On the speed of convergence in the central limit theorem of sequential log-likelihood ratio processes. (English) Zbl 0808.62068

Basu, A. K.; Bhattacharya, Debasis

Sequential Anal. 13, No. 2, 97-111 (1994).

Let \(X_ 1, X_ 2,\dots\) be a general dependent sequence of random variables with distribution depending on a parameter vector \(\theta\). Let \(P_{m\theta}\) denote the joint distribution of \(X_ 1, X_ 2,\dots, X_ m\) and \(\Lambda_ m (\theta^*, \theta_ 0)=\log dP_{m\theta^*}/ dP_{m\theta_ 0}\) the log-likelihood ratio statistic, where \(\theta_ 0\) is the true value of \(\theta\) and \(\theta^*= \theta_ m^*\) is a close alternative. Let \((\nu_ n)\) be a sequence of stopping times such that \(\nu_ n/n\) tends to 1 in probability and consider the randomly stopped process \(\Lambda_{\nu_ n} (\theta^*, \theta_ 0)\).
Under a large set of regularity assumptions uniform and nonuniform bounds are derived for the rate of convergence of the suitably normalized sequence \(\overline{\Lambda}_{\nu_ n} (\theta^*, \theta_ 0)\) to the standard normal distribution. A first order autoregressive model and the discrete Ornstein-Uhlenbeck process are studied as particular examples.

Reviewer: T.F.Móri (Budapest)

MSC:

62L10	Sequential statistical analysis
60F05	Central limit and other weak theorems
60G40	Stopping times; optimal stopping problems; gambling theory
62L12	Sequential estimation

Keywords:

CLI; martingale; dependent sequence of random variables; log-likelihood ratio statistic; randomly stopped process; uniform and nonuniform bounds; rate of convergence; first order autoregressive model; discrete Ornstein- Uhlenbeck process

References:

[1]	Akritas M.G., Papers dedicated to Prof.Illevls 70th Anniversary 2 pp 90– (1984)
[2]	Anscombe, F.J. Large Sample Theory of Sequential Estimation. Proc.Camb.Phil.Soc. Vol. 48, pp.600–607. · Zbl 0047.13401 · doi:10.1017/S0305004100076386
[3]	Basu A.K., Appeared In abrldged form In the proceedings of Statistics section of the Indian Science Congress (1989)
[4]	Basu A.K., Cal.Statist.Assoc.Bull. 37 pp 143– (1988)
[5]	Basu A.K. Bhattacharya D. On the speed of convergence In the Central Limit Theorem of log-likelihood ratio process Tech. Report # 3/91. Calcutta Unlverslty 1990a To appear in the J.Theor.Prob
[6]	Bolthausen E., Ann.Prob. 10 pp 672– (1982) · Zbl 0494.60020 · doi:10.1214/aop/1176993776
[7]	Haeusler E., Ann.Prob. 16 pp 1699– (1988) · Zbl 0656.60034 · doi:10.1214/aop/1176991592
[8]	Inagaki V., Ann.Inst.Statist.Math. 27 pp 391– (1975) · Zbl 0381.62023 · doi:10.1007/BF02504659
[9]	Jeganathan P. Some asymptotic theory with appllcations to tlme series models Tech.Report # 166.University of Michigan 1988
[10]	Lecam, L. and Yang, G.L. 1990. ”Asymptotlcs in Statlstfcs,some basic concepts”. Springer-Verlag.
[11]	Mlchel R., Wahrsch.Und.Verw.Gebiete. 18 pp 73– (1971) · doi:10.1007/BF00538488
[12]	Roussas, G.G. 1972. ”Contiguity fn probability measures”. Oxford Unlverslty press. · Zbl 0265.60003 · doi:10.1017/CBO9780511804373
[13]	Roussas G.G., Z. Wahrsch. Verw. Geblete pp 31– (1979)