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- Unpaywall
- ISSN
- 0091-0635
- DOI
- 10.6028/jres.053.047
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Abstract
--- -- ----- Journal of Research of the Nationa l Bureau of Standards Vol. 53, No. 6, December 1954 Research Paper 2556 Bounds on a Distribution Function That Are Functions of Moments to Order Four Marvin Zelen Explicit ex pressions are presented [or bound s on a distribution function wh e n mom ents to order rolll' are known. Thes e inequali t ies are given in a f o rm s uitabl e [or applicat ions . 1. Introduction and Statement of Problem Tch ebych en' [7] 2 in J874 propose d a problem t 11at can b e stated as follows: Let F (y ) be an unknO\\'n di stribution f unction oyer t he clo se d inter val 3 [a, b], and satis fying th e conditions F (CL - O) = 0 (j = o, J , ... , k). If t h e moments m j for j = O,I , ... , k arc known , th en for a giv en value of x, (a < x < b) , wh at are t h e (s harp ) llpp er ancllow cl' hounds on F (x )? T ch ebych cA' presen te d without proo f a solution to t h e above problem , which is som et imes call ed t he reduced-moment problem. Proofs were l ate!" given by ;'1arko ff [ I ], Posse [2], and Stielt j es [5,6]. ' 1'1 1(' book by Sh ohat fLnd TfLluarkin [4 ] gives an. acco un t of some of th e moclern day treatments of t h e subje ct . T his paper presents t he ex pli cit express ions for solutions of t h e moment problem (often referred to as t h e T ch ebychcfT"-1[arkoA' inequ alit ies) for t h e cases k = 2,3,4. Inequalities t h at a r c functions of moments to order two were give n b y Tcheb ych eA' [7] for distributions over th e interval [O,b]. Inequa li t ies th at a re functions of moments to order thr ee IVere g iv en b y Po sse [2]. P osse also solved th e case of four moments for di stribu t ions over t h e interval [a , ro). 2. Explicit Expressions for Bounds This section pr es ents without p roo f th e expli cit expressions for bounds on a di stribu tion function. Proofs may b e found in [9]. These are derived as special cases of th e T ch ebych eff Markoff inequaliti es . In all that follows it will be assumed tha t ( 1) Th is willl" esult in no lo ss of generalit y, as any distribution fun ct ion can be mad e to conform to t h ese conditions by the li se of a linear t ransform ation . The assumption (1) implies t h at a, b satisfy the inequaliti es (2) This fo llows from t he necessary condi t ions for the solu t ion of t h e momen t problem (d. Shohat and Tamarkin [4]). I A condensation of certain res ul t s obtained by the author in a thesis submitted to the University of North Caroli na in June 1951 in pa rti al fulfillment of th e requ irements for t110 Mas ter of Arts d eg ree. 2 Fig ures in brac kets indicate the li terature references at the end of t hi s paper. 3 Throughout thi s paper it will be understood that a di s tr ibu t ion func tion over the inte rval [ao, b] is one where the range of the random variabl~ is [a, b l. and the end point s will belong to that inter val unl ess the end points are - co or + co. 377 2 .1. Bounds for Two Moments Let F (y ) be a distrib ution function on [a, b] wi t h known moments t h en for a giy en x, (a < x < b) , . 