1938 (Ph.D) 1940 (D.Sci) 1940 (Kai-Lai Chung) Lebesgue-Stieltjes [6] ( [22]) 1942 (1941 ) 1945 J. Neyman H. Hotelling ( ) (University of Cali

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1 () () (E. A. Poe) (Osgood, ) ( ) A note on the indices and numbers of nondegenerate critical points of biharmonic functions, (University College London) J. Neyman ( ) Fisher-Behrens [15] [16] [21] F [19]( ) 1

2 1938 (Ph.D) 1940 (D.Sci) 1940 (Kai-Lai Chung) Lebesgue-Stieltjes [6] ( [22]) 1942 (1941 ) 1945 J. Neyman H. Hotelling ( ) (University of California, Berkeley) (Columbia University) (University of North Carolina, Chapel Hill) [3] Neyman Neyman (A. Wald) ( [14] 283 ) E. L. Lehmann ( ) Lehmann [8] 1947 H. Hotelling ( ) W. L. Deemer I. Olkin Biometrika [7] H. Robbins 1947 [23] Robbins ( ) [3] Robbins 1948 Robbins (T. L. Lai) 1947 J. Neyman (Neyman [14])

3 1948 (A. N. Kolmogorov, ) (1933 ) 1949 [1] Am happy after liberation 1950 E. C. Titchmarch Introduction to the Theory of Fourier Integrals (Khintchine, , ) () (Gnedenko) ( ) (Stepanov) ( ) ( ) 1951 ( ) () 1947 () 1955 ( ) 1956 ( [10]):

4 ( ) ( ) 50 ( ) ( ) Fisz Urbanik Dynkin Prokhorov () ( ) ( ) 1956 W. Feller [33] 4 4

5 ( ) ( )[31] ()[32] ( ) (1983) G. Hunt (1978 ) 1958 ( 5

6 ) ( ), [13] 1962 [9] [25] [28][29] ( ) ([28] 1 1 [29] 3 2 ) ( ) ( ) [12] (The Annals of Statistics) [3] E. L. Lehmann [4] T. W. Anderson [5] K. L. Chung ( ) [6] 1981 [34] 1983 Springer-Verlag 6

7 [35](Pao-Lu Hsu Collected Papers) K. L. Chung 1997 (Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present) (N. J. Johnson S. Koty ) 114 (1) [15] 1938 Behrens- Fisher X 1, X 2,..., X n Y 1, Y 2,..., Y m N(µ 1, σ 2 1 ) N(µ 2, σ 2 2 ) σ 1 σ 2 H 0 : µ 1 = µ 2 Behrens-Fisher S 2 1 = U = ( X Ȳ )2 /(A 1 S A 2 S 2 2), X = 1 n X i, Ȳ = 1 m (X i X) 2, S2 2 = m Y i, m (Y i Ȳ )2, A 1 A 2 A 1 = A 2 = (m + n)/[(n + m 2)nm] U t u 1 A 1 = 1/(n(n 1)), A 2 = 1/(m(m 1)) U Behrens- Fisher u 2 U {U > C} θ = σ 2 1 /σ2 2 λ = (µ 1 µ 2 ) 2 /( 1 n σ m σ2 2 ) H. Scheffe (1970) λ = 0 θ u 1 u 2 H 0 (n = m ) u 2 θ (2) [16] y = Aβ + ε y = (y 1,..., y n ) T (T ) β = (β 1,..., β p ) T A n p ( p) ε = (ε 1,..., ε n ) T Eε i = 0, Eε 2 i = σ 2, Eε 4 i = α i σ 4 (i = 1,..., n), σ α i σ Q = Q(y 1, y 2,..., y n ) y 1, y 2,..., y n (i) β σ, EQ = σ 2, (ii) Q β (iii) (i) (ii) Q 1 Q Q 1, Q σ 2 7

