ALMOST SURE CONVERGENCE OF THE BARTLETT ESTIMATOR
|
|
- Ophelia Barker
- 5 years ago
- Views:
Transcription
1 Periodica Mathematica Hungarica Vol. 51 1, 2005, pp ALMOST SURE CONVERGENCE OF THE BARTLETT ESTIMATOR István Berkes Graz, Budapest, LajosHorváth Salt Lake City, Piotr Kokoszka Logan Qi-man Shao Eugene, Singapore Dedicated to Endre Csáki Pál Révész on the occasion of their 70 th birthdays Abstract We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a. s. behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator. 1. Introduction If {Y i } is a weakly stationary sequence, then under weak conditions which quantify short-range weak dependence 1 n Var Y i σ 2 := γ j, 1 i n j= as n,whereγ j =CovY 0,Y j. Inference for time series modeled by weakly dependent processes requires estimation of the asymptotic variance σ 2.Oneofthe Mathematics subject classification number: 60F15, 62F12. Key words phrases: weak dependence, long-range dependence, variance of the mean, cumulants, increments of partial sums. Research supported by the Hungarian National Foundation for Scientific Research, Grants T T Research supported by NSF grant INT NATO grant PST.EAP.CLG Partially supported by NSF grant DMS , grants R R at the National University of Singapore /2005/$20.00 Akadémiai Kiadó, Budapest c Akadémiai Kiadó, Budapest Springer, Dordrecht
2 12 i. berkes et al. most popular estimators is the Bartlett estimator defined as s 2 n =ˆγ 0 +2 ω j qnˆγ j, j qn where ˆγ j = 1 n 1 i n j Yi Ȳn Yi+j Ȳn 1.2 are the sample autocovariances ω j q are the Bartlett weights defined by ω j q =1 j q The estimator s 2 n has also been used for long-range dependent observations Y i,see Lo In the weakly dependent case, σ 2 =2πf0, where f is the spectral density of {Y i }, so s 2 n /2π is also an estimator for f0. An extension of the estimator 1.1 to arbitrary frequencies has been studied extensively in the spectral domain, so most results focused on L 2 convergence. Giraitis et al proved that under regularity conditions quantifying weak dependence, s 2 n P σ 2. They also considered the in probability behaviour of s 2 n when {Y i } exhibit long range dependence. In the present paper we establish the almost sure consistency of the variance estimator s 2 n under the conditions used by Giraitis et al in the case of weak dependence as well as in the long memory case. Theorem 1.1 below plays a crucial role in Berkes et al who developed a procedure for distinguishing between a sequence of long-range dependent observations a sequence of weakly dependent observations with a change point. To lighten the notation, we assume in the following that EY i =0. Recall the definition of the fourth order cumulant κh, r, s givenby κh, r, s =E[Y k Y k+h Y k+r Y k+s ] γ h γ r s + γ r γ h s + γ s γ h r. 1.4 We will also work with the quantity νh, r, s =CovY k Y k+h,y k+r Y k+s = E[Y k Y k+h Y k+r Y k+s ] γ h γ r s = κh, r, s+γ r γ h s + γ s γ h r. 1.5 Theorem 1.1. Suppose {Y k } is a fourth order stationary sequence with EY i =0 γ j =CovY 0,Y j. Suppose the sequence qn is nondecreasing q 2 k+1 sup k 0 q 2 k <. 1.6
3 almost sure convergence of the bartlett estimator 13 Then, i Suppose, in addition, that γ j <, 1.7 sup h j κh, r, s <, 1.8 r,s qn qnlog n 4 = On. 1.9 s 2 n σ2 := j= γ j a.s ii Assume 1 <H< γ k c 0 k 2H for some c 0 > 0. Assume also that qn qn =O nlog n 7/4 4H 1.13 Then sup h qn n r,s n qn 1 2H s 2 n c 2 H = κh, r, s = O n 2H c 0 H2H 1 a.s Remark 1.1. The proof of Theorem 1.1 remains valid if the Bartlett weights 1.3 are replaced by arbitrary weights which in addition to 1.6 satisfy ω j q =0 for >q,0 ω j q 1 another condition which is different for parts i ii. For part i, it must be required cf. 2.5 that lim ω jq =1 foreachj q For part ii, it must be required that lim c 0 q q1 2H ω j q γ j = 2H 1H q
4 14 i. berkes et al. Define also Proof of part i. Let γ j = 1 n n 2. Proofs S k,l = Y i Y i+, l Y i. i=k <n. Then ˆγ j γ j = Ȳn 2 1 n nȳns 1,n + S +1,n =:δ j. As in the proof of Theorem 3.1 of Giraitis et al. 2003, decompose s 2 n as where It suffices to show that v n,1 = v n,2 = s 2 n = v n,1 + v n,2, qn qn γ j, qn+1 δ j. qn+1 v n,1 σ 2 a.s. 2.1 v n,2 0 a.s. 2.2 We first verify the easier relation 2.2. By 1.7 Theorem of Stout 1974 S 1,n = o n 1/2 log 2 n a.s. 2.3 consequently Hence, by 1.9, qn max S i,j = o n 1/2 log 2 n 1 i j n a.s. Ȳn 2 qn+1 n 2qn+1Ȳ n 2 = o qn/nlog 4 n = o1 a.s.