1 If a<x<-~ (3) - b l + bx < F (x) <1- l + ax (4) (a- b) (a - x ) - - (b- a)(b- x) --<F(x) < 1 (5) 1 + x - - For any- distribution defined over (- 00, (0) inequalities (3) and (5 ) hold for x< O and x> O, respectively. 2 .2 . Bounds for Three Moments Let F (y ) be a distribution function on [a, b] with known moments m3-(a + b) w= l + ab l + ex{3 A ( ex,{3,'Y) = (-y _ ex ) ('Y - (3) ' then for a given x, (a < x < b) (6) ° ~F(x) ::::;A( b, Z2, x ) if x< O, g(x ) ;::: ° if g(x) ~ O , x::::;w (7) A(x, b, Z2) ~F(x) ~A(x, b, Z2) + A(b , Z2 , x) if g(x ) ~O , x;:::w (8) 1 - A (a , Zl , x) ~F(x) ::::; l if g(x) ;::: 0, x> 0. (9) Inequalities (7) and (9) hold for any distribution F(y ) on [a , (0) . Inequaliti es (6) and (8) hold for any distribution F (y ) on (- 00, b]. Note t h at none of t h e inequalit i es (6) to (9) holds for distributions over (- 00, (0). 2 .3 . Bounds for Four Moments L et F(y ) be a d istribution funct ion on [a, b] with moments Let m m m m c( )= 1 2+ [ 'Y - 3+ 'Y ('Y 3-'"!'.4) ] 1 + ['Y 3- 4+(m - 'Y )2]. 9 y ,'Y Y 1 + 'Y (m3- 'Y) . Y 1 + 'Y(m3- 'Y) l_ ~ U (y ) = g(y ,a ), V(y )= g(y , b), Z (y )= g(y , x), and let Ul<U2, Vl<V2, ZI<Z2 b e th e di tinct zeros of U( y ), V (y ), Z(y ) , respectively, t b en a <vl < ul <,U2<U2< b. D efine m 3 (a + b + x) - m 4 - ab - ax- bx Z3 abx + a + b+ x - m 3 A m4 - m N- l ( 1 + x ) (m4-m~- 1 ) + (x2 - m3x - 1)2 m 3- (a + (3 +z3) - a(3 z3 B ( a ,(3 ,'Y ) ('Y- ) ('Y - (3 ) ('Y - 3) a Z then for a given yalue of x, (a <x< b), O ~F(x) ~A (10) if a<x~vl' ( 11 ) B (b,x,a ) ~ F(x) ~B (b ,x,a) + B (a ,b,x) if Vl~X~ Ul x z 1 + 2 < F (x) < l +xz2 +.11 (12) (ZI-X) (ZI- Z2) - - (ZI-X)(ZI- Z2) I - B (a,b ,x) - B (a ,x, b) ~ F(x) ~ I - B (a ,x, b) (13) l -A ~ F(x) ~ l (14) For any distribution defined over (- en, en) inequalities (10), (12), and (14) h old , r esp ec tively, for X<ZI' ZI<X<Z2 , Z2<X . How ever, the ordering of x in relat ion to ZI, Z2 is equiv alent to t h e followin g. L et g (x) =x2- m 3x- 1, t h en x> o , g (x» O if , and only if , Z2<X. ( 15) (16) if , and only if, ZI>X x< O, g (x» ° (17) g(x)< O Us ing (15 ) to ( 17) , t h e applications of tb e T ch ebych efI-MarkoiI inequ alities for t h e case wher e F (y ) is defin ed over (- en, OJ) are mad e part icularly easy. 3 . Application of the Tchebycheff-Markoff Inequalities L et F (y ) b e a distribution function whose fir t foUl' moments coincid e with t ho e of the standard normal distribution , i. e., mo= l , m l= O, m2= 1, m 3= 0 , m 4=3. The T ch eb ycheff Markoff inequalities will b e used to find bounds for F(x ) when x = 2, 3. Bounds using two moments: Since x> 0, inequality (5) is applicable, and we have .8000 ~ F (2) ~ 1 .9000 ~ F(3) ~ 1. B ounds using jour moments : Since X> O, 9(2» 0, 9(3» 0, inequality (14) is applicable. ubstituting t h e appropriate values, we h ave t h erefore, .8947 ~ F(2 ) ~ 1 .9777 ~F(3) ~ 1. Note that th ere are no inequ aliti es applicabl e using only moments to order tluee . 