Directional Statistics K V Mardia & P E Jupp Wiley, Chichester, 2000 Errata to 1st printing 8 4 Insert of after development 17 3 Replace θ 1 α,, θ 1 α θ 1 α,, θ n α 19 1 = (2314) Replace θ θ i 20 7 Replace 13, 13 13, 0 22 4 = (2410) The right hand side should be c 3 1 n (x i x) 3 + O(c 5 ) 22 5 = (2411) The right hand side should be 1 2c 2 1 n (x i x) 2 + 2c4 3 1 n (x i x) 4 + O(c 6 ) 22 5 Replace m 2 = 0383 + 0491i m 2 = 0383 0030i 22 3 Replace ŝ = 0322 ŝ = 0196 32 3 = (353) Replace ψ ψ θ θ T 38 4 = (3522) Replace κ 1/2 (θ µ) κ 1/2 (θ µ) 46 14 = (3548) Replace the term φ(θ; 0, Σ) in the denominator φ(µ; 0, Σ) 47 1 = (3555) Replace e 2πix e ix 51 13 Replace e a t itµ e a t +itµ 51 6 = (3570) Replace φ p = ρ p, α p = ρ p cos µ, β p = ρ p sin µ φ p = ρ p e ipµ, α p = ρ p cos pµ, β p = ρ p sin pµ 58 5 6 Replace square-summable square-integrable 58 13 Replace square-summable square-integrable 1
67 3 = (4412) Replace = 80 1 Replace nvar R nvar ( R) 89 9 Insert (up to addition of a constant) after is 89 8 = (543) The right hand side should be 90 10 Replace n 2 log ( 1 µ 2) log ( 1 µ T ) x i ˆρ 2 = 1 1 ˆµ 2 1 + ˆρ = 1 1 ˆµ 2 1 ˆµ 2 ˆµ 91 6 4 The model with density (554) is not a special case of (552) It is symmetrical about 0 and, for large enough values of µ, it has modes at ±µ 95 7 Replace O(n 1/2 ) O(n 1 ) 95 13 Replace O(n 1 ) O(n 2 ) Replace 1999 2001 95 4 = (638) This should be w = 2n {ˆκ R log I 0 (ˆκ) } = 2n {ˆκA(ˆκ) log I 0 (ˆκ)} 95 2 The right hand side should be 2n {A(ˆκ) + ˆκA (ˆκ) A(ˆκ)} 108 12 Delete the 113 5 Replace j=1 j i 117 2 = (643) The right hand side should be k p j S j j=1 2
117 4 = (644) The right hand side should be 1 n k ( Sj S ) 2 pj j=1 123 10 = (7223) The right hand side should be 2κ(n R) + 2κ(R C) 139 11 23 This subsubsection should be moved to the end of subsection 742 on page 141 (between lines 141 11 and 141 10 ), replacing ANOVA Based on a (in line 139 11 ) A, replacing asymptotic large-sample (in line 139 8 ) high-concentration asymptotic, renumbering equations as follows: (7420) (7417) (7421) (7418) (7422) (7419) (7423) (7420) (7424) (7421) (7425) (7422) (7417) (7423) (7418) (7424) (7419) (7425) and replacing equation references as follows: 139 8 (7417) (7423) 141 19 (7423) (7420) 141 17 (7423) (7420) 138 4 This should be x i x 2 = n(1 R 2 ) 139 9 = (7419) The first equation should be d i = 1 n i n i j=1 d ij 140 12 = (7424) The first term on the right hand side should be q ( ) 2 w i g 2 Ri 3
142 14 Delete is after Thus 140 1 = (753) This equation should be S = s2 c nv c (ˆκ) + s2 s nv s (ˆκ), 143 2 Replace na(ˆκ) ni 2 (ˆκ)/I 0 (ˆκ) 143 4 6 These equations should be v c (κ) = I 0(κ) 2 + I 0 (κ)i 4 (κ) 2I 2 (κ) 2 2I 0 (κ) 2 (I 0(κ)I 3 (κ) + I 0 (κ)i 1 (κ) 2I 1 (κ)i 2 (κ)) 2 2I 0 (κ) 2 (I 0 (κ) 2 + I 0 (κ)i 2 (κ) 2I 1 (κ) 2 ) v s (κ) = (I 0(κ) I 4 (κ))(i 0 (κ) I 2 (κ)) (I 1 (κ) I 3 (κ)) 2, 2I 0 (κ)(i 0 (κ) I 2 (κ)) 143 7 Replace approximations to their variances their conditional variances 151 10 Insert Assume that there are no ties before Let s i be 151 7 4 Replace these lines d 1 = 1 n 2,, d s1 = s 1 n 2, d s1+1 = s 1 + 1 n 2 1 n 1,, d s2 = s 2 n 2 1 n 1, d s2+1 = s 2 n 2 2 n 1,, d n = n 2 n 2 n 1 n 1 Hence, from (837) and some algebraic manipulation we obtain { Un 2 1,n 2 = 1 n1 ( r i n ) 2 ( i n 1 r n(n ) } 2 1 + 1) + n + n 1 (838) nn 2 n 1 2n 1 12nn 1 4