5 almost sure convergence of the bartlett estimator 15 1 Ȳn S1,n + S +1,n qn+1 n qn 2qn+1 o log 2 n n 1/2 log 2 n n n 1/2 qnlog 4 n = o = o1 a.s. n This proves 2.2. We now turn to the verification of 2.1. Set Since z n = nv n, Ev n,1 = Ev n,1 = qn qn qn+1 n γ j qn+1 n Yi Y i+ γ. 2.4 j= γ j = σ 2, 2.5 it suffices to verify that z n = on a.s. In the study of z n, we replace qn with q n, where q n is constant on large intervals. This replacement will be done in two steps. First we replace qn in the limit of summation then in the sum itself. We thus define q n =q2 k, if 2 k <n 2 k+1 introduce z n = ẑ n = We will show that q n q n qn+1 q n+1 n n Yi Y i+ γ, 2.6 Yi Y i+ γ. 2.7 z n z n = on a.s., 2.8 We will use the relation z n ẑ n = on a.s. 2.9 sup h ẑ n = on a.s νh, r, s < 2.11 r,s
6 16 i. berkes et al. which follows immediately from Clearly, where n,2 = z n z n = n,1 = We first show that q n< qn qn+1 n Yi Y i+ γ = n,1 + n,2, 2.12 q n< qn q n< qn For any 2 k <m<n 2 k+1,wewrite n, m,1 = Using 2.11, we have E[a 1n, m] 2 = = q n< qm q n< qm q n< qm n m, j qm q n< qm + qm< qn q n+1 n 1 q n+ 1 qn+1 Yi Y i+ γ n Yi Y i+ γ. n,1 = on a.s n Yi q Y i+ γ n+1 i=m +1 q n+1 n Yi Y i+ γ =: a 1 n, m+a 2 n, m q n< j qm q n< j qm q n< j qm l 4n m <i n m <i n m<i<n m<i <n ν, l, l+ j m j <i n j m j <i n j CovY iy i+, Y i Y i + j ν, i i, i i + j ν, i i j +, i i + Cn mqm. 2.15
7 almost sure convergence of the bartlett estimator 17 Applying 2.11 again, we obtain E[a 2 n, m] 2 qm< qn qm< j qn n n j i =1 CovY i Y i+,y i Y i + j = n qm< qn qm< qn qm< j qn qm< j qn n l 4n n j i =1 ν, i i, i i + j ν, l, l+ j Cqn qmn Combining , we conclude that E[ n, m,1 ] 2 C[n mqn+qn qmn] = C C n i=m+1 n i=m+1 [qn+qi qi 1n] [ q 2 k+1 +qi qi 12 k+1] Using Problem 5 of Billingsley 1968, p. 102, cf. Móricz et al. 1982, we get [ ] 2 E max 2 k+1, m,1 2 k <m 2 k+1 C [ 2 k+ 2 k q 2 k+1 + q 2 k+ q 2 k 2 k+1] log 2 k+1 2, therefore the Chebishev inequality 1.6 yield { } P max 2 k+1, m,1 k 7/4 q 1/2 2 k 2 k/2 2 k <m 2 k+1 C [ 2 k+ 2 k q 2 k+1 + q 2 k+ q 2 k 2 k+1] k 2 k 7/2 q 2 k 2 k C k 3/2. The Borel Cantelli lemma 1.9 give max 2 k+1, m,1 a.s. = o 2 k k <m 2 k+1
8 18 i. berkes et al. Similarly to 2.15, we have, using 2.11 E 2 n,1 = = n q n<, j qn q n<, j qn q n<, j qn q n<, j qn 1 i n 1 i n 1 i,i n l 4n 1 i n j 1 i n j Cov Y i Y i+, Y i Y i + j ν, i i, i i + j ν, i i j +, i i + ν, l, l+ j Cnqn Hence the Chebishev inequality 1.6 give P { 2 k+1,1 k 7/4 q 1/2 2 k 2 k/2} C 2k+1 q2 k+1 k 7/2 q2 k 2 k C k 7/2, so the Borel Cantelli lemma 1.9 yield 2 k+1,1 a.s. = o2 k Now 2.13 follows from Next we show n,2 = on a.s Observe that n,2 = qn q n qn+1 q n< qn q n< qn q n+1 For any 2 k <m<n 2 k+1,wewrite n,2 m,2 = q n< qn q n< qm n q n+1 q n+1 n Yi Y i+ γ. n q n+1 Yi Y i+ γ Yi Y i+ γ m Yi Y i+ γ
9 almost sure convergence of the bartlett estimator 19 = q n< qm + qm< qn q n+1 q n+1 = a 3 n, m+a 4 n, m. Following the proofs of , we obtain n i=m +1 n Yi Y i+ γ Yi Y i+ γ E[a 3 n, m] 2 Cn mqm, 2.22 E[a 4 n, m] 2 Cqn qmn 2.23 E 2 n,2 Cnqn Just as relations 2.15, implied 2.13, relations 2.22, imply Relation 2.8 follows from Observing that z n ẑ n = qn q n qn+1 q n q n+1 n Yi Y i+ γ, following the proof of 2.21, one can easily verify that 2.9 holds. Also, similarly to , for any 2 k <m,n 2 k+1,wehave E[ẑ n ẑ m ] 2 Cn mq n Eẑ 2 n Cnq n. Hence, relation 2.10 follows from Problem 5 of Billingsley 1968, the Chebyshev inequality, 1.6 the Borel Cantelli lemma. Proof of part ii. The idea of the proof is similar to that used in part i but different bounds are needed. We use the same notation as in the proof of part i. As verified on p. 291 of Giraitis et al qn 1 2H Ev n,1 = qn 1 2H qn ω j qn γ j c 0 H2H 1 = c2 H We first show that qn 1 2H v n,2 0 a.s. Observe that by 1.12 it follows easily ESk,l 2 Cl k +1 2H
