## Doctoral Theses

### On the Asymptotic Theory of Quantiles and L-Statistics.

6-28-1978

6-28-1979

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Let {Xi} be a sequence of r.v.s. nth stage we At the define e.d.f. as Fn (x) = (# Xi, â‰¤ x :â‰¤ i â‰¤ n)/n and the ttn sample quantile as Qnt = inf {x : Fn (x) â‰¥ t} for t > 0 and = en(0+) for t=0. Most of the techniques of studying the process Qnt: 0 â‰¤ tâ‰¤ 1} consist of relating {Qnt} with some suitable linear statistics. The following are some of the commonly used methods for studying quantiles :(i) The Direct Methods In The Independent Case. Here, one can actually find the exact dis tribution of quantiles and investigate their properties. This mothod is commonly found in the older literature on order statistics. Recently, Reiss (1976 )applied this method to get Edgeworth expansion of the distributions of quantiles. This procedure is not at all flexible if the underlying r.v.s are dependent.(ii) Methods Using Set Inequalities. It is easy to see that (Fn (x) > t} âŠ† {Qntâ‰¤ x} âŠ† {Fn(x) â‰¥ t}.These set inoqualities are used to find weak and strong laws for quantiles in the independent as well as in the weakly dependent cases (see section 1,8 for definitions of weak-dependence structures that we will be dealing with). Reiss (1974) uses this technique and obtains the Berry-Essean bound for quantiles in the i.1.d. case. The main drawback of this method lies in tho fact that it does not wo rk when we have a linear combination of quantilos.(iii) Methods Using A Property of Unifom Distribution.Let and {Uj} be a sequence of i.1.d. U[0,1] r.v.s and = U1tlog(1/Ui). If Ukn denotes the kth order statistics at the nth stÄ…ge, then (1,1.1) Uk,n âˆ‘ ki=1 Uit /âˆ‘ i=1n+1Uit, k=1,3,.....,n.The statistic in tho r.h,s. behaves moro or less like a linear statistic. Further, if ,n denote the k th order statistics(at the nth stago) of i.i,d, observations (Xi with a continuous d.f. F as the underlying d.f., then Xk,n=F-1(Uk,n), k = 1,2,...,n (we take the left continuous version of F-1). Thus the relation- ship (1.1.1) providos a mothod for investigating the probabilistic behaviours of order statistics and their functions. Chernoff et al (1967), Bjerve (1977), CsÃ³rgo and RÃ©vÄ›sz (1977), among others, use this technique. Just like the direct methoÃ¡, this also has the drawback in that it depends heavily on the independence of the underlying r.v.s.(iv) Methods Using The Bahadur-Kiefer Repres entation of Quantiles.Let {Xi} be ah i.i.d. sequenee of r.v.s with the underlying d.,r. F. Bahadur (1966) provod that if F is twice differentiable in a neighbourhood of F-1(t), o < 1, with the first derivative bounded away from zoro and the second bounded then (1,1,2) Qnt- F-1(t) = [t - - F-1(t))1/P (F(t)) + O(n-3/4)(log n 1/2)1/2 (log log n)1/4 (log a.s. Many of the asymptotic properties of quantiles are immediate from (1,1,2). Given. below is a survey of the literature on this area of the investigation.

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