161 15 11 Replace this paragraph The projection (911) distorts the lower hemisphere more than the upper hemisphere If the data are spread over both hemispheres then it is helpful to project the two hemispheres onto separate discs in the plane, using (911) on the upper hemisphere and the variant which replaces θ π θ on the right hand side of (911) on the lower hemisphere 162 1 4 Replace tangent vectors x T µ = 0 tangent vectors z to the sphere at µ are wrapped onto the sphere z cos ( z ) µ + sin ( z ) z (913) z where z T µ = 0 164 16 14 The equation array should be S(a) = 1 n x i a 2 = 2(1 x T a) = 2(1 R x T 0 a) (923) 164 13 Replace middle expression last line 166 2,4 Replace x 0 x (3 times) 168 19 Replace κ 2 sinh κ exp { κµ T x } κ sinh κ exp { κµ T x } 169 5 Replace the integral in the numerator 1 1 teκt (1 t 2 ) (p 3)/2 dt 170 3 = (9312) Replace I (p 1)/2 (κ) I p/2 1 (κ) 172 10 Replace (1 y 2 ) 1/2 proportional to (1 y 2 ) 1/2 173 12 Replace M p (µ 2, κ) M q (µ 1, κ) 185 4 Replace x 0 R x 0 R 185 5 Replace µ 0 µ 5
185 10 Replace (958) (956) 185 11 This should be { (x1,, x n ) : x = R, x T µ = C } 185 14 15 These lines should be Integration over { (x 1,, x n ) : x T µ = C } gives the density of C as g( C; µ, κ) = c(κ) n exp{nκ C} g( C; µ, 0), and so the conditional density of R C is g ( R, C; µ, 0) g( C; µ, 0)), (9511) 185 1 = (9513) The right hand side should be c(κ) n c(nκ R) 1 f( R, R 1,, R q ; µ, 0) 186 3 = (9514) This should be 186 9 This should be f( R 1,, R q R; κ) = f( R, R 1,, R q ; µ, 0) h n (n R) ( p ) p E Xi 2 j=1 X 2 j = 1 186 7 This should be E[X 4 i ] = E [ (Xi ) ] 4 + X j 2 j i 190 17 Delete and n t 192 7 Replace tr A κ 1 193 8 Replace a b a b 198 14 15 = (1035) This should be { I ν (κ) = (2πκ) 1 2 e κ 1 4ν2 1 + (4ν2 1)(4ν 2 } 9) 8κ 2(8κ) 2 +O(κ 3 ) 6
203 6 This should be ( 12 l(a; ±x 1,, ±x n ) = n {tr(a T) log 1 F 1, p2 )}, A 207 10 Replace O(n 1/2 ) O(n 1 ) 207 7 Replace O(n 1 ) O(n 2 ) Replace 1999 2001 214 8 = (10426) The equation should be Pr(x T µ cos δ) = 1 α 215 2 = (10431) The left hand side should be sin δ 220 13,11 Replace 1 µ T 1 µ 2 1 x T 01 x 02 220 8 Replace x T i µ i x T 0i µ i 225 10 The left hand side should be x i x 2 225 1 = (10619) This should be ( q n R i i 2 n R 2 )/(q 1)(p 1) (n q n R i i 2 )/(q 1)(p 1) 226 10 = (10621) Replace ( q n ν log R ) i ν ( q n ν log R ) i ν 226 8 Replace ν i = 2(n i 1), ν = 2(n q) ν i = (p 1)(n i 1), ν = (p 1)(n q) 232 3 Replace p 2 + 6p + 20 12(p + 4) 2p 2 + 3p + 4 6(p + 4) 232 2 Replace (p 2 + 3p + 8) 3(p + 4)(p 2 + p + 2) (4p 2 + 3p 4) 3(p + 4)(p 2 + p + 2) 7
