Bayesian Sequential Estimation of the Inverse of the Pareto Shape Parameter

Mohamed Tahir

doi:10.5539/ijsp.v5n3p49

Abstract

The problem addressed is that of developing a sequential procedure for estimating the inverse of the shape parameter of the Pareto distribution under the squared loss, assuming that the shape parameter is the value of a random variable having a density function with compact support and that the cost per observation is one unit. A stopping time is proposed and a second-order asymptotic expansion is obtained for the Bayes regret incurred by the proposed procedure.

Highlights

Let X1, ..., Xn denote independent observations to be taken sequentially from the Pareto distribution with p.d.f. θ fθ ( x) = xθ +1 if x ≥ 1 (1)0 if not, where θ is an unknown positive number
The problem addressed is that of developing a sequential procedure for estimating the inverse of the shape parameter of the Pareto distribution under the squared loss, assuming that the shape parameter is the value of a random variable having a density function with compact support and that the cost per observation is one unit
The sample size n is not chosen in advance; instead, data are analyzed as they become available and whether to stop taking observations is decided according a stopping time t, say. t is a stopping time means that t takes on the values 1, 2, ... and has the properties that P{t < ∞} = 1 and that {t = n} ∈ DDnn for each integer n ≥ 1, where DDnn is the sigma-algebra generated by X1, ..., Xn

Summary

Introduction

Let X1, ..., Xn denote independent observations to be taken sequentially from the Pareto distribution with p.d.f. Throughout this paper, it is assumed that θ (the shape parameter) is a value of a random variable Θ having (prior) density function ξ with compact support in (0, ∞) and the objective is to determine a stopping time t for which the Bayes regret (see (5) below) of the procedure (t,δt ) is as small as possible for large a. Since na depends on the unknown value of θ, there is no fixed-sample-size procedure that attains the minimum risk Ra* in practice. We propose to use the sequential procedure (t,δt ) which stops the sampling process after observing Y1, ..., Yt and estimates δ (θ ) by δt , where. Note that the estimate of the population mean is where t is given by (4)

Preliminary Results Lemma 1

The Main Result

Conclusions