379 ----- - 4. Appendix Statements and proofs of t h e Tchebycheff-Markoff inequaliti es can be fou n d in Shoh at and T amarkin [4], Uspensky [8 ], and Ro yden [3] . This section con tains a statem en t of t h e Tchebycheff-Markoff inequalities as the above sources do no t give t h e theorem in full gen erality , and it is not readily available in the literature. B efor e stati ng the t h eorem i t will be con venient to define th e following: Let T n(Y), U n( y ) , I1n(Y ) , W n(y ) b e polynomials of d egree n defined by (18) i T n(Y) 8n - 1 (y )dF(y ) = 0 i U (y )8 _ (y )(y - a )dF(y ) = 0 (19 ) n n l (2 0) f: I1n(y)8 n- l (y ) (b - y )dF(y ) = 0 i b W n(y )8n_ 1 (y ) (y - a) (b- y )dF(y) = 0 , (2 1) where en- ley) is any polynomial of d egr ee ::::;n - l , and the coefficient for y n in T n(Y) , Un( y ) Vn(Y ) , W n(y ) is unity. Tch ebychejJ-jt1ark ojJ In equali ties: Let F(y) b e any distribution function on [a,b] with moments mo, ml , .. . ,m k mj= i y idF(y) (j = O,I , .. . , k ), and let x b e a given number (a < x< b), then wher e p( z) = ( b q( z) - q(y) dF (y ), Ja z- y and YI < Y2< ... < x < .. . are th e zeros of the polynomial q(y) of degTee l' defined b y q(y ) = (y - x)w( y ) if k = 2n, Un (x) I1n(x) > 0, (22) q(y ) = (y - b) (y - a) (y -x)w(y ) if k = 2n , Un (x) I1n(x) < 0, (23) q(y) = (y - a ) (y -x)w(y ) if k = 2n - l , T n (x) W n_l(x» 0 , (24 ) q(y) = (y - b) (y -x)w(y ) if k = 2n - l , T n (x) W n_1 (x) < 0, (25} where r = n + 2 for eq (23), r = n + 1 for (22), (24 ), (25 ), and w(y) is determin ed b y i = O, 1, . . , n - l for (22) i= O, 1,. ., n-2 for (23), (24), (25 ) . COROLLARY : For eq (22) the i n equali ties hold jar any distribution over ( - 00 , CD) with moments m o, ml,. . ., m 2n. Th e i nequali ties jar eq (24) hold JOT any distributi on oveT [a, 0) with moments mo, m l , .. . ,m2n- !. Th e inequali ties JOT eq (25) hold JOT any distributi on over (- ro, b] with. moments m o, m l, . .. , m2n- !. 380 I wish to cxpress m y t hanks to Professor Was sily Ho effdin g , Univcrsity of S O l'Lh Carolina , for hi s invaluable guid an ce during t h e preparation of t his work. 5 . References [1) A . Markoff, D emo nstration d e cer taine s in ega li ti 6s d e ~ . Tchebycheff, Math . Ann . XXIV, 172- J 80 (18 4) . [2) C . P osse, S ur quelqu es app li ca t ion s des fraction s continues a lgeb riq ues (Acade mi c Im p eria le d es Sciences, St. P eters burg, 1886). [3) H . L. Royd e n, Bound s on a di st ri bution function \\"he ll its .fir st n mom e nt s arc g il'en , A nn . :V[ath . Stat. 24, 361- 376 (1953). [4) J . A. Shohat and J . D . Tama rkin. The problems of moments (Am . :V[ath . Soc. , NelV York , 1943). [5) T . J . Stieltjes, Sur l'el'aluation a pp roc hee des integra les, Compt. re nd XCVII, 740- 742 , 79 799 (1883 ) . [6) T . J . Stieltj es, R eche r ches s ur les fraction s co ntinu es, Ann . facu lte scicnce s Tou lou se VIII, 1- 22 (1894); IX, 45- 4 7 (1895 ) . [7) P . T chebycheff, Sur les v a le u rs limites des integra les, J . de Math. [2) XIX, 157- 160 (18 74 ). [8) J . V . U spen s ky, In t rod uct io n to mathematical p rob ab ili t.l' ( YlcGra"'-Hill Book Co ., New York , N. Y. , 1937 ) . [9) M . Zelen, B o und s o n a di stribution function which are fun ction s of moments, unpubli s h ed mas te rs th es is (Un i l'er s it.1' of ~orth Car olina Librar y , 195 1). W ASHINGTON, Noyember 20 , 1953.
Journal
Journal of research of the National Bureau of Standards – Unpaywall
Published: Dec 1, 1954