10 20 i. berkes et al. hence E max 1 l n S 0,l 2 Cn 2H 2.26 by a maximal inequality of Billingsley 1968, p. 94. We will now verify that 2.26 implies max S 0,l = o n H log n a.s l n Fix 1/2 <p<1 note that for any ɛ>0 P max S 0,l >ɛ2 kh k p 1 1 l 2 k ɛ 2 1 k< 1 ɛ 2 1 k< 1 k< = C 1 ɛ 2 1 k< [ 2 kh k p] 2 E max 1 l 2 k S 0,l 2 [ 2 kh k p] 2 C2 2kH k 2p <. Hence, by the Borel Cantelli lemma max S 0,l = o 2 kh k p 1 l 2 k a.s. as k. Now, for any n, choosingk such that 2 k 1 <n 2 k,weobtain max S 0,l max S 0,l = o 2 kh k p = o n H log n a.s., 1 l n 1 l 2 k establishing Using the definition of v n,2 2.27, it is easy to see that qn 1 2H v n,2 = o1qn 2 2H log 2 nn 2H 2 [ ] 2 2H qn = o1 log n1/1 H a.s. n Therefore, assumption 1.13 implies that qn 1 2H v n,2 0 a.s. We now show that qn 1 2H v n, Ev n,1 0 a.s. This will be accomplished by showing that z n z n = o nqn 2H 1 a.s., 2.28 z n ẑ n = o nqn 2H 1 a.s ẑ n = o nqn 2H 1 a.s., 2.30 with z n, z n ẑ n defined, respectively, by 2.4,
11 almost sure convergence of the bartlett estimator 21 We obtain some inequalities for the second moments of z n z n, z n ẑ n ẑ n their increments on the intervals 2 k, 2 k+1]. The inequality in Problem 5 of Billingsley 1968, p. 102 cf. also Corollary 3.1 in Móricz et al with the Borel Cantelli lemma then yields First we use the decompositions in Similarly to 2.15, for 2 k <m<n 2 k+1, we have, using the first relation of Lemma 2.1, E[a 1 n, m] 2 n m ν, l, l+ j qm j qm l 4n Cn mqn 2H n 2H As in 2.16, we have, using the second relation of Lemma 2.1, E[a 2 n, m] 2 n ν, l, l+ j qm< qn qm< j qn l 4n C qn qmn 2H qn 2H Hence, using also 1.13 we get { P max 2 k+1, m,1 ɛ2 k+1 [ q 2 k+1] } 2H 1 2 k <m 2 k+1 Ck2 ɛ 2 Ck2 ɛ 2 2 k+ 2 k[ q 2 k+1] 2H 2 2H 1k + q 2 k+ q 2 k[ q 2 k+1] 2H 1 2 2kH q 2 k+1 2 k+1 2 2k [q2 k+1 ] 4H 2 2 2H Ck 3/2 ɛ 2, 2.33 so by the Borel Cantelli lemma we have max 2 k+1, m,1 = o 2 k+1 [ q 2 k+1] 2H 1 2 k <m 2 k+1 a.s Similarly, E 2 n,1 Cnqn 2H, so by 1.6, 1.13 the Borel Cantelli lemma we have a.s. = o 2 k+1 q 2 k+1 2H 1. Hence we conclude that 2 k+1,1 n,1 = o nqn 2H 1 a.s. 2.35
12 22 i. berkes et al. Similarly, along the lines of the proof of , we get n,2 = o nqn 2H 1 a.s., 2.36 completing the verification of The proofs of are similar to that of 2.28 are therefore omitted. We conclude this section with Lemma 2.1 which was used in the proof of Theorem 1.1. The proof of Lemma 2.1 uses Lemma 2.2 which follows the proof of Lemma 2.1. Lemma 2.1. Suppose {X k } is a fourth order stationary sequence. If 1.12 is satisfied with 1/2 <H<1 assumptions hold, then νh, r, r + s = O qn 2H n 2H h qn s qn ν h,r,r+ s qm< h qn qm< s qn = O qn qmn 2H 1 qn 2H Proof. We note that by 1.6, relation 1.14 remains valid if the summation domain n r, s n is changed to 4n r, s 4n. Thus κh, r, r + s = O qnn 2H 1, h qn s qn by Lemma 2.2 [ γr γ h r+ s + γr+ s γ h r ] h qn s qn qn sup h s qn γ h r+ s γ r +sup h γ h r s qn γr+ s = O qn 2H n 2H 1. Since H>1/2, we have qn =o qn 2H, so 2.37 follows from the identity 1.5.
