T A F (,,, q) T (,, ). (55) A q

13 Rear 3: There is anoher syeric generalizaion of he -disribuion when we se λ in (). Is pdf is given as λ + + iq + iq Γ( ) Γ( ) + ( ) T( q,,, λ, λ) Tq (,,, λ, λ) ( + ) cosh( qarcan ). (56) Γ( ) π Therefore, o suarize he las secion we in fac considered he hree following paricular sub-cases of he general disribuion (), a) q ib and λ λ / ; syeric case b) λ ; asyeric case c) λ λ ; syeric case References [] Arfen, G., 985, Maheaical Mehods for Physiciss. (New Yor: Acadeic Press) [] de Barra, G., 98, Measure Theory and Inegraion. (New Yor: Ellis Horwood) [3] Chihara, S. A., 978, An Inroducion o Orhogonal Polynoials. (New Yor: Gordon and Breach) [4] Masjed-Jaei, M.,, Three finie classes of hypergeoeric orhogonal polynoials and heir applicaion in funcions approxiaion, J. Inegral Transfors and Special Funcions, 3, [5] Masjed-Jaei, M., 4, Classical orhogonal polynoials wih weigh funcion p ax + b (( ax + b) + ( cx + d) ) exp( q arcan ) ; x (, ) and a generalizaion of T and F cx + d disribuions, J. Inegral Transfors and Special Funcions, 5, [6] Niiforov, A. F. and Uvarov, V. B., 988, Special Funcions of Maheaical Physics (Basel-Boson: Birhäuser) [7] Press, W. H., Flannery, B. P., Teuolsy S. A. and Veerling, W. T., 99, Bea Funcion, T Suden disribuion and F-Disribuion. In: Nuerical Recipes in Forran: The Ar of Scienific Copuing. (Cabridge: Cabridge Universiy Press), Second ediion, Secion 6., 9-3 [8] Suden, 98, The probable error of a ean. Bioeria, 6, -5 [9] Walpole, R. E. and Freund, J. E., 98, Maheaical Saisics. (Englewood Cliffs, NJ: Prenice Hall) 3