8 Q = y T Λy Λ n Q σ 2 Λ Λ = MD τ M, M = I A(A T A) 1 A T I D τ = τ τ τ n, τ 1, τ 2,..., τ n F = i,j µ ij τ i τ j n m iiτ i = 1 µ ij = (α k 3)m 2 ki m2 kj + 2m2 ij k=1 (i = 1,..., n; j = 1,.., n), m ij M = (m ij ) n n α i = σ 4 Eε 4 i (i = 1,..., n) σ 2 S 2 0 = 1 n p y A ˆβ 2 ( ˆβ β z z ) (n p) (α i 3)m ii m 2 ik = m kk (α i 3)m 2 ii. (3) [17] Hotelling T 2 T 2 χ 2 χ 2 T 2 θ i W = (1 θ i ) V = θ i /(1 θ i ) [18] [19] Y 1, Y 2,..., Y m, Z 1,..., Z n p(y 1,..., y m, z 1,..., z n ) = ( 2πσ) (m+n) exp{ 1 m 2σ 2 [ (y i η i ) zi 2 ]},

9 η 1,..., η m m σ ( ) n 1 m H 0 : η 1 = η 2 =... = η n1 = 0, F = n1 y2 i n1 y2 i + n z2 i W 0 = {(y 1,..., y n1, z 1,..., z n ) : F F α }, F α P ( F F α H 0 ) = α F = n 1 Y i 2/( n 1 Y i 2 + n Z2 i ) W 0 H 0 λ β 0 (λ) λ = 1 2σ 2 n 1 W (y 1,..., y m, z 1,..., z n ) W α W λ β(λ) λ > 0 β(λ) β 0 (λ) H 0 W 0 α λ F ( H. B. Mann Analysis and Design of Experiments, 1949) Simaika (1941) Hotelling T 2 Lehmann Scheffe [4] (4) S X 1, X 2,..., X N N(0, Σ) A (N 1)S = Wishart W (Σ, N 1) η 2 i., N (X α X)(X α X) T (1) α=1 [20] Wishart [5] [21] A B Wishart W (Σ, m) W (Σ, n) m p, n p (p ) θ 1 θ 2... θ p A θ(a + B) = 0 9

10 θ 1,..., θ p p θ 1 2 (m p 1) i p (1 θ i ) 1 2 (n+p 1) ) p p j=i+1 (θ i θ j ). (1) A Σ ( ) ( A 11 A 12 A =, Σ = A 21 A 22 Σ 11 Σ 12 Σ 21 Σ 22 ), A 11 Σ 11 p 1 A 22 Σ 22 p 2 λa 11 A 12 A 21 λa 22 = 0, λσ 11 Σ 12 Σ 21 λσ 22 = 0. ( [35] ) A B A Wishart B Wishart A φb = 0 ( [35] ) X 1, X 2,..., X N T = Q/S Q = N a ij (X i X)(X j X), S = i,j=1 N (X i X) 2. a ij N N T ( [35] ) f(ū 1,..., ū k ) ū 1,..., ū k f( ) ( 0) (5) [6] [22] Berry ξ 1, ξ 2,..., ξ n n Eξ i = 0, Eξ 2 i = 1 (i = 1,..., n) ( 0 1) ξ = 1 n ξ i, η = 1 n (ξ i ξ) 2, Φ(x) ξ η ( n )Cramér F n (x) P ( n ξ x) F n (x) = Φ(x) + ψ(x) + R(x), 10

11 ψ(x) R(x) ξ 1 n lim n R(x) = 0 Berry F n (x) Φ(x) F n (x) Φ(x) Aβ 3 n 1 2 ( x), β 3 = E ξ 1 3 A ( n ξ 1 ) [22] Berry Cramér η = 1 n n (ξ i ξ) 2 ξ = 1 n n ξ i G n (x) = P ( n(η 1)/( α 4 1) x) (α 4 = Eξ1 4) α 6 = Eξ1 6 < α 4 1 α3 2 0 (α 3 = Eξ1 3) G n (x) Φ(x) A ( n α 4 1 α3 2 ) 3 2 ( x), A Eξ 2k < ( k > 3) G n (x) (6) {ξ n, n 1} µ Robbins [23] ε > 0 α 6 P ( 1 ξ k µ ε) <. (2) n n=1 k=1 ( 1 n n k=1 ξ k 1 µ) (2) [23] 1 n n k=1 ξ k µ [23] (2) P. Erdös (7) Levy, Feller, Kolmogorov Gnedenko [24] Gnedenko Kolmogorov Gnedenko() (8) X F (x) f(t) = + 11 e itx df (x).

