Open Journal of Statistics, 2015, 5, 334-339 Published Online June 2015 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/10.4236/ojs.2015.54034
On the Approximation of Maximum Deviation Spline Estimation of the Probability Density Gaussian Process Mukhammadjon S. Muminov1*, Kholiqjon S. Soatov2 1
Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan Tashkent University of Information Technologies, Tashkent, Uzbekistan * Email:
[email protected]
2
Received 23 March 2015; accepted 22 June 2015; published 26 June 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In the paper, the deviation of the spline estimator for the unknown probability density is approximated with the Gauss process. It is also found zeros for the infimum of variance of the derivation from the approximating process.
Keywords Spline-Estimator, Distribution Function, Gauss Process
1. Introduction The present work is a continuation of the work [1], that’s why we use notations admitted in it. We shall not turn our attention to more detailed review because it is given [1]. Let X 1 , X 2 , , X n be a simple sample from the parent population with the probability density f ( t ) concentrated and continuous on the segment [ 0,1] . Let Sn ( x ) be a cubic spline interpolating values yk = Fn ( tk ) at the points xk = kh , k = 0, , N with the boundary conditions
= Sn′ ( 0 )
y1 − y0 y N − y N −1 = , Sn′ (1) h h
where N = N ( n ) , h = 1 N , h → 0 , nh → ∞ as n → ∞ . Remind that *
Corresponding author.
How to cite this paper: Muminov, M.S. and Soatov, K.S. (2015) On the Approximation of Maximum Deviation Spline Estimation of the Probability Density Gaussian Process. Open Journal of Statistics, 5, 334-339. http://dx.doi.org/10.4236/ojs.2015.54034
M. S. Muminov, K. S. Soatov
ξ n = nh max
0 ≤ x ≤1
Sn′ ( x ) − f ( x )
σ N ( x) f ( x)
ξ n* ( x ) = nh max
Sn′ ( x ) − ESn′ ( x )
σ N ( x) f ( x)
0 ≤ x ≤1
ηn ( x ) =
,
,
1
1
σ N ( x) h
∫ WN ( x, y ) dωn ( y ) , 0
11 2 WN ( x, y ) dy , WN ( x, y ) is the kernel of the spline, see [1], {ωn ( t ) , t ∈ [ 0,1]} is a sequence h ∫0 of Wiener processes. Denote by Qn ( y ) the distribution function of the random variable where σ N2 ( x ) =
ln max ξ n* ( x ) − ln2 , 0 ≤ x ≤1
(1)
and by Qn
( y)
the distribution functions of the random variable
ln max ηn ( x ) − ln2 , 0 ≤ x ≤1
where
ln = 2 log N (1 + g n ) , lim g n =0 .
(1)
n →∞
In the second section of the work, Theorem 2 and 3 are proven: D ln max ξ n* ( x ) − ln2 → ln max ηn ( x ) − ln2 , 0 ≤ x ≤1
0 ≤ x ≤1
and D lnξ n − ln2 → ln max ξ n* ( x ) − ln2 , 0 ≤ x ≤1
And it is also stated (Theorem 5) that
inf Var ηn′ ( x ) = 0.
0 ≤ x ≤1
2. Formulation and Proof the of Main Results It holds the following Theorem 1. Let ξ and η be random variables, in addition P ( ξ − η > ε ) < δ for some ε > 0 , δ > 0 . Then for any x P (η < x − ε ) − δ ≤ P (ξ < x ) ≤ P (η < x + ε ) + δ .
The proof of this statement is easy, therefore we omit it. Theorems 2 and Theorem 3 will be proved by the mthods given in [2]. Theorem 2. Let h = n −δ , 0 < δ < 1 , and there exist a constant γ > 0 such that 3
n −γ ≥ log n 2 n
−
1−δ 2
3
∨ log n 2 n
−
δ 2
.