13 almost sure convergence of the bartlett estimator 23 Using 1.14 again, we have qm< h qn qm< s qn κ h,r,r+ s = O qn qmn 2H 1 by Lemma 2.2, qm< h qn qm< s qn γr γ h r+ s =2 qm< h qn 2 qm< h qn qm<s qn i 2qn γ r γ i r γr γ h s r Cqn qmn 2H 1 qn 2H 1. Similarly, by 1.12 Lemma 2.2, qm< h qn = qm< h qn qm< s qn qm< s qn γr+ s γ h r Cqn qm r 4n 4n r 1 Cqn qmn 2H 1 qn 2H 1. 0 r 4n r 2H 2 r 2H 2 + 4n r 1 qm< h qn qm< s qn γ r+ s γ h r γ h r γr+ s Lemma 2.2. Suppose {X k } is a fourth order stationary sequence. If 1.12 is satisfied with 1/2 <H<1, then uniformly in v =0, ±1, ±2,... k γ i+v Ck 2H 1
14 24 i. berkes et al. Proof. Let v 1, then k k γ i+v C i + v 2H 2 k C i 2H 2 Ck 2H 1. Let v 0. If v k 1thenwehave k k k γ i+v C i + v 2H 2 C i k 1 2H 2 C If k 1 v 0, we have k i 2H 2 Ck 2H 1. k γ i+v 2C 2k i 2H 2 Ck 2H 1. References I. Berkes, L. Horváth, P. Kokoszka Q-M. Shao, On discriminating between longrange dependence changes in mean, Technical Report, Utah State University, P. Billingsley, Convergence of probability measures, Wiley, New York, L. Giraitis, P. S. Kokoszka, R. Leipus G. Teyssière, Rescaled variance related tests for long memory in volatility levels, Journal of Econometrics , A. W. Lo, Long-term memory in stock market prices, Econometrica , F. A. Móricz, R. J. Serfling W. F. Stout, Moment probability bounds with quasi-superadditive structure for the maximum partial sums, Ann. Probab , W. F. Stout, Almost sure convergence, Academic Press, New York, Received: April 25, 2005 István Berkes Department of Statistics Graz University of Technology Steyrergasse 17, A-8010 Graz Austria A. Rényi Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127, H-1364 Budapest Hungary
15 almost sure convergence of the bartlett estimator 25 Lajos Horváth Department of Mathematics University of Utah 155 South 1440 East Salt Lake City, UT USA Piotr Kokoszka Department of Mathematics Statistics Utah State University 3900 Old Main Hill Logan, UT USA Qi-man Shao Department of Mathematics University of Oregon Eugene, OR USA Department of Statistics Applied Probability National University of Singapore Singapore Singapore
Monitoring Shifts in Mean: Asymptotic Normality of Stopping Times 1
Monitoring Shifts in Mean: Asymptotic Normality of Stopping Times Alexander Aue 2 Lajos Horváth 2 Piotr Kokoszka 3 Josef Steinebach 4 Abstract: We consider a sequential procedure designed to detect a possible
More informationAn almost sure invariance principle for additive functionals of Markov chains
Statistics and Probability Letters 78 2008 854 860 www.elsevier.com/locate/stapro An almost sure invariance principle for additive functionals of Markov chains F. Rassoul-Agha a, T. Seppäläinen b, a Department
More informationON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY
J. Korean Math. Soc. 45 (2008), No. 4, pp. 1101 1111 ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY Jong-Il Baek, Mi-Hwa Ko, and Tae-Sung
More informationWeak invariance principles for sums of dependent random functions
Available online at www.sciencedirect.com Stochastic Processes and their Applications 13 (013) 385 403 www.elsevier.com/locate/spa Weak invariance principles for sums of dependent random functions István
More informationMi-Hwa Ko. t=1 Z t is true. j=0
Commun. Korean Math. Soc. 21 (2006), No. 4, pp. 779 786 FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MULTIVARIATE LINEAR PROCESSES GENERATED BY DEPENDENT RANDOM VECTORS Mi-Hwa Ko Abstract. Let X t be an m-dimensional
More informationLimiting behaviour of moving average processes under ρ-mixing assumption
Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 30 (2010) no. 1, 17 23. doi:10.1285/i15900932v30n1p17 Limiting behaviour of moving average processes under ρ-mixing assumption Pingyan Chen
More informationNOTES AND PROBLEMS IMPULSE RESPONSES OF FRACTIONALLY INTEGRATED PROCESSES WITH LONG MEMORY
Econometric Theory, 26, 2010, 1855 1861. doi:10.1017/s0266466610000216 NOTES AND PROBLEMS IMPULSE RESPONSES OF FRACTIONALLY INTEGRATED PROCESSES WITH LONG MEMORY UWE HASSLER Goethe-Universität Frankfurt
More informationHan-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek
J. Korean Math. Soc. 41 (2004), No. 5, pp. 883 894 CONVERGENCE OF WEIGHTED SUMS FOR DEPENDENT RANDOM VARIABLES Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek Abstract. We discuss in this paper the strong
More informationLocation Multiplicative Error Model. Asymptotic Inference and Empirical Analysis
: Asymptotic Inference and Empirical Analysis Qian Li Department of Mathematics and Statistics University of Missouri-Kansas City ql35d@mail.umkc.edu October 29, 2015 Outline of Topics Introduction GARCH
More informationStrong Approximation for the Sums of Squares of Augmented GARCH Sequences
Strong Approximation for the Sums of Squares of Augmented GARCH Sequences Alexander Aue 1 István Berkes Lajos Horváth 3 Abstract: We study so called augmented GARCH sequences, which include many submodels
More information4 Sums of Independent Random Variables
4 Sums of Independent Random Variables Standing Assumptions: Assume throughout this section that (,F,P) is a fixed probability space and that X 1, X 2, X 3,... are independent real-valued random variables
