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2 incoe incoe 2 adaptive histogra Kogure 987 Terrell and Scott 992 Lecoutre 987 Scott 992
3 {X,X 2,...,X n} F n x δ ν j [x + jδ (δ/2), x + jδ +(δ/2)), j=,±,±2,... (2.) N j nx i= I (X i ν j), j=,±,±2,... I(A) A {ν j} (2.2) H(x δ) Nj nδ Fn (x + jδ +(δ/2)) Fn (x + jδ (δ/2)) =, x ν j δ F n( ) {X i} F n(x) n nx i= I(X i x), <x< {X i} (2.3) Y j F n (j/) inf{x : F n(x) j/},,2,..., Y in i n {Xi}, Y ax {Xi} i n Π j (2.4) Π j [Y j,y j),,2,..., ; Π [Y,Y ] M j nx i= I (X i Π j) [n/] [n/]+ [ ] {Π j} (2.5) H % (x ) M j n(y j Y j ), x Πj
4 MISE f(x) df (x)/dx MISE ean integrated squared error MISE E[( b f(x) f(x)) 2 ]dx MISE Jones 99 b f MISE 2 ISB integrated squared bias ISB (E[b f(x)] f(x)) 2 dx IV integrated variance IV = Var( b f(x))dx MISE = ISB + IV 2.3 δ E[H(x δ)] f(x)+ (j )δ + ISB (2.6) ISB δ2 x + O(δ 2 ), x ν j, δ (E[H(x δ) f(x)]) 2 dx δ2 2 R(f ) Scott, 992 R( ) g L 2 R(g) g(x) 2 dx Freedan and Diaconis 98 R(f) < E[H % (x )] f Lecoutre 987 F j /2 R(f /f) + O( 2 ), x Π j, f 2 (x) dx < f(x) ISB ISB (E[H % (x ) f(x)]) 2 dx f (2.7) 2 R 2 f ( )
5 379 MISE Equal Bins Percentile Bins n = MISE VS. MISE Equal Bins Percentile Bins n = MISE VS. Scott 992, p f R(f /f)= x 2 dx =! puzzling Scott
6 MISE 3 4 Scott [ 4,4] n = n = ISE ( ˆf(x) f(x)) 2 dx MISE ISE δ =8/ Scott MISE 2.4 IV (2.8) Z IV Var(H(x δ)) f(x) nδ, Var (H(x δ))dx nδ Z x νj f(x)dx = nδ Scott, 992 Var(H % (x )) j /2 n f F IV (2.9) IV Var (H % (x ))dx n R(f) 2, x Π j Lecoutre, 987 R(f) < IV ISB 2.5 MISE MISE MISE = ISB + IV δ2 2 R(f )+ nδ MISE δ n /3 MISE n 2/3 MISE n 2/3 MISE MISE = ISB + IV 2 R(f /f)+ 2 n R(f) MISE n /3 MISE n 2/3
7 38 3. Lecoutre 987 R(f /f) < ISB 2.7 R(f /f)= ISB 2.7 R(f /f)= 2 ISB (3.) y j F (j/),, 2,..., y y F. (3.2) f(x)=2x, x R(f /f) R(f /f)= x dx = 2 F (z)= z, z E[H % (x )] = F (yj) F (yj ) y j y j = / p j/ p (j )/, x Π j y = y = 3.2 ISB ISB = γ (E[H n(x)] f(x)) 2 dx = γ 3/2 + O( 2 ), γ = b 4(x)x 7/2 dx, b 4(x) 4 A R(f /f)= ISB 3/2 2.7 MISE n 2/5 MISE n 3/5 2. (3.3) f(x)=e x, x> R(f /f)= dx =
8 F (z)= log( z), E[H n(x)] = z< / log ( (j )/) log( j/) = log( + /( j)) ISB = (E[H n(x)] f(x)) 2 dx X =» ( j)+/2 2 log ( + /( j)) log( + x) x» ( j)+/2 ISB 2 X /( j) = 2, R(f /f)= ISB R(f /f) ISB 2 R(f /f)= ISB 2 Scott 992 ISB 4. 3 R(f /f) < R(f /f) < 4. {X,X 2,...,X n} (4.) bf(x h) n nx i= K h (x X i), <x<, K h ( ) K h (u) K(u/h)/h, <u< K K(u)du =, K(u) du < h = h n li hn =, li n n nhn = {h n} K 2 K
9 383 {ν j} binned kernel estiator (4.2) ef(x h,δ) n X j= N jk h (x g j), <x<, N j 2. {g j} ν j g j x + jδ, j =,±,±2,... Fan and Marron 994 Hall and Wand 996 (4.3) ef % (x h,) n X D jk h (x Y j), <x<, {Y j} 2.3 {D j} D j Mj + Mj+,,2,..., 2 M j 2.4 {Y j} 3 3. {y j} M j [y j,y j) E[ e f% (x h,)] = n X E[D j]k h (x y j)= 4. E[ b f(x h)] = K h (x y)f(y)dy = c j j/ /2 (j =,2,...,) X K h (x y j)= Z cj+ X c j X K h (x y j) K h x F (z) dz K h x F (j/) dz (4.4) E[ f% e (x h,)] E[ f(x h)] b X Z cj+ = c j Z c K h (x F (j/)) K h (x F (z))dz K h (x F (z)) dz c K h (x F (z))dz
10 X + Z cj+ Z c c j K h (x F (j/)) K h (x F (z))dz K h (x F (z))dz + B 4.4 X Z cj+ c j 2 Z c c K h (x F (z))dz K h (x F (j/)) K h (x F (z))dz h K h (x F (z))dz + K h (x F (z))dz c K Z K (u) du <, sup K < K (u) du h sup K x (h) h 2 E[ b f(x h)] f(h)=o(h 2 ) Scott, 992 h 3 E[ e f% (x h,)] f(x)=e[e f% (x h,)] E[ b f(x h)] + E[ b f(x h)] f(h)=e[ b f(x h)] f(h)+o(h 2 ) MISE MISE MISE K(u)= (/ 2π)e u2 /2 Scott, 992, 6.7 h =.6sn /5 s n = 7 6 n MISE
11 385 MISE Histogra Percentile n = MISE VS. MISE Ker Histogra Percentile n = MISE VS. 5.