12 [25] f(t) ( δ, +δ)( δ > 0) X β M β (F ) = + x β df (x) < [26] f(t) (, + ) ( δ, +δ) Gnedenko ( δ, +δ) (, + ) f(t) (Ū) (Ū) F (x) F (x) p(x) p(x) = O[exp( x )] ( x ), ψ( x ) ψ(x) (lnx) λ, (lnx)(lnlnx) λ,... (λ > 1) (9) [27] A B = P A( P ) 1, P P P A B A B [27] ( ) [28] A 1 A 2 A 1 B 1 = P A 1 Q, A 2 B 2 = P A 2 Q, P Q (A 1, A 2 ) (B 1, B 2 ) ( ) [29] ( ) () A 1 A 2 (A 1, A 2 ) A 1 B 1 = P A 1 ( P ) T, A 2 B 2 = P A 2 P T, P T (A 1, A 2 ) (B 1, B 2 ) 12

13 (i) A 1 A 2 (A 1, A 2 ) ( 7 ) (ii) A 1 A 2 (A 1, A 2 ) ( 8 ) (10) [30] n X F X Borel p(t, x, E) s x t + s E (x X, t > 0, E F) [30] p(t, x, E) t p(t, x, E) Austin p ij (t) ([31] [32]) [31] m (PBIB) PBIB [32] X 1,..., X n F (x) ξ (n) 1... ξ n (n) [32] ξ (n) k n (k n, k n /n λ [0, 1)) ξ (n) k n φ 1 (x) = 1 2π x e t2 2 dt 0 x 0 φ 2 (x) = 1 αlnx+β 2π e t 2 2 dt x > 0, α > 0 1 x 0 φ 3 (x) = 1 αln x +β 2π e t 2 2 dt x < 0, α > 0 F (x) φ i (x) (i = 1, 2, 3) 13

14 [1] (1980). [2] (1981). [3] Anderson, T. W., Chung, K. L. and Lehmann, E. L., Pao-Lu Hsu , Ann. Statist, 1979, 7: [4] Lehmann, E. L., Hsu s Work on Inference. Ann. Statist. 1979, 7: [5] Anderson, T. W., Hsu s Work in Multivariate Analysis. Ann. Statist, 1979, 7: [6] Chung, K. L., Hsu s Work in Probability. Ann. Statist, 1979, 7: [7] Deemer, W.L. and Olkin, I.(1951), The Jacobians of Certain Matrix Transformations Useful in Multivariate Analysis, Based on Lectures of P.L.Hsu at the University of North Carolina, Biometrika 38: [8] Lehmann, E. L., Reminiscences of a Statistician: The Company I Kept, 2008 Springer,. [9]. [10]. [11].( ) (1994). [12] [13]. [14] (C. Reid): (1987). [15] Contribution to the theory of Student s t-test as applied to the problem of twosamples. Statist. Res. Mem, 2(1938), [16] On the best unbiased quadratic estimate of the variance. Statist. Res. Mem. 2(1938), [17] Notes on Hotelling s generalized T. Ann. Math. Statist.9(1938), [18] On generalized analysis of variance. Biometrika. 31(1940),

15 [19] Analysis of variance from the power function standpoint. Biometrika. 32(1941), [20] A new proof of the joint product moment distribution. Proc. Cambridge Philos. Soc. 35(1939), [21] On the distribution of roots of certain determinantal equations. Ann. Eugenics. 9(1939), [22] The approximate distribution of the mean and variance of a sample of independent variables. Ann. Math. Statist 16(1945), [23] Complete convergence and the law of large numbers. (with H. Robbins) Proc. Nat. Acad. Sci. U.S.A. 33(1947), [24] A general weak limit theorem for independent distributions. (Appendix III in Limit Theorems of Sums of Independent Random Variables by B. V. Gnedenko and A. N. Kolmogorov, translated by K. L. Chung. revised edition, Addison-Wesley 1968). [25]. 1(1951) [26]. 4(1954) [27]. 5(1955) [28]. ( ) 1(1955) [29]. ( ) 3(1957) [30]. ( ) 4(1958) [31] ( ). 7(1964) [32] ( ). 14(1964) [33] (1982). [34] (1981). [35] Pao-Lu Hsu Collected Papers, Springer-Verlag(1983). New York. 15

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