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# Let D be the set of continuous densities, defined on the sphere S; we estimate f, an element of D, from a sample of size n, denoted by X

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(1)

22,4 (1995), pp. 427–446

1

n

S

k(n)

k

k,r

k

k,r

k→∞

r∈Rk

k,r

k→∞

r∈Rk

k,r

k→∞

r∈Rk

k,r

r∈Rk

k,r

n→∞

k

n,r

i

k,r

(2)

n

k

k,r

n

nr

k,r

k,r

k,r

n

r∈Rk

k,r

−1

1

1

1

2

1

2

3

1

2

2

1

1

k,r

k,r1,r2

k,0

2

k,1,r2

2

2

2

2

k,r1,r2

1

1

2

2

1

2

k,k,r2

2

2

2

2

k,k+1

2

k,r

k,r

2

2

2

i

s−1

1

1

i

i−1

j=1

j

i

(3)

s

s−1

j=1

j

s

0

i

i=1,...,s−1

i0

i=1,...,s−1

2

0

s−1

i=1 i−1

j=1

j

j0

2

i

i0

i

i+1

s−1

0

0

S0 s−1

i=1

mi

i

i

i

q

π/2

0

2q+1

q

π 0

2q

i

k,r

s−1

i=1

ri−1

ri

0k

ri

αri

αri−1

mi

i

i

i

i

i

qi

α 0

2qi+1

i

i

qi

qi

qi

ri

qi

ri

i

qi

i

αri

αri−1

2qi+1

i

i

qi

(4)

i

i

i

qi

α 0

2qi

i

i

qi

qi

ri

qi

ri

i

qi

i

αri

αri−1

2qi

i

i

qi

i

rs−1−1

rs−1

s−1

s−1

s−1

qi

qi

i

s−1

k,r

s−1

i

i+1

s−2

s−1

1

i

k,r

1

2

3

k,r

r1−1

r1

2

2

3

3

k,r

2

3

r1

r1

2

r1

1

k,0

1/3

k,k+1

1/3

k,1,0

1/3

1

k,1,k+1

1/3

1

(5)

k,1,1,r3

1/3

1

3

3

3

n,4

2

3

n,s

s−1

s−2

s−1

s

n

n,x

n

n→∞

n

n,x

n,x

s

s−1n

s−1n

s

(s−1)/2

n,x

i

n,x

n

nx

s

s−1n

n

1−sn

1−sn

n

n

1

n

n

n

n

n

n

(6)

n

kn,0

n,kn

n,0

n

kn,0

n,0

n

i

kn,0

n

n,kn+1

n,0

n

kn+1,0

kn,0

kn+1,0

n

n

n,0

n

kn,0

n

n,0

n

kn+1,0

kn+1,0

kn,0

n,kn+1

n

kn,0

kn+1,0

n,kn

n

n

n,kn

n,kn+1

n,kn

n,kn+1,

kn+1,0

kn,0

n

kn,0

kn+1,0

kn+1,0

kn,0

n

n,kn

n,kn+1

n

n

n

n

1−sn

1

n→∞

nx

(7)

n→∞

s

s−1n

n

n→∞

s

s−1n

1−sn

1−sn

0

i

s−1

n

kn,r

0kn

0

kn,r

kn,r

kn,r

s−1

i=1

ri−1

ri

0kn

kn,r

ri−1/2

αri−1/2

0

mi

i

i

i

qi

i

qi

rs−1−1/2

s−1

s

kn,r

kn,r

0

r∈R

0

kn

kn,r

kn,r

1/2

n

0k

n

kn,r

kn,r

1/2

n

n

1/2

n

0k

n

kn,r

kn,r

n

n,¯xkn,r

0

0kn

0

ns−1

0

0

n

n→∞

n

kn,r

0

i

n,¯xkn,r

n

kn,r

n

n→∞

r∈R0

kn

n,r

n

(8)

n,r

n

i

kn,r

n→∞

r∈R0kn

n,r

n

1

n

i∈I

i

i∈I

i

i

n→∞

r∈R0kn

n,r

n

0

n→∞

r∈R0

kn

kn,r

n

kn,r

ns−1

s−2n

0

s−1n

n→∞

ns−1

0

s−1n

n

n

1−sn

+

∞ 0

(s−3)/2

n

n→∞

n

n

s−1n

K,s

n

n i=1

i

2n

(9)

i

K,s

n

1−sn

S

2n

K,s

n

K,s

n

(s−1)/2

2/h

2n

0

2

2n

(s−3)/2

n→∞

K,s

n

(s−1)/2

∞ 0

(s−3)/2

[0,1/2]

K,s

n

(s−1)/2

1/2

0

2

2n

(s−3)/2

K,s

n

1−sn

(s−1)/2

2 arcsin h

n/2 0

s−2

s−1n

K,s

n

n,x

n

y→∞

∞ y

(s−3)/2

n

1−sn

1−sn

(10)

n

1−sn

1

1

1−sn

1

i

s−1

∞ M

(s−3)/2

∞ 0

(s−3)/2

n

n

n

1

n

n

i

n

n

n

n

n

n

n

−2(2M )(s−1)/2Cs

1

Hn

n

Hn

n

s

(s−1)/2

s−1n

n

n

n

n

n

(s−1)/2

s

s−1n

K,s

n

2/h

2n

%2n/(2h2n)