More informationOn the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables
On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables Deli Li 1, Yongcheng Qi, and Andrew Rosalsky 3 1 Department of Mathematical Sciences, Lakehead University,
More informationAsymptotic Normality under Two-Phase Sampling Designs
Asymptotic Normality under Two-Phase Sampling Designs Jiahua Chen and J. N. K. Rao University of Waterloo and University of Carleton Abstract Large sample properties of statistical inferences in the context
More informationKrzysztof Burdzy University of Washington. = X(Y (t)), t 0}
VARIATION OF ITERATED BROWNIAN MOTION Krzysztof Burdzy University of Washington 1. Introduction and main results. Suppose that X 1, X 2 and Y are independent standard Brownian motions starting from 0 and
More informationConditional independence, conditional mixing and conditional association
Ann Inst Stat Math (2009) 61:441 460 DOI 10.1007/s10463-007-0152-2 Conditional independence, conditional mixing and conditional association B. L. S. Prakasa Rao Received: 25 July 2006 / Revised: 14 May
More informationPACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION
PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION DAVAR KHOSHNEVISAN AND YIMIN XIAO Abstract. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced
More informationSpectral representations and ergodic theorems for stationary stochastic processes
AMS 263 Stochastic Processes (Fall 2005) Instructor: Athanasios Kottas Spectral representations and ergodic theorems for stationary stochastic processes Stationary stochastic processes Theory and methods
More informationConfidence Intervals for the Autocorrelations of the Squares of GARCH Sequences
Confidence Intervals for the Autocorrelations of the Squares of GARCH Sequences Piotr Kokoszka 1, Gilles Teyssière 2, and Aonan Zhang 3 1 Mathematics and Statistics, Utah State University, 3900 Old Main
More informationSelf-normalized Cramér-Type Large Deviations for Independent Random Variables
Self-normalized Cramér-Type Large Deviations for Independent Random Variables Qi-Man Shao National University of Singapore and University of Oregon qmshao@darkwing.uoregon.edu 1. Introduction Let X, X
More information1. A remark to the law of the iterated logarithm. Studia Sci. Math. Hung. 7 (1972)
1 PUBLICATION LIST OF ISTVÁN BERKES 1. A remark to the law of the iterated logarithm. Studia Sci. Math. Hung. 7 (1972) 189-197. 2. Functional limit theorems for lacunary trigonometric and Walsh series.
More informationAsymptotic efficiency of simple decisions for the compound decision problem
Asymptotic efficiency of simple decisions for the compound decision problem Eitan Greenshtein and Ya acov Ritov Department of Statistical Sciences Duke University Durham, NC 27708-0251, USA e-mail: eitan.greenshtein@gmail.com
More informationARTICLE IN PRESS Statistics and Probability Letters ( )
Statistics and Probability Letters ( ) Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro The functional central limit theorem
More informationarxiv: v1 [math.pr] 7 Aug 2009
A CONTINUOUS ANALOGUE OF THE INVARIANCE PRINCIPLE AND ITS ALMOST SURE VERSION By ELENA PERMIAKOVA (Kazan) Chebotarev inst. of Mathematics and Mechanics, Kazan State University Universitetskaya 7, 420008
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationRobustní monitorování stability v modelu CAPM
Robustní monitorování stability v modelu CAPM Ondřej Chochola, Marie Hušková, Zuzana Prášková (MFF UK) Josef Steinebach (University of Cologne) ROBUST 2012, Němčičky, 10.-14.9. 2012 Contents Introduction
More informationZeros of lacunary random polynomials
Zeros of lacunary random polynomials Igor E. Pritsker Dedicated to Norm Levenberg on his 60th birthday Abstract We study the asymptotic distribution of zeros for the lacunary random polynomials. It is
More informationSoo Hak Sung and Andrei I. Volodin
Bull Korean Math Soc 38 (200), No 4, pp 763 772 ON CONVERGENCE OF SERIES OF INDEENDENT RANDOM VARIABLES Soo Hak Sung and Andrei I Volodin Abstract The rate of convergence for an almost surely convergent
More informationERDŐS AND CHEN For each step the random walk on Z" corresponding to µn does not move with probability p otherwise it changes exactly one coor dinate w
JOURNAL OF MULTIVARIATE ANALYSIS 5 8 988 Random Walks on Z PAUL ERDŐS Hungarian Academy of Sciences Budapest Hungary AND ROBERT W CHEN Department of Mathematics and Computer Science University of Miami
More informationStochastic volatility models: tails and memory
: tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier Plan Model assumptions; Limit theorems for partial sums and
More informationα i ξ t i, where ξ t i.i.d. (0, 1),
PROBABILITY AND MATHEMATICAL STATISTICS Vol. 31, Fasc. 1 (2011), pp. 300 000 NONLINEARITY OF ARCH AND STOCHASTIC VOLATILITY MODELS AND BARTLETT S FORMULA BY PIOTR S. KO KO S Z K A (LOGAN) AND DIMITRIS
More informationOnline Appendix. j=1. φ T (ω j ) vec (EI T (ω j ) f θ0 (ω j )). vec (EI T (ω) f θ0 (ω)) = O T β+1/2) = o(1), M 1. M T (s) exp ( isω)
Online Appendix Proof of Lemma A.. he proof uses similar arguments as in Dunsmuir 979), but allowing for weak identification and selecting a subset of frequencies using W ω). It consists of two steps.