(2)
Then under our assumption a) and b) concerning f ( x ) , there exists a constant C18 > 0 such that for sufficiently large n
(
(
)
)
1 1 Qn( ) x − C18 n −γ − C18 n −γ < Qn ( x ) ≤ Qn( ) x + C18 n −γ + C18 n −γ .
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M. S. Muminov, K. S. Soatov
Proof. By the main Theorem from [1], ln max ξ n( ) ( x ) + ξ n( 1
2)
0 ≤ x ≤1
ξ n(1) ( x ) ( x ) − max 0 ≤ x ≤1
≤ ln max ξ n(
2)
0 ≤ x ≤1
( x) ,
(3)
and for any ε > 0
(
)
P ln max ηn ( x ) + ηn( ) − max ηn ( x ) > ε ≤ 1
0 ≤ x ≤1
0 ≤ x ≤1
{
}
C4 h + C5ε −1 h exp −C6ε 2 h −1ln−2 . log N
(4)
Set = ε ε= C20 n log N ⋅ ln . n Theorem 2 follows now from Theorem 1, relations (2) from [1], inequalities (3) and (4), and the fact that the random variables ln max ξ n( ) ( x ) − ln2 and ln max ηn ( x ) + ηn( ) ( x ) − ln2 have the same distribution. 1
1
0 ≤ x ≤1
0 ≤ x ≤1
Theorem 3. If conditions of Theorem 2 hold and
(
) x ) P (l ξ (=
1 < δ < 1 , then for sufficiently large n 3
(
)
Qn x − C1* n −γ − C1* n −γ < Gn ( x ) ≤ Qn x + C1* n −γ + C1* n −γ ,
where C1* > 0 is a constant, Gn Proof. From the interpolation condition
n n
)
− ln2 < x , γ is defined in (2).
= Sn ( xk ) F= 0, N n ( xk ) , k we have
= ESn ( xk ) F= ( xk ) , k 0, N . One can easily note that MSn ( x ) is a cubical spline interpolating of = yk F= ( xk ) , k 0,1, , N ,
in the points of interpolation xk = kh , k = 0, N . On the other hand ESn′ ( x ) = ( ESn ( x ) )′ . By Theorem 9 from the monograph [3] we get ESn′ ( x ) − f ( x ) ≤ 5hβ n ,
(5)
where
= βn
2 3 3 ω ( f ′, h ) + max f ′ ( 0 ) , f ′ (1) , 5 5 5
ω (= f ′, h ) max f ′ ( x ) − f ′ ( y ) , 0 ≤ x, y ≤ 1 . x− y ≤h
The relation (5) implies that for arbitrary ε > 0
(
P ln ξ n − max ξ n* ( x ) > ε 0 ≤ x ≤1
)
S′ ( x) − f ( x) S ′ ( x ) − ESn′ ( x ) = P ln nh max n − nh max n >ε 0 ≤ x ≤1 0 ≤ x ≤1 σ N ( x) f ( x) σ N ( x) f ( x) ESn′ ( x ) − f ( x ) ≤ P ln nh max > ε ≤ P ln nhC20 h > ε . 0 ≤ x ≤1 x f x σ ( ) ( ) N
(
)
It remains to choose ε = 2C20 ln nh3 and using Theorem 1 [1]. Theorem 3 is proved. Relations lim Ei′, j ( t ) = lim Ei′+1, j ( t ) , lim Ei′′, j ( t ) ≠ lim Ei′′+1, j ( t ) , i = 1, , N − 1 imply t ↑ xi
t ↓ xi
t ↑ xi
t ↓ xi
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M. S. Muminov, K. S. Soatov
