Directional Statistics by K. V. Mardia & P. E. Jupp Wiley, Chichester, Errata to 1st printing

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1 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, = (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 = i m 2 = i 22 3 Replace ŝ = 0322 ŝ = = (353) Replace ψ ψ θ θ T 38 4 = (3522) Replace κ 1/2 (θ µ) κ 1/2 (θ µ) = (3548) Replace the term φ(θ; 0, Σ) in the denominator φ(µ; 0, Σ) 47 1 = (3555) Replace e 2πix e ix 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 Replace square-summable square-integrable Replace square-summable square-integrable 1

2 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 Replace n 2 log ( 1 µ 2) log ( 1 µ T ) x i ˆρ 2 = 1 1 ˆµ ˆρ = 1 1 ˆµ 2 1 ˆµ 2 ˆµ 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 ) Replace O(n 1 ) O(n 2 ) Replace = (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)} Delete the Replace j=1 j i = (643) The right hand side should be k p j S j j=1 2

3 117 4 = (644) The right hand side should be 1 n k ( Sj S ) 2 pj j= = (7223) The right hand side should be 2κ(n R) + 2κ(R C) This subsubsection should be moved to the end of subsection 742 on page 141 (between lines and ), replacing ANOVA Based on a (in line ) A, replacing asymptotic large-sample (in line ) 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: (7417) (7423) (7423) (7420) (7423) (7420) This should be x i x 2 = n(1 R 2 ) = (7419) The first equation should be d i = 1 n i n i j=1 d ij = (7424) The first term on the right hand side should be q ( ) 2 w i g 2 Ri 3

4 Delete is after Thus = (753) This equation should be S = s2 c nv c (ˆκ) + s2 s nv s (ˆκ), Replace na(ˆκ) ni 2 (ˆκ)/I 0 (ˆκ) 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)) Replace approximations to their variances their conditional variances Insert Assume that there are no ties before Let s i be Replace these lines d 1 = 1 n 2,, d s1 = s 1 n 2, d s1+1 = s 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 ) } ) + n + n 1 (838) nn 2 n 1 2n 1 12nn 1 4

5 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 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 = 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) Replace middle expression last line 166 2,4 Replace x 0 x (3 times) Replace κ 2 sinh κ exp { κµ T x } κ sinh κ exp { κµ T x } Replace the integral in the numerator 1 1 teκt (1 t 2 ) (p 3)/2 dt = (9312) Replace I (p 1)/2 (κ) I p/2 1 (κ) Replace (1 y 2 ) 1/2 proportional to (1 y 2 ) 1/ Replace M p (µ 2, κ) M q (µ 1, κ) Replace x 0 R x 0 R Replace µ 0 µ 5

6 Replace (958) (956) This should be { (x1,, x n ) : x = R, x T µ = C } 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) = (9513) The right hand side should be c(κ) n c(nκ R) 1 f( R, R 1,, R q ; µ, 0) = (9514) This should be 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 = This should be E[X 4 i ] = E [ (Xi ) ] 4 + X j 2 j i Delete and n t Replace tr A κ Replace a b a b = (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

7 203 6 This should be ( 12 l(a; ±x 1,, ±x n ) = n {tr(a T) log 1 F 1, p2 )}, A Replace O(n 1/2 ) O(n 1 ) Replace O(n 1 ) O(n 2 ) Replace = (10426) The equation should be Pr(x T µ cos δ) = 1 α = (10431) The left hand side should be sin δ ,11 Replace 1 µ T 1 µ 2 1 x T 01 x Replace x T i µ i x T 0i µ i The left hand side should be x i x = (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) = (10621) Replace ( q n ν log R ) i ν ( q n ν log R ) i ν Replace ν i = 2(n i 1), ν = 2(n q) ν i = (p 1)(n i 1), ν = (p 1)(n q) Replace p 2 + 6p (p + 4) 2p 2 + 3p + 4 6(p + 4) Replace (p 2 + 3p + 8) 3(p + 4)(p 2 + p + 2) (4p 2 + 3p 4) 3(p + 4)(p 2 + p + 2) 7

8 232 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) Replace O(n 3/2 ) O(n 2 ) Replace (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 ) This line should be ĉ 11 = 1 n ( ) x T 4 i t 1 ĉ pp = 1 n ( ) x T 4 i t p (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) The right hand side should be 1 n q n i ( ) x T 4 ij t p j= = (10740) The right hand side should be 1 n q n i ( ) x T 4 ij t 1 j=1 8

9 239 4 Replace (10740) (10739) = (1124) Replace sin 4 (πn) in the lower line sin 4 (π/n) Replace U n = 0398 U n = Replace 10% 5% Replace 459 and and Replace accepted rejected 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 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 = Replace Replace r 2 = 664 r 2 = Replace nr 2 = 664 nr 2 = 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, Replace A T A = I 2 and Ab = 0 A T A = ( 1 b 2) I 2, A T b = 0 and b Replace is a is an = (1233) This should be F (x) = 1 exp( ˆκx) This should be Replace φ i θ i ˆκ = n 1 n (1 cos θ i ) = n 1 n ( 1 R ) φ i sin θ i Replace Replace Replace The left hand side should be just ˆf T 9

10 287 8 Replace O(n 1/2 ) O(n 1 ) Replace O(n 1 ) O(n 2 ) Replace Replace Fisher matrix matrix Fisher Replace in in in Replace O(n 1/2 ) O(n 1 ) 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) 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) 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) Replace O(n 1 ) O(n 2 ) Replace Delete If H is invertible 10

11 306 5 Delete a before mouse Replace O(n 1/2 ) O(n 1 ) Replace Replace O(n 1 ) O(n 2 ) = (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) 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 = (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) = (A10) Replace the integral in the numerator 1 1 teκt (1 t 2 ) (p 3)/2 dt Replace reconstruction reconstructions Insert matrix before Langevin Replace Matsuita Matsusita Replace likelhood likelihood Replace Replace Replace (1999) (2001) Replace Submitted for publication J Multivariate Anal, 77, Insert matrix before distribution 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