ρがゼロでないときのγの分布について
8
0
0
全文
(2) Vol. 18, No. 2 Journal of Hokkaido University of Education (Section D: A) March, 1968. On the Distribution of r When p is not Zero Masahiro OB A The Study of Mathematics, Iwamizawa Branch, Hokkaido University of Education. ^^WX : p ^-^'P ^'/^^^ toro^^pr'o^-c. Although the exact distribution of r for samples from a correlated parent population was found by C43 Fisher, using a geometrical method, this method is very difficult. Sawkins has given us its analitical treatment. But this was a special case, where the means of variates were zero. The following states the most general case, on the supposition that two variates are normal and linear, giving an orthogonal linear transformation.. The linear and most general probability density function is given by the following 1 ^ ^x-^__ ^x-Wy - ^~)_ _^ ^y-^T. ^,y")=——x, „ ^ 2(1-p2) I rfi2 ^2 ' ^2. V1" 2^i^/T^re. Writing ;; = a;—ri^ u= ^~ct . the joint frequency distribution of t and % is a\. '. al. '. l—,—{.i'i-2Ptu+us}. ^ u)=—J- ^ ^ 2C1-P2) "' ^J 2^i/T^or e. i _ - ^_^ {^-po2+a-w}. 2V 1-^. Hence, if u= /\ r\ the joint distribution of t and v is given by 1/ l-pi. ^__p_. Kt- ^-^^^=^=e~^-7^e~^ n. t and v are therefore independent normal standard variates, and ^uj is distributed as i=]. X2 with n degrees of freedom. By the following transformation from the v, to new variates Zi,. 1. 1/^~ V, =. 1. Z1+. 1. I •2 1. 7T21 - 1 •2. ILz.. +v. ~n^. J^.+ +v ~3-I^. •z;+\ "2^3-zs. zz+/\. '^3zi. 73. •i. 3-4 '. 1_... (ra-l>. (ra-l>.
(3) Masahiro Oba. ^ „ ^=2, - 2^/-JLZ3 + J^—^ +......+ J_^_^^Z, y3 ~V~n,~' "V 9..-=i~3 ' V ?i.4~4 ' ' V (n--\^n~r' y4 = /-—Z1 ~ <^A/ ^ . Z4 -t- •••••• -t- ^/ ^—"_^__Z^. I/. n. '. "'V. 3.4. ". 'V. (re-l)ra. Vn = -L=Z, - (n - lV-, 1,, ^, I/ n v ( ra — i )n n. n. we have ^uj= Szt- ^ course, Zs. are normal standard variates, and ^zj is distributed 1=1. t=]. ^=1. as X2 with n degrees of freedom, and this transformation is an orthogonal linear transformation. n_. Choosing z^ =—J=Svt> we have 1/ n ^ _»__ 71.-nt. 1 Sui-i°S^. v ui,~ ptt i ?^i ~ ' ^T '~~^—t /—T- —5— == 7=— n i~=\ \/ 1- pi ,y/ n y \—pi. 1 nu — pnt -i / n (u — pt) V~n, V 1-p7- ~ V 1-p2 '. and n _ TO , n ,,2_On,,.^._|_ ^2 1 , _»»^ , ra_ _»» .2 _-Sr',,Z _'sT' "». ~ ^fju'il/i ~T~ ^ t'i i (V ,,Z _ o^^.sT'fl/y- J_ ^V^/L2(. 'A =2-^ =2-i—^—"^ 'Y r " =-^ ^ ^ ^2-iut--^io2-,^?t+io".2-;r^ ^ ~ t^ " f^\ l~Pi l—^'i^T" -1 ^T ~ " 1 f^T n. n. n. ^ EC^ - ^)2 - 2^SC^ - u)0i - 0 + P'SO. - D2 — p'- T=\ ~ ~ ' i=\ ~ ~ ~ ~ ' i=] +n(ut-2put+ptt1')}. J-^EC^-^)z-2^SC^-^)Oz-D+^±0.-02+<^-^. l—pi 't^'" "' " "' t^}'~ " "~ " " ' ^1'. -l-^W-2prS,S^p^+z,i, p". n. SC^-UX^-D. where we put S^ = ^^ - u)2, S,1 = ^t,-t-)2, r = -^—„-„-—. •t=1. (=1. -. »^J]. Therefore we have. S>t = -T^T ^2Z - 2prS^ + p7S^ ,. ^2 ~ i~P". and this is distributed as %2 with n—\ degrees of freedom.. SO<-DU. Choosing z^ =-i^t — ^ —^ we have '1. n. n. _. __.__.. SO, - DCU< - ^0 SC?< -D C^ - <°^) - (.nt - nt~) Cu - pO. zz =—^wT^7=-5,7^^7 74 —.