Theorem 4. First order mean square derivations of the Gauss process ηn ( x ) = 0 are continuous in [0, 1], and second order mean square derivations do not have discontinuity in the points of the spline interpolation. and Tk [ = Let now= xk kh = , k 0,1, , N be points of the cubical spline interpolation, = xk −1 , xk ] , k 1, , N be a uniform partition of the interval [0, 1]. Is is valid the following Theorem 5. 1) The variance of mean square derivations of the Gauss process 1 1 2σ h ηn ( x ) = ∫ WN ( x, y ) dωn ( t ) vanishes in the intervals T1 and TN at the points t1 = − 1 − σ 2 and σ N ( x) h 0 ( )
(1 + σ ) h , respectively; 2
= t N xN −1 +
(1 − σ )
2
2) If the variance vanishes also in intervals T2 , , TN −1 , then there will be not more than two roots in each interval. Proof. At the beginning of the proof of the theorem, we proceed as in [2]. Let ht , hs ∈ T1 . Then using the relation ([4], p. 28) 2
∑ a jbj = j
we get for
∑ a 2j ∑ b2j − 2 ∑ ( ak b j − a j bk ) 1
j
j
2
,
k, j
( t , s ) ∈ [0,1]
2
ρ n2 ( th, sh ) = ( cov (ηn ( th ) ,ηn ( sh ) ) ) N −1
= 1−
∑
k , j =0
2
E1, k ( th ) E1, j ( sh ) − E1, j ( th ) E1, k ( sh )
2
(6)
.
2σ N2 ( th ) σ N2 ( sh )
Substituting into (6) E1,0 ( th ) E1,1 ( sh ) − E1,1 ( th ) E1,0 ( sh ) = D1,0 ( th ) D1,1 ( sh ) − D1,0 ( sh ) D1,1 ( th ) + D1,1 ( sh ) − D11 ( th )
and taking into account that D1,0 ( th ) D1,1 ( sh ) − D1,0 ( sh ) D1,1 ( th ) = 0 , we obtain E1,0 ( th ) E1,1 ( sh ) − E1,0 ( sh ) E1,1 ( th ) = D1,1 ( sh ) − D1,1 ( th )
or E1,0 ( th ) E1,1 ( sh ) − E1,0 ( sh ) E1,1 ( th ) 2 9 2 −1 −1 −1 . = + ( s + t ) A1,1 − A1,2 ( s − t ) ( 2 − t − s ) A0,1−1 − A0,2 4 2
(
)
(
)
We find analogously
E1, k ( th ) E1, j ( sh ) − E1, k ( sh ) E1, j ( th ) 2 9 2 = ( s − t ) ( 2 − t − s ) A0,−1j − A0,−1j +1 + ( t + s ) A1,−1j − A1,−1j +1 , 4 2
(
)
(
)
and also E1, N −1 ( th ) E1, N − 2 ( sh ) − E1, N −1 ( sh ) E1, N − 2 ( th ) 2 9 2 = ( s − t ) ( 2 − t − s ) A0,−1N −1 + ( t + s ) A1,−1N −1 . 4
Generalizing the obtained results, we have
337
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M. S. Muminov, K. S. Soatov
1 − ρ n2 (= th, sh )
2 9 N −2 −1 −1 −1 −1 ∑ ( 2 − s − t ) A0, j − A0, j +1 + ( t + s ) A1, j − A1, j +1 2 j =1
(
)
(
+ ( 2 − t − s ) A0,−1N −1 + ( t + s ) A1,−1N −1
(
Denote K n ( t ,τ ) = WN ( t ,τ )
2
} ⋅ (σ
)
( th ) ⋅ σ N ( sh ) ) ⋅ ( s − t ) −2
N
2
.
