Remarks on a parameter estimation for von Mises--Fisher distributions

0.07 MB·English

by  Árpád Baricz

REMARKS ON A PARAMETER ESTIMATION FOR VON
MISES–FISHER DISTRIBUTIONS⋆
3
1 A´RPA´DBARICZ
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2

n
a
Abstract. Wepointoutanerrorintheproofofthemainresultofthepaper
J
of Tanabe et al. (2007) concerning a parameter estimation for von Mises–
2
Fisherdistributions,wecorrecttheproofofthemainresultandwepresenta
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shortalternativeproof.

] Recently,Tanabeetal. (2007)proposedaniterativealgorithmusingfixedpoints
A
toobtainthemaximumlikelihoodestimateforoneoftheparametersofthep-variate
C vonMises–Fisherdistributiononthep-dimensionalunithypersphere. Intheirstudy
. Tanabe et al. (2007) arrived at the equation
h
t 1
a (1) =R,
m rp−1(κˆ)
2
[ where rp−1(κˆ) = Ip−1(κˆ)/Ip(κˆ), Iν is the modified Bessel function of the first
2 2 2
1 kind of order ν, and R = ||x1 +x2 +···+xn||/n is the mean length of the data
v vector(x1,x2,...,xn). Inordertosolve(1)Tanabeetal. (2007)usedafixedpoint
4 iterationmethod and for this first derivedthe following main result: if ν ≥1, then
4 the function Φ2ν :(0,∞)→R, defined by Φ2ν(x)=Rxrν−1(x), has a unique fixed
2
point.
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The purposeofthis note is threefold: to pointoutanerrorin the proofofabove
.
1
main result of Tanabe et al. (2007), to provide a correct proof and to show that
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the above mentioned result is almost immediate by using some known results on
3
1 the ratio 1/rν−1.
: In what follows we list our comments concerning the above mentioned result:
v
i 1. Let
X Sν(x)=Iν2(x)−Iν−1(x)Iν+1(x)
r
be the Tur´anian of the modified Bessel function of the first kind. In the
a
proof of the above mentioned result Tanabe et al. (2007) considered the
Tur´an type inequalities
1
(2) 0<S (x)< ·I2(x)
ν ν+x ν
and attributed these inequalities to Nasell (1974), and Thiruvenkatachar
and Nanjundiah (1951). We wouldlike to point out that there is no Tur´an
type inequality proved by Nasell (1974) and the right-hand side of (2)
cannotbe found inthe paper of ThiruvenkatacharandNanjundiah (1951).
File: correction.tex, printed: 2013-01-23, 2.08
Key words and phrases. von Mises–Fisher distribution, Maximum likelihoodestimate, Modi-
fiedBesselfunctions ofthefirstkind,Tura´n–type inequalities.
⋆Research supported by Romanian National Research Council CNCS-UEFISCSU, project
numberPN-II-RU-TE190/2013.
1
2 A´RPA´DBARICZ
In their paper there is a Tur´an type inequality of this kind, but with ν+1
instead of ν +x. Incidentally the right-hand side of (2) can be found in
the paper of Joshi and Bissu (1991), but it was pointed out very recently
by Baricz (2012) that this inequality is not valid. This means that the
right-hand side of the inequality
1 1 1
′
− r (x)<r (x)< − r (x)
x ν ν ν+x+1 x ν
is not valid, i.e. the Lemma in Tanabe et al. (2007) is not true and hence
the proof of the main Theorem is not correct.
2. The proof of the main Theorem of Tanabe et al. (2007) was based on the
Banach fixed point theorem, and therefore they wanted to prove that the
functionΦ2ν isacontractionmapping. Forthisitwasenoughtoprovethat
′
0 < Φ (x) < 1 for each x > 0 and ν ≥ 1. However, since the right-hand
2ν
′
side of (2) is not true, the proof of the inequality Φ (x) < 1 presented in
2ν
Tanabe et al. (2007) is not correct. All the same, this result is true. For
this observe that
xS (x)
′ ′ ν
(3) Φ2ν(x)=R rν−1(x)+xrν−1(x) =R(cid:20) I2(x) (cid:21).
(cid:2) (cid:3) ν
′
Since R ∈ (0,1) to prove Φ (x) < 1 it would enough to show the Tur´an
2ν
type inequality
1
(4) S (x)< ·I2(x).
ν x ν
But, this inequality is valid for ν ≥ 1 and x > 0, as it was shown re-
2
cently by Baricz (2012), and is equivalent to the fact that the function
x7→xI′(x)/I (x)−x is strictly decreasing on (0,∞) for ν ≥ 1, which was
ν ν 2
proved by Gronwall (1932). It is important to note here that Hamsici and
Martinez(2007)usedalsotheright-handsideof (2)inmodelingthedataof
two spherical-homoscedastic von Mises-Fisher distribution. The correction
of their result based on the right-hand side of (2) was made also by using
(4), see Baricz (2012) for more details. We also note that for ν ≥ 1 and
2
x>0the Tur´antypeinequality (4)canbe improvedas(seeBaricz(2012))
1
S (x)< ·I2(x),
ν ν
x2+ν2− 1
4
q
′
but the contraction constant 1 in the inequality Φ (x) < 1 cannot be
2ν
improved. For this consider the Tur´an type inequality (see Segura (2011))
1
·I2(x)<S (x),
ν ν
ν+ 1 + x2+ ν+ 1 2
2 2
q
(cid:0) (cid:1)
which is valid for all ν ≥ 0 and x > 0. Combining this inequality with (3)
and (4) it is easy to see that for ν ≥ 1 the expression xS (x)/I2(x) tends
2 ν ν
to 1 as x tends to infinity. This shows that indeed the constant 1 in the
′ ′
inequality Φ (x) < 1 cannot be improved. Now, recall that Φ (x) > 0
2ν 2ν
for all ν > 0 and x > 0, and combining this with the above inequality,
we obtain that if ν ≥ 12, then the function Φ2ν is a contraction mapping
and therefore has a unique fixed point. This correctsthe proofofthe main
VON MISES–FISHER DISTRIBUTION 3
Theorem of Tanabe et al. (2007) and improves the range of validity of the
parameter ν = p.
2
3. Finally,wepresentashortproofforthefollowingresult,whichimprovesthe
range of validity of the main result of Tanabe et al. (2007): if ν >0, then
the function Φ2ν has a unique fixed point. Observe that this statement
is equivalent to the fact that the equation 1/rν−1(x) = R has a unique
solution on (0,∞) for ν > 0. It is known that (see Yuan and Kalbfleisch
(2000))thefunctionx7→1/rν−1(x)=Iν(x)/Iν−1(x)isincreasingon(0,∞)
for ν ≥ 1, while for ν ∈ 0,1 is increasing first to reach a maximum and
2 2
then decreasing. Moreov(cid:0)er, 1/(cid:1)rν−1(x) → 1 as x → ∞ for each ν > 0, and
the graph of x 7→ 1/rν−1(x) approaches the asymptote from above when
ν ∈ 0,1 ,andfrombelowwhenν ≥ 1.Now,sinceR<1,theaboveresults
2 2
impl(cid:0)y tha(cid:1)t the graph of x7→1/rν−1(x) intersects only once the horizontal
line y = R for each ν > 0, and indeed the equation 1/rν−1(x) = R has a
unique solution on (0,∞) for ν >0.
References
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Departmentof Economics, Babe¸s-BolyaiUniversity,400591Cluj-Napoca,Romania
E-mail address: [email protected]