More informationPacking-Dimension Profiles and Fractional Brownian Motion
Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 Packing-Dimension Profiles and Fractional Brownian Motion By DAVAR KHOSHNEVISAN Department of Mathematics, 155 S. 1400 E., JWB 233,
More informationUseful Probability Theorems
Useful Probability Theorems Shiu-Tang Li Finished: March 23, 2013 Last updated: November 2, 2013 1 Convergence in distribution Theorem 1.1. TFAE: (i) µ n µ, µ n, µ are probability measures. (ii) F n (x)
More informationThe Codimension of the Zeros of a Stable Process in Random Scenery
The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar
More informationWeakly dependent functional data. Piotr Kokoszka. Utah State University. Siegfried Hörmann. University of Utah
Weakly dependent functional data Piotr Kokoszka Utah State University Joint work with Siegfried Hörmann University of Utah Outline Examples of functional time series L 4 m approximability Convergence of
More informationASYMPTOTIC NORMALITY UNDER TWO-PHASE SAMPLING DESIGNS
Statistica Sinica 17(2007), 1047-1064 ASYMPTOTIC NORMALITY UNDER TWO-PHASE SAMPLING DESIGNS Jiahua Chen and J. N. K. Rao University of British Columbia and Carleton University Abstract: Large sample properties
More informationON THE STRONG LIMIT THEOREMS FOR DOUBLE ARRAYS OF BLOCKWISE M-DEPENDENT RANDOM VARIABLES
Available at: http://publications.ictp.it IC/28/89 United Nations Educational, Scientific Cultural Organization International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
More informationInvariance principles for fractionally integrated nonlinear processes
IMS Lecture Notes Monograph Series Invariance principles for fractionally integrated nonlinear processes Xiaofeng Shao, Wei Biao Wu University of Chicago Abstract: We obtain invariance principles for a
More informationGrouped Network Vector Autoregression
Statistica Sinica: Supplement Grouped Networ Vector Autoregression Xuening Zhu 1 and Rui Pan 2 1 Fudan University, 2 Central University of Finance and Economics Supplementary Material We present here the
More informationGoodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach
Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach By Shiqing Ling Department of Mathematics Hong Kong University of Science and Technology Let {y t : t = 0, ±1, ±2,
More informationComplete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions
Proc. Indian Acad. Sci. Math. Sci.) Vol. 124, No. 2, May 214, pp. 267 279. c Indian Academy of Sciences Complete moment convergence of weighted sums for processes under asymptotically almost negatively
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationRANDOM WALKS WHOSE CONCAVE MAJORANTS OFTEN HAVE FEW FACES
RANDOM WALKS WHOSE CONCAVE MAJORANTS OFTEN HAVE FEW FACES ZHIHUA QIAO and J. MICHAEL STEELE Abstract. We construct a continuous distribution G such that the number of faces in the smallest concave majorant
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationComplete Moment Convergence for Weighted Sums of Negatively Orthant Dependent Random Variables
Filomat 31:5 217, 1195 126 DOI 1.2298/FIL175195W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Complete Moment Convergence for
More informationMakarov s LIL for SLE
Makarov s LIL for SLE Nam-Gyu Kang Department of Mathematics, M.I.T. Workshop at IPAM, 2007 Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, 2007 1 / 24 Outline 1 Introduction and Preliminaries
More informationOn singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime. Title
itle On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime Authors Wang, Q; Yao, JJ Citation Random Matrices: heory and Applications, 205, v. 4, p. article no.
More informationDISTRIBUTION OF EIGENVALUES OF REAL SYMMETRIC PALINDROMIC TOEPLITZ MATRICES AND CIRCULANT MATRICES
DISTRIBUTION OF EIGENVALUES OF REAL SYMMETRIC PALINDROMIC TOEPLITZ MATRICES AND CIRCULANT MATRICES ADAM MASSEY, STEVEN J. MILLER, AND JOHN SINSHEIMER Abstract. Consider the ensemble of real symmetric Toeplitz
More informationOn rate of convergence in distribution of asymptotically normal statistics based on samples of random size
Annales Mathematicae et Informaticae 39 212 pp. 17 28 Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics, University of Debrecen, Debrecen, Hungary, August
More informationA note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series
Publ. Math. Debrecen 73/1-2 2008), 1 10 A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series By SOO HAK SUNG Taejon), TIEN-CHUNG
More informationThe Kadec-Pe lczynski theorem in L p, 1 p < 2
The Kadec-Pe lczynski theorem in L p, 1 p < 2 I. Berkes and R. Tichy Abstract By a classical result of Kadec and Pe lczynski (1962), every normalized weakly null sequence in L p, p > 2 contains a subsequence
More information1 Probability theory. 2 Random variables and probability theory.
Probability theory Here we summarize some of the probability theory we need. If this is totally unfamiliar to you, you should look at one of the sources given in the readings. In essence, for the major
More informationSOME CONVERSE LIMIT THEOREMS FOR EXCHANGEABLE BOOTSTRAPS
SOME CONVERSE LIMIT THEOREMS OR EXCHANGEABLE BOOTSTRAPS Jon A. Wellner University of Washington The bootstrap Glivenko-Cantelli and bootstrap Donsker theorems of Giné and Zinn (990) contain both necessary
More informationA strong consistency proof for heteroscedasticity and autocorrelation consistent covariance matrix estimators
A strong consistency proof for heteroscedasticity and autocorrelation consistent covariance matrix estimators Robert M. de Jong Department of Economics Michigan State University 215 Marshall Hall East
More informationMean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
J. Math. Anal. Appl. 305 2005) 644 658 www.elsevier.com/locate/jmaa Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
More informationEstimates for probabilities of independent events and infinite series
Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences
More informationMaximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments
Austrian Journal of Statistics April 27, Volume 46, 67 78. AJS http://www.ajs.or.at/ doi:.773/ajs.v46i3-4.672 Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments Yuliya
More informationA Note on the Strong Law of Large Numbers
JIRSS (2005) Vol. 4, No. 2, pp 107-111 A Note on the Strong Law of Large Numbers V. Fakoor, H. A. Azarnoosh Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Iran.