12 MISE Equal Bins Percentile Bins n = MISE VS. MISE Equal Bins Percentile Bins n = MISE VS. Minnotte 996, 998 Sagae and Scott 997, Sagae and Kogure 2
13 387 ISB = = A (E[H n(x)] f(x)) 2 dx f(x) 2 dx 2 X = 4 3 3/2 X ( p j + p j ). p j/ p (j )/ Euler-Maclaurin Lange, 999, Proposition 6.2. X p Z h i j = xdx + /2 + + h i / h i Z 5/2 + 5 b 4(x)x 7/2 dx Z = 2 3 3/2 + 2 /2 + γ /2 92 5/2 5 b 4(x)x 7/2 dx 28 = 2 3 3/2 + 2 /2 + γ 2 + O( /2 ). 3/2 X ( p j + p j ) = 4 3 3/2 γ + O( 2 ). B Freedan and Diaconis 98 Lea 2.22 j Z cj+ c j K h (x F (j/)) K h (x F (z))dz 4.4 X Z cj+ = h c j Z cj X 2 Z c = Z cj K h (x F (j/)) K h (x F (z))dz K h(x F (z)) c j x F (z) K h K h (x F (z))dz + Z F (/2) sup K h h K h (x y)f(y)dy + Z F (/2) f(f (z)) dz c j K h(x F (z)) f(f (z)) dz = h c K h (x F (z))dz F ( /2) f(y)dy + f(y)dy F ( /2) K (u) du K h (x y)f(y)dy h sup K f(f (z)) dz
14 Fan, J. and Marron, J. S Fast ipleentation of nonparaetric curve estiators, Journal of Coputational and Graphical Statistics, Freedan, D. and Diaconis, P. 98. On the histogra as a density estiator: L 2 theory, Zeitschrift für Wahrscheinlichkeitstheorie und Vervandte Gebiete, 57, Hall, P. and Wand, M. P On the accuracy of binned kernel density estiators, Journal of Multivariate Analysis, 56, Jones, M. C. 99. The roles of ISE and MISE in density estiation, Statistics & Probability Letters, 2, Kogure, A Asyptotically optial cells for a histogra, Annals of Statistics, 5, Lange, K Nuerical Analysis for Statisticians, Springer, New York. Lecoutre, J. P The histogra with rando partition, New Perspectives in Theoretical and Applied Statistics, , John Wiley and Sons, New York. Minnotte, M. C The bias-optiized frequency polygon, Coputational Statistics,, Minnotte, M. C Achieving higher-order convergence rates for density estiation with binned data, Journal of the Aerican Statistical Association, 93, Sagae, M. and Kogure, A. 22. Maxiu entropy density estiator, Proceedings of JSM22, Joint Statistical Meetings 22, New York. Sagae, M. and Scott, D. W Bin interval ethod of locally nonparaetric density estiation, Tech. Report, Departent of Statistics, Rice University, Houston, Texas. Scott, D. W Multivariate Density Estiation: Theory, Practice, and Visualization, Wiley, New York. Terrell, G. R. and Scott, D. W Variable Kernel density estiation, Annals of Statistics, 2,
15 Proceedings of the Institute of Statistical Matheatics Vol. 53, No. 2, (25) 389 Estiating Probability Density fro Percentiles Atsuyuki Kogure and Masahiko Sagae 2 Faculty of Policy Manageent, Keio University 2 Faculty of Engineering, Gifu University We consider the proble of estiating density fro percentiles. Prior results such as Lecoutre (987) and Scott (992) suggest that a histogra with percentiles suffers large biases. We re-exaine the bias proble and show that the kernel ethod alleviates it by asyptotic arguents. To conr the theoretical results we give siple siulation studies. Key words: Histogra, percentiles, bin frequency, kernel estiator, bias.
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