(4) On the Distribution of r When p is not Zero. n. -. n. n. ^t,-t\u,-pt^-^t.,-t\u-pt') SO.-D \^-pt^-(u-pt~)\ "5,. SG(-DKUr. -WT 7". V~t^ n. n. sc^:-DCUi-u). 1=1. ~^~T-^pr. ~s7. ^ l/^or. <=1. ~S^=1-Z7-. 1. n. so. :-ocu. -")-pSGr ~tJ. ~^-p(ti-t~)\. n. sc^--D2. -P —5T •i=]. 1 ^. -7T7"'Ji. -pS,\.. Therefore we have ,2 __rlSJ-2rpS^+ 1-<°2. p^. and n. n. .?=^Vz_^__JL_(c2_o^<? ^- ^ozt _'^Si~2rpSjS]+ p2Sfj Si(l-r'i^). Jzi = ^z'i ~ Z2 = l _„? i6'2 — ^|0^2>'> 1 + ^' IS — —'—1' _\t —'—~ = -1 _„? " •. T^s. T^i. ~. i.—P". "". '. -. •. .. .-. i.~P'. 1—^cr. This is distributed as X2 with n-2 degrees of freedom. n. Moreover, as 5'f = SC^ — O2 ls independently distributed as X2 with n—\ degrees 1=). of freedom, if we name a== z,.. _rS,-pS^. 1-p. »-is ^_ sja-o. 2^~i 2(1 -p2~) '. c-ilp.-oi-^, a, b, and c are respectively independent, a is A^(0, 1) distribution, 26 is distributed as X2 with %-2 degrees of freedom, and 2c is distributed as X2 with %-1 degrees of freedom. Using the above relations, we have. /C", b, c)=/00/CQ/Cc)^00-2/C26>2/C2c). As ^-2. _i. _. 26. -,. ^-4. fW = 2- ^2 ' —C26) 2 'e 2 ^—^—6 2. 2^rln^2} r{n-^\ ~^] . '. e. ~-2} F2~. n-1. _-,. 2c. ^. n-3. KC-) -2 • ^rTJ——C2c)-2--l e~^-=—L—c~^e~c. 2^r(-B^1) ~ ' r("^). we have. — 75 —.
(5) Masahiro Oba. n-3 -c. n-4: -b. Ka, b, c)=-4 V 2'T. 2 e .. " rf"^2V. ra-1. r. 2). ^+t+c)t sia-r^ \^s sn^ Xi^Q ^ l~^-<. ~c 2 e. /^-rln~1} v/2^(!. r(n~2}'. On the other hand, the relations. -°L^^_ ?-^o2, 5z2a-^ , 5?a-^) 5z+5j-2^^ ~^+b+c= \(l1^ +-2CT-^) + ~2C1-/) =~~' 2C1--^)L-' %—4. fsia-rDi^ i ^L^ „ s,'-3s,"-4a-.o^. \~2(l-p^i \~Ti ~ 2n^7- . ^n=±. 2 2 a-^). are constant, so we have w-4. Si2+S22-2iorSi52. /(a, b, c) =e 2(1-,2). ^—^-Q ^-3 c ^-4. (l-rz) ^ S^ " " St n-4: 2n-7. a-,o-^2^v/-2Tr(n^)r('^2). Next, we must find a probability density function f(r, Si, S^. That is. /(?-, 5,, S^^f^a, b, c). jfa, b, c. s,sl. -/(a, b, c>. [r, S^, S.,. _Si2+Ssi-2prS^. 2(1-<o2). a-^r2 w—4. a-'-o 2 5,. n-2 ^ n-2. n—1. a_p^,/T2"-3rf'^l)r("^2 because of (a, b, c \r, S^, ^/. 1. -f^2- .V~T^pr -s/T^ •rS^. 5,0-rO. l-pi. l-pi. 0. 0. ^2 a-^o. ^ r^-rSpS.-S^g-r^S, __ -S,Sj_. 3. 3. a-p9)2 a-^)2 — 76 —. -rSi 0 ^Cl-'-O.