232 1 Replace p 2 4 3(p + 4)(p 2 + p + 2)(p 2 + p + 6) 4(p 2 4) 3(p + 4)(p 2 + p + 2)(p 2 + p + 6) 233 3 Replace O(n 3/2 ) O(n 2 ) Replace 1999 2001 235 11 12 (10715) and (10716) should be w b n(p2 1) { tr( T 2 ) t 2 1 (1 t 1 ) 2 /(p 1) } (10715) 2(1 2 t 1 + ĉ 11 ) w g n(p2 1) { tr( T 2 ) t 2 p (1 t p ) 2 /(p 1) }, (10716) 2(1 2 t p + ĉ pp ) 235 14 This line should be ĉ 11 = 1 n ( ) x T 4 i t 1 ĉ pp = 1 n ( ) x T 4 i t p 235 9 8 (10717) and (10718) should be n(p 2 1) { tr( T 2 ) t 2 1 (1 t 1 ) 2 /(p 1) } 2(1 2 t 1 + ĉ 11 ) n(p 2 1) { tr( T 2 ) t 2 p (1 t p ) 2 /(p 1) } 2(1 2 t p + ĉ pp ) χ 2 (p+1)(p 2)/2 (10717) χ 2 (p+1)(p 2)/2, (10718) 239 13 The right hand side should be 1 n q n i ( ) x T 4 ij t p j=1 239 6 = (10740) The right hand side should be 1 n q n i ( ) x T 4 ij t 1 j=1 8
239 4 Replace (10740) (10739) 247 1 = (1124) Replace sin 4 (πn) in the lower line sin 4 (π/n) 248 3 Replace U n = 0398 U n = 673 248 4 Replace 10% 5% Replace 459 and 46 57 and 58 248 5 Replace accepted rejected 249 7 Replace 2(r cc r ss r cs r sc )r 1 r 2 2(r cc r ss + r cs r sc )r 1 r 2 249 8 Replace r 2 cc = 0974, r 2 cs = 0213, r 2 sc = 0152, r 2 ss = 0933 r cc = 0993, r cs = 0646, r sc = 0719, r ss = 0960 249 7 Replace 0714 0713 Replace r 2 = 664 r 2 = 187 249 5 Replace nr 2 = 664 nr 2 = 187 254 10 This should be S = 1 n x ix T i y i x T i x i y T i y i y T i = S 11 S 21 S 21 S 22, 262 8 Replace A T A = I 2 and Ab = 0 A T A = ( 1 b 2) I 2, A T b = 0 and b 1 264 16 Replace is a is an 269 4 = (1233) This should be F (x) = 1 exp( ˆκx) 269 2 This should be 270 8 Replace φ i θ i ˆκ = n 1 n (1 cos θ i ) = n 1 n ( 1 R ) φ i sin θ i 271 2 Replace 0530 0389 271 6 Replace 0953 0885 271 9 Replace 0250 0490 279 5 The left hand side should be just ˆf T 9
287 8 Replace O(n 1/2 ) O(n 1 ) 287 4 Replace O(n 1 ) O(n 2 ) Replace 1999 2001 289 5 Replace Fisher matrix matrix Fisher 293 10 Replace in in in 294 11 Replace O(n 1/2 ) O(n 1 ) 294 5 Replace p 2 (p 2 + p 2) + 2r(p r)(p 2 + 4p 20) 12r(p r)(p 2)(p 1)(p + 4)(p + 2) 2p 2 (p 1)(p + 2) r(p r)(5p 2 + 2p + 8) 6r(p r)(p 2)(p + 4) 294 4 Replace [ p 2 (p 2 + p 2) r(p r)(p 2 ] 2p + 16) 3r(p r)(p 2 + p + 2)(p 2)(p + 4) [ 4p 2 (p 1)(p + 2) r(p r)(13p 2 ] + 10p 8) 3r(p r)(p 2 + p + 2)(p 2)(p + 4) 294 3 Replace (p 2r) 2 (p 1)(p + 2) 3r(p r)(p 2)(p + 4)(p 2 + p + 2)(p 2 + p + 6) 4(p 2r) 2 (p 1)(p + 2) 3r(p r)(p 2)(p + 4)(p 2 + p + 2)(p 2 + p + 6) 294 1 Replace O(n 1 ) O(n 2 ) Replace 1999 2001 300 8 Delete If H is invertible 10
306 5 Delete a before mouse 323 7 Replace O(n 1/2 ) O(n 1 ) 323 6 Replace 1999 2001 323 3 Replace O(n 1 ) O(n 2 ) 325 13 = (1452) This line should be f(x; λ, κ) = {1+κ(λ T x+1)} exp{κ(λ T x 1)}, x S 2, λ = 1, (1452) 325 12 Replace and λ is the mean shape where λ is the shape of (µ 1, µ 2, µ 3 ) and κ = 3 (µ i µ) 2 /4σ 2 349 6 5 = (A4) Replace (4p2 1)(4p 2 9) 2(8κ) 2 (4p2 1)(4p 2 9) 2!(8κ) 2 Replace (4p2 1)(4p 2 9)(4p 2 25) 2(8κ) 3 (4p2 1)(4p 2 9)(4p 2 25) 3!(8κ) 3 350 10 = (A10) Replace the integral in the numerator 1 1 teκt (1 t 2 ) (p 3)/2 dt 398 19 Replace reconstruction reconstructions 398 14 Insert matrix before Langevin 398 13 Replace Matsuita Matsusita 398 5 Replace likelhood likelihood 398 2 Replace 53 54 402 11 Replace 1980 1981 404 26 Replace (1999) (2001) 404 27 Replace Submitted for publication J Multivariate Anal, 77, 1 20 406 3 Insert matrix before distribution 408 16 Replace A-s 86 AS 86 I am very grateful to all those who have pointed out errors in the text PEJ 21/2/14 11