2

2n

(s−3)/2

2n

2n

n

n

(s−1)/2

s

s−1n

K,s

n

∞ M

(s−3)/2

(11)

n

n

s

s−1n

(s−1)/2

0

(s−3)/2

K,s

n

n→∞

n

n→∞

K,s

n

(s−1)/2

∞ 0

(s−3)/2

n

n

14

1

n

1−sn

1−sn

1−sn

1

1

1−sn

1

s−1

s−3

1

s−3

S

s

s

s

s

s

s

0

s

0

s

s

y→∞

∞ y

(s−3)/2

0

0

∞ M

(s−3)/2

∞ 0

(s−3)/2

0

s

s

0

0

s

0

s

0

(12)

0

0

s

s

s

0

0

s

s

∞ M

(s−3)/2

∞ 0

(s−3)/2

n

−1n

s

s

1/(s−1)

n

n−1

s

s

1/(s−1)

s

s−1n

s

1

s−1n

0

0

s

n

1

n

n

n

s

s−1n

s

s−1n

n

n,t

n

n→∞

n

n

n

1

j

1

j

n

n

1

j

n,t

1

j

1−sn

n

0

00

n

0

(13)

0

00

n

i

i

n

1

j

n

n

1

j

n

s

s−1n

n

s−1n

s

j r=1

n,tr

j r=1

n,tr

n

n

1

j

s−1n

K,s

n

S−C

2n

s−1n

K,s

n

C−

jr=1n,tr

2n

n

s−1n

s

n

n

s−1n

s

s−1n

K,s

n

S−C

2n

0

s−1n

K,s

n

C−

jr=1n,tr

2n

00

0

00

0

00

S−

jr=1n,tr

s−1n

K,s

n

2n

(14)

1

n,t1

kn,0

s−1n

K,s

n

S−

jr=1n,tr

2n

s−1n

K,s

n

S−

jr=1n,tr

1

2n

1

s−1

00

1

2n

K,s

n

(s−1)/2

D00

(s−3)/2

n,t1

00

2n

2n

(s−1)/2

K,s

n

%2n/(2h2n)

(s−3)/2

2n

2n

s

s

2/(s−1)

2n

2n

n2

2n

s

s

2/(s−1)

2n

2n

(s−1)/2

K,s

n

∞ M

(s−3)/2

∞ M

(s−3)/2

∞ 0

(s−3)/2

∞ M

(s−3)/2

∞ 0

(s−3)/2

(s−1)/2

0

(s−3)/2

K,s

n

(15)

0

0

23

+

m

m

m

m,j

m

m

+m

m

+

m

m

m+1

+m+1

+m

(s−1)/2

0

(s−1)/2

m→∞

∞ 0

(s−3)/2

+m

m→∞

∞ 0

(s−3)/2

m

∞ 0

(s−3)/2

1−sn

n

+m

∞ j=0

mj

Imj

m

∞ j=0

0mj

Imj

s−1n

K,s

n

n i=1

∞ j=0

0mj

Imj

i

2n

n

s−1n

K,s

n

n i=1

∞ j=0

mj

Imj

i

2n

(16)

Imj

i

2n

m

i

2n

m

i

n,m,j+1,x

n,m,j,x

n,m,j,x

n,m,j,x

n,m,j+1,x

n

m−1

1/2

n

n

m−1

1/2

n

n,m,j

n,m,j,x

s

s−1n

n,m,j+1

n,m,j+1,x

s

s−1n

n

m−1

(s−1)/2

s−1n

K,s

n

m i=1

Imj

i

2n

m−1

(s−1)/2

s−1n

K,s

n

n,m,j+1,x

n,m,j,x

s

K,s

n

(s−1)/2

n,m,j+1

(s−1)/2

n,m,j

s

K,s

n

∞ j=0

0mj

m−1

(s−1)/2

(s−1)/2

n,m,j+1

(s−1)/2

n,m,j

n

s

K,s

n

∞ j=0

mj

m−1

(s−1)/2

(s−1)/2

n,m,j+1

(s−1)/2

n,m,j

(17)

n

s

K,s

n

∞ j=0

m,j

(m−1)(s−1)/2

(s−1)/2

n,m,j+1

(s−1)/2

n,m,j

s

K,s

n

∞ j=0

m,j

(m−1)(s−1)/2

(s−1)/2

(s−1)/2

n→∞

K,s

n

(s−1)/2

∞ 0

(s−3)/2

s

(s−1)/2

∞ 0

+m

(s−3)/2

∞ j=0

mj

(m−1)(s−1)/2

(s−1)/2

(s−1)/2

0

0

∞ 0

+m

(s−3)/2

## R

∞ 0

(s−3)/2

0

0

0

(s−1)/2

j6∈J

mj

(s−1)/2

(s−1)/2

0

n

s

K,s

n

j∈J

mj

(m−1)(s−1)/2

(s−1)/2

n,m,j+1

s

K,s

n

j∈J

mj

(m−1)(s−1)/2

(s−1)/2

n,m,j

s

K,s

n

n

(18)

1

2

2

1

(19)

1

2

2

1

12

−u

n

s

(20)

### Sci., Wiley, 1983.

MONIQUE BERTRAND-RETALI LARBI AIT-HENNANI

UNIVERSIT ´E DE ROUEN

UFR DES SCIENCES MATH ´EMATIQUES ANALYSE ET MOD `ELES STOCHASTIQUES URA CNRS 1378

76821 MONT SAINT AIGNAN CEDEX, FRANCE

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