Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Babu, Gutti Jogesh (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

This thesis deals with an asymptotic study of estimatore in some discrete time and continuOus time models.The firet part deele with time eeriee date modelled by moving sverage and eutoregreseive proceseee. Higher order as ymptotioe beyond. as ymptotio normality, of the usual astimatore heve been dealt by Phillipe (1977,1978), Durbin (198o), Oohi (1983), Tanaka (1983), Fujikoshi and Ochi (1984). o(n-1/2) or o(n-1) expenaions ara 'obteined by these authore under the assumption of normality of orrors. The reason for this wes the lack of suitable. expaneions for normalized sums of dependent random vectora. Recently Gotza and Hipp (1983) heve been able to solve thie problem. They have obtained Edgevorth expanaion -for nor- malized suma of weakly dependent random vectore under feirly goneral conditione.We start by showing that these recent reaults of Gotze and Hipp (1983) enable us to dorive general asymptutio expaneione for the distri- bution of autocovariances from oertain linear processes of the form where (εi) are 1.1.d. with sufficiently high order momenta. The main conditions imposed are oxponential decay of the 8equence (δr) and Cramer's condition on (ε1 ,ε12). These results are applied to moving average and autoregreseive processee (of any order) under atability conditions. Edgeworth expansions of any order (depending upon the existence of momenta of ε1) for the distribution of the usual 6s timators follow from the main result. The Barry-Esseen bounds in all these eituations hold as e corollary. These are. the.contents of Chapter l.Having proved that the normal approximation le of the order o(n-1/2), it is natural to enqóire whether there ie a better approxima- tion. This idea etema from the recent concept of bootatrap.The bootstrep Le beeicelly a non-paromotric procedure and wan Introduced by Efron (1979 1982). Since then, it has also boon used in parametric aituations. Ita performance on eimulatad data, both in para- metric and non-perametric aituatione is quite encouraging. See B.g. Efron (1979,1982,1985), Blckel and Freedman (1983.), Freadman and Petere (1984a, bye) ate, Simultanecusly, various authors have tried' to provide theoretical justification as to uhý thie method parforma well. The main worke in this direction are by Blokol and Freedman (1980, 1981), Singh (1981), Beren (1982) and Babu and Singh (1984). These results deal with eccuraoy of the bootstrap approximation in various seneee (a.g. asymptotic norma- lity, Edgeworth axpeneione etc.) mainly for sample mean type etatis tice (or their functionals), quantiles stc. in the i.i.d. situation, where the basic asymptotic distribution theory is normal. For nice functionale, the bootstrap opproximation out-performs the normal approxination. It was anticipated by various autnore that the bootstrap would work (in the sense of yialding the samo ae ymptotic distribution as for the original statistic) even in dependent aituations provided the resempling takas cars of the dependence proparly. Freedman (1984) con- firma this by shouing that it does work for two stage least aqueres eeti- mates in linear autoragressione with possible oxoganeous varisbles ortho- gonal to errors.


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