)
hσ N ( t ) . The equality
1 − ρ n ( t ,= s)
2 11 K n ( t ,τ ) − K n ( s,τ ) 2 ∫0
implies
∂ρ n ( t , s ) ∂ρ n ( t , s ) = = 0. ρ n ( t , t ) 1, = ∂t ∂s =t s =t s On the other hand, = Var ηn′ ( th )
BN ,1 ( t , t ) ∂ 2 ρ n ( th, sh ) = ∂t ∂s 2σ 4 ( th ) s =t
where 2 2 N −2 BN ,1= ( t , t ) 18 ∑ (1 − t ) A0,−1j − A0,−1j +1 + t A1,−1j − A1,−1j +1 + (1 − t ) A0,−1N + tA0,−1N −1 . j =1
(
σ N2 ( th ) Obviously,=
N −1
∑ E1,2 j ( th ) > 0 . j =0
) (
The point t1 =−h ⋅
σ Var ηn′ ( th ) = 0 . Recall that =
)
2σ
(1 − σ )
2
∈ T1 will be a solution of the equation
3 − 2 . Like the case of i = 1 , we can act analogously in the case of i = N , 1+ σ 2 i.e. at ht + xN −1 ∈ TN , Var ηn′ ( th ) = 0 when = tN h + xN −1 ∈ TN . 2 (1 − σ ) The first part of Theorem 5 is proved. Let pass to the proof of the second part. Both in the case of i = 1 , i.e. when ht , hs ∈ T1 , and in the case of = i 2, , N − 1 , the equality
ρ n2 ( th + xi −1 , sh + xi −1 ) =1 − ( t − s )
2
BN ,i ( t , s )
4σ N4 ( th + xi −1 )
is valid for ht + xi −1 , hs + xi −1 ∈ Ti . The explicit form of BN ,i ( t , s ) is given in Muminov (1987), and it is very cumbersome. BN ,i ( t , t ) Note, in this case Var ηn′ ( th ) = also. 2σ 4 ( th + xi −1 )
One can easily see that Bn ,i ( t , t ) is the sum of second powers of quadratic trinomials with respect to t ∈ [ 0,1] , and it has not more than two real roots if they exist in [0, 1]. The first part of Theorem 5 is proved. At last, Theorems 2 and 3 imply that limit distributions of the random variables lnξ n − ln2 and ln max ηn ( x ) − ln2 0 ≤ x ≤1
coincide. However, the Gauss process ηn ( x ) does not have second order mean square derivatives in the interpolation points for the spline, and inf Var ηn′ ( x ) = 0 . Therefore one can not apply results of the works [5]-[7] 0 ≤ x ≤1
to investigate the distribution of the maximum of ηn ( x ) . This deficiency has been removed in [8].
References [1]
Muminov, M.S. and Soatov, Kh. (2011) A Note on Spline Estimator of Unknown Probability Density Function. Open
338
Journal of Statistics, 157-160. http://dx.doi.org/10.4236/ojs.2011.13019
M. S. Muminov, K. S. Soatov
[2]
Khashimov, Sh.A. and Muminov, M.S. (1987) The Limit Distribution of the Maximal Deviation of a Spline Estimate of a Probability Density. Journal of Mathematical Sciences, 38, 2411-2421.
[3]
Stechkin, S.B. and Subbotin, Yu.N. (1976) Splines in Computational Mathematics. Nauka, Moscow, 272p.
[4]
Hardy G.G., Littlewood, J.E. and Polio, G. (1948) Inequalities. Moscow. IL, 456 p. http://dx.doi.org/10.1007/BF01095085
[5]
Berman, S.M. (1974) Sojourns and Extremes of Gaussian Processes. Annals of Probability, 2, 999-1026. http://dx.doi.org/10.1214/aop/1176996495
[6]
Rudzkis, R.O. (1985) Probability of the Large Outlier of Nonstationary Gaussian Process. Lit. Math. Sb., XXV, 143154.
[7]
Azais, J.-M. and Wschebor, M. (2009) Level Sets and Extrema of Random Processes and Fields, Wiley, Hoboken, 290p.
[8]
Muminov, M.S. (2010) On Approximation of the Probability of the Large Outlier of Nonstationary Gauss Process. Siberian Mathematical Journal, 51, 144-161. http://dx.doi.org/10.1007/s11202-010-0015-6
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