More information3 Integration and Expectation
3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ
More informationRandom Bernstein-Markov factors
Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit
More informationA COMPLETE ASYMPTOTIC SERIES FOR THE AUTOCOVARIANCE FUNCTION OF A LONG MEMORY PROCESS. OFFER LIEBERMAN and PETER C. B. PHILLIPS
A COMPLETE ASYMPTOTIC SERIES FOR THE AUTOCOVARIANCE FUNCTION OF A LONG MEMORY PROCESS BY OFFER LIEBERMAN and PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1247 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
More informationLecture 1: Review of Basic Asymptotic Theory
Lecture 1: Instructor: Department of Economics Stanfor University Prepare by Wenbo Zhou, Renmin University Basic Probability Theory Takeshi Amemiya, Avance Econometrics, 1985, Harvar University Press.
More informationSpatial autoregression model:strong consistency
Statistics & Probability Letters 65 (2003 71 77 Spatial autoregression model:strong consistency B.B. Bhattacharyya a, J.-J. Ren b, G.D. Richardson b;, J. Zhang b a Department of Statistics, North Carolina
More informationOn a class of additive functionals of two-dimensional Brownian motion and random walk
On a class of additive functionals of two-dimensional Brownian motion and random walk Endre Csáki a Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 27, H-364, Hungary.
More informationBahadur representations for bootstrap quantiles 1
Bahadur representations for bootstrap quantiles 1 Yijun Zuo Department of Statistics and Probability, Michigan State University East Lansing, MI 48824, USA zuo@msu.edu 1 Research partially supported by
More informationAsymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½
University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 1998 Asymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½ Lawrence D. Brown University
More informationSupplementary Material for Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data
Supplementary Material for Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data Raymond K. W. Wong Department of Statistics, Texas A&M University Xiaoke Zhang Department
More informationA NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES
Bull. Korean Math. Soc. 52 (205), No. 3, pp. 825 836 http://dx.doi.org/0.434/bkms.205.52.3.825 A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES Yongfeng Wu and Mingzhu
More informationThe Convergence Rate for the Normal Approximation of Extreme Sums
The Convergence Rate for the Normal Approximation of Extreme Sums Yongcheng Qi University of Minnesota Duluth WCNA 2008, Orlando, July 2-9, 2008 This talk is based on a joint work with Professor Shihong
More informationSCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIII 1992 FASC. 2 SCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS BY JACEK D Z I U B A Ń S K I (WROC
More informationResearch Article Almost Sure Central Limit Theorem of Sample Quantiles
Advances in Decision Sciences Volume 202, Article ID 67942, 7 pages doi:0.55/202/67942 Research Article Almost Sure Central Limit Theorem of Sample Quantiles Yu Miao, Shoufang Xu, 2 and Ang Peng 3 College
More informationRandom Graphs. EECS 126 (UC Berkeley) Spring 2019
Random Graphs EECS 126 (UC Bereley) Spring 2019 1 Introduction In this note, we will briefly introduce the subject of random graphs, also nown as Erdös-Rényi random graphs. Given a positive integer n and
More information6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series. MA6622, Ernesto Mordecki, CityU, HK, 2006.
6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series MA6622, Ernesto Mordecki, CityU, HK, 2006. References for Lecture 5: Quantitative Risk Management. A. McNeil, R. Frey,
More information1 Probability space and random variables
1 Probability space and random variables As graduate level, we inevitably need to study probability based on measure theory. It obscures some intuitions in probability, but it also supplements our intuition,
More informationLARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS*
LARGE EVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILE EPENENT RANOM VECTORS* Adam Jakubowski Alexander V. Nagaev Alexander Zaigraev Nicholas Copernicus University Faculty of Mathematics and Computer Science
More informationRobust Backtesting Tests for Value-at-Risk Models
Robust Backtesting Tests for Value-at-Risk Models Jose Olmo City University London (joint work with Juan Carlos Escanciano, Indiana University) Far East and South Asia Meeting of the Econometric Society
More informationProofs for Large Sample Properties of Generalized Method of Moments Estimators
Proofs for Large Sample Properties of Generalized Method of Moments Estimators Lars Peter Hansen University of Chicago March 8, 2012 1 Introduction Econometrica did not publish many of the proofs in my
More informationEXPLICIT MULTIVARIATE BOUNDS OF CHEBYSHEV TYPE
Annales Univ. Sci. Budapest., Sect. Comp. 42 2014) 109 125 EXPLICIT MULTIVARIATE BOUNDS OF CHEBYSHEV TYPE Villő Csiszár Budapest, Hungary) Tamás Fegyverneki Budapest, Hungary) Tamás F. Móri Budapest, Hungary)
More informationCLASSICAL PROBABILITY MODES OF CONVERGENCE AND INEQUALITIES
CLASSICAL PROBABILITY 2008 2. MODES OF CONVERGENCE AND INEQUALITIES JOHN MORIARTY In many interesting and important situations, the object of interest is influenced by many random factors. If we can construct