(6) On the Distribution of r When p is not Zero. The frequency function for r is given by integrating with respect to Si and Sz from 0 to oo. This integration is not so easy. Fisher found an ingenious transformation from 5i and 5s to new variables a and ^ such as ,3. _13. S-^ = ,-y a e2, S^= ,^/ a e 2.. By the following relation f9. _. ^. ^. _/3. S^Si-2prS,S,=(y/~ae^y+^/~ae ^y-2pr^~^e^~)^/~ae 2) = 2a ie-^e—) - 2pra = 2<COS h^ - pr~), we have n—4:. a (cos ,2/3—pr). rn 2 a-^2). /0,5,,50=e 1-P2. n-2. n-1. a-^.^2"~3 r(n^\r(n-^ Next, moreover we must find a function /(r, a, ;3). we have. f(r, ^ P^-Kr, S,, 5,). 'r kjz. a, P. -Kr, S,, 5Q-1. 2. n—4. a^cosh^—pr"). =l-e. 2. l-p2. n-1. QL-rQ 2 a. n-2. CI_^^-,/T a"-3 rf"^l)r("^2. because of. 10. J_^-2p~2 a 2 e '1> U7. \ _. a,?. 1_ -A. -^a 2'eg 2 ~2a. 1 J_^T^Y -a. 2 e :. 1-a^ -4 2 e 2. Seeing that the limits of a are from 0 to oo and those of i3 from -oo to oo, we have. ^. %—4. a'(cosA/3-pr) a'(cosA/3—pr). Cl-r2) 2 a _e~ "V~1:-'P2 "^ _Cl-^-2) ^-Jo°°-^-—l^~. 2. n—1. 77 —. -da.. ("^. —^-3 ^/ra-l\ ^/ra-2. Ci-^) 2 s/^~2" 0 r. When we put a(.c-oshP-pr~) _^^ ^ ^^. n—2. r.
(7) Masahiro Oba. ^ _ ^-^"~2__^ , da- ___!,-/ , dr.. Ccos^-pr)'*-2"' ' "~ cosA/i-p. "". and then we have n—4:. Kr. »-—..—K^-^-. ^~~~9~ /~^cn~^ r^in-V\ n/ra-2\^___ ,_o __.^»-1. 2Ci-^) 2 -s/^2" " r(^o-!l) r(—7^)(cos^-^)'. .CC, _y ^_^. XJne ' r" " dr w—4 n—1. a-^o 2 a-^) 2 ^c"-i). 2^2"-3 rC^r^V^kr-pr-)"-1 On the other hand, 2%-a rfre-l) rfra^) = s./^-F(re-2) is constant owing to. p(n-l} r(n~2} _ra-3 ra-4 7i-5 n-6 n-7 ra-8............ 3 ^ J_rfJ_. 2 / \ 2 I 2 2 2 2 2 2 222~\2. 1. 2n~3. <^-3)C^-4)C^-5)C"-6) ......... 3-2-1- v/"-. -^^C»-2).^, therefore we have n—4:. n—1. n—4:. f(r, »- a-^U-pO^r^H^ ^ _,,; 2-rF^n- 2XCOS hp-prf 27t. n-\. x Cl -9rS~2~- — _1 ,,_i . CcosA/9—^or) Finally we get .co. ^_Q. %-4. f(.r')-f_f^ W^2JJ(.r, ^=n^za-r1^ ^—. a-pz)-2-C—dp. 0 ^-___ LQ __.-^-1. , cos. — 78.
(8) On the Distribution of r When p is not Zero References C 1 ~] Anderson, T. W, (1958) Introduction to Multivariate Statistical Analysis, Wiley, New York. P. 66-74.. C 2 3 Fisher, R. A. (1915) Biometrika. Cambridge-London. Vol. 10, P. 507. ["3D Fisher, R. A. (1921) Metron. Ferare-Rovigo-Roma. Vol. 1, pt. 4, p. 14. C4^ Kenney, J. F. (1951) Mathmatics of Statistics. Van Nostrand Company, pt. 2, P. 217-219, C 5 '] Sawkins, D. T, (1944) J. and Proc. R. Soc. N. S. W. Vol. 77, p. 85-95.. — 79.
(9)
関連したドキュメント
This paper derives a priori error estimates for a special finite element discretization based on component mode synthesis.. The a priori error bounds state the explicit dependency
Sreenadh; Existence and multiplicity results for Brezis-Nirenberg type fractional Choquard equation, NoDEA Nonlinear Differential Equations Applications Nodea., 24 (6) (2016), 63..
2 Combining the lemma 5.4 with the main theorem of [SW1], we immediately obtain the following corollary.. Corollary 5.5 Let l > 3 be
It should be noted that all these graphs are planar, even though it is more convenient to draw them in such a way that the (curved) extra arcs cross the other (straight) edges...
Some new results concerning semilinear differential inclusions with state variables constrained to the so-called regular and strictly regular sets, together with their applications,
[56] , Block generalized locally Toeplitz sequences: topological construction, spectral distribution results, and star-algebra structure, in Structured Matrices in Numerical
The study of the eigenvalue problem when the nonlinear term is placed in the equation, that is when one considers a quasilinear problem of the form −∆ p u = λ|u| p−2 u with
In the case of the KdV equation, the τ -function is a matrix element for the action of the loop group of GL 2 on one-component fermionic Fock space, see for instance [10, 20, 26]..