More informationA Quadratic ARCH( ) model with long memory and Lévy stable behavior of squares
A Quadratic ARCH( ) model with long memory and Lévy stable behavior of squares Donatas Surgailis Vilnius Institute of Mathematics and Informatics onatas Surgailis (Vilnius Institute of Mathematics A Quadratic
More informationReducibility of generic unipotent standard modules
Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric
More informationOn the Kolmogorov exponential inequality for negatively dependent random variables
On the Kolmogorov exponential inequality for negatively dependent random variables Andrei Volodin Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A, Canada e-mail: volodin@math.uregina.ca
More informationA CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin
A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN Dedicated to the memory of Mikhail Gordin Abstract. We prove a central limit theorem for a square-integrable ergodic
More informationAn almost sure central limit theorem for the weight function sequences of NA random variables
Proc. ndian Acad. Sci. (Math. Sci.) Vol. 2, No. 3, August 20, pp. 369 377. c ndian Academy of Sciences An almost sure central it theorem for the weight function sequences of NA rom variables QUNYNG WU
More informationSupermodular ordering of Poisson arrays
Supermodular ordering of Poisson arrays Bünyamin Kızıldemir Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore
More informationLecture 3 - Expectation, inequalities and laws of large numbers
Lecture 3 - Expectation, inequalities and laws of large numbers Jan Bouda FI MU April 19, 2009 Jan Bouda (FI MU) Lecture 3 - Expectation, inequalities and laws of large numbersapril 19, 2009 1 / 67 Part
More informationON IDEALS OF TRIANGULAR MATRIX RINGS
Periodica Mathematica Hungarica Vol 59 (1), 2009, pp 109 115 DOI: 101007/s10998-009-9109-y ON IDEALS OF TRIANGULAR MATRIX RINGS Johan Meyer 1, Jenő Szigeti 2 and Leon van Wyk 3 [Communicated by Mária B
More informationEmpirical Processes: General Weak Convergence Theory
Empirical Processes: General Weak Convergence Theory Moulinath Banerjee May 18, 2010 1 Extended Weak Convergence The lack of measurability of the empirical process with respect to the sigma-field generated
More informationKybernetika. István Berkes; Lajos Horváth Approximations for the maximum of stochastic processes with drift
Kybernetika István Berkes; Lajos Horváth Approximations for the maximum of stochastic processes with drift Kybernetika, Vol. 39 (2003), No. 3, [299]--306 Persistent URL: http://dml.cz/dmlcz/135532 Terms
More informationEötvös Loránd University, Budapest. 13 May 2005
A NEW CLASS OF SCALE FREE RANDOM GRAPHS Zsolt Katona and Tamás F Móri Eötvös Loránd University, Budapest 13 May 005 Consider the following modification of the Barabási Albert random graph At every step
More informationMath 6810 (Probability) Fall Lecture notes
Math 6810 (Probability) Fall 2012 Lecture notes Pieter Allaart University of North Texas September 23, 2012 2 Text: Introduction to Stochastic Calculus with Applications, by Fima C. Klebaner (3rd edition),
More informationMOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES
J. Korean Math. Soc. 47 1, No., pp. 63 75 DOI 1.4134/JKMS.1.47..63 MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES Ke-Ang Fu Li-Hua Hu Abstract. Let X n ; n 1 be a strictly stationary
More information1 Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type - part I
1 Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type - part I after F. L. Nazarov [3] A summary written by Christoph Marx Abstract A classical
More informationEstimating GARCH models: when to use what?
Econometrics Journal (2008), volume, pp. 27 38. doi: 0./j.368-423X.2008.00229.x Estimating GARCH models: when to use what? DA HUANG, HANSHENG WANG AND QIWEI YAO, Guanghua School of Management, Peking University,
More informationConditional moment representations for dependent random variables
Conditional moment representations for dependent random variables W lodzimierz Bryc Department of Mathematics University of Cincinnati Cincinnati, OH 45 22-0025 bryc@ucbeh.san.uc.edu November 9, 995 Abstract
More informationAN IMPROVED MENSHOV-RADEMACHER THEOREM
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996 AN IMPROVED MENSHOV-RADEMACHER THEOREM FERENC MÓRICZ AND KÁROLY TANDORI (Communicated by J. Marshall Ash) Abstract. We
More informationCHANGE DETECTION IN TIME SERIES
CHANGE DETECTION IN TIME SERIES Edit Gombay TIES - 2008 University of British Columbia, Kelowna June 8-13, 2008 Outline Introduction Results Examples References Introduction sunspot.year 0 50 100 150 1700
More informationDetecting instants of jumps and estimating intensity of jumps from continuous or discrete data
Detecting instants of jumps and estimating intensity of jumps from continuous or discrete data Denis Bosq 1 Delphine Blanke 2 1 LSTA, Université Pierre et Marie Curie - Paris 6 2 LMA, Université d'avignon
More informationORTHOGONAL SERIES REGRESSION ESTIMATORS FOR AN IRREGULARLY SPACED DESIGN
APPLICATIONES MATHEMATICAE 7,3(000), pp. 309 318 W.POPIŃSKI(Warszawa) ORTHOGONAL SERIES REGRESSION ESTIMATORS FOR AN IRREGULARLY SPACED DESIGN Abstract. Nonparametric orthogonal series regression function
More information