1. Introduction

The Type I Pareto distribution is a continuous distribution with probability density function (p.d.f.)

(1)

where θ is a positive number.

The Pareto distribution was first formulated in the late 1800s by the Italian economist Vilfredo Pareto, who used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe the distribution of income. The Pareto distribution is not limited to describing wealth or income. It can be used as the distribution for insurance loss, oil reserve in a oil field, standardized price return on an individual stock, area burnt in a forest fire, etc…

Let X₁, , …, X_n denote independent observations to be taken sequentially up to a predetermined stage n from the Pareto distribution with p.d.f. (1). It is desired to estimate the parameter θ, subject to the loss function

(2)

where c is a known cost per observation and is the maximum likelihood estimator of θ. Since the log-likelihood function is for observed values x₁ > 1, …, x_n > 1, of X₁, , …, X_n, it follows that the maximum likelihood estimator of θ is

where with Y_i = lnX_i, i = 1, …, n, and where the random variables Y₁, …, Y_n are independent with common distribution the Exponential distribution with mean Expanding h(y) = 1/y about y = 1/θ and substituting in the obtained expansion yields for large n. It follows that the risk incurred by estimating θ by under the loss (2) is

for large n. The approximate risk is minimized with respect to n by choosing n as the greatest integer less than or equal to

(3)

The minimum risk is

for sufficiently small c. Since n* depends on the unknown value of θ, there is no fixed-sample-size procedure that attains the minimum risk Therefore, we propose to use the sequential procedure which stops the sampling process after observing Y₁, …, Y_t and estimates θ by , where

(4)

with m being a positive integer. The performance of the procedure is measured by its regret, which is defined as

(5)

for c > 0.

In this paper we propose a stopping time, t, based on (3), and provide a second-order asymptotic expansion, as c → 0, for the regret incurred by the sequential procedureunder the loss (2). We also show that the estimator is asymptotically unbiased for the shape parameter θ.

The problem of sequentially estimating the mean of a population was studied by many authors. Starr and Woodroofe[7] considered the case in which X₁, X₂, … are i.i.d. Normal random variables and showed that the regret of his procedure is O(1). Then, Woodroofe[10] showed that the regret is 0.5 + o(1) if m ≥ 4. Martinsek[8] extended Woodroofe’s[10] result to the nonparametric case. For the distribution-free case, Ghosh and Mukhopadhyay[3] and Chow and Yu[5] established asymptotic risk efficiency under certain conditions. Tahir[9] proposed a class of bias-reduction estimators of the mean of the one-parameter exponential family and provided an asymptotic second-order lower bound for the regret.

2. Asymptotic Expansion for the Regret of the Sequential Procedure

Theorem 1: Let t be as in (4) with m ≥ 1, then as c → 0.

Proof: The stopping time t can be rewritten as

(6)

Furthermore, let for i = 1, 2, … and let Then, with g(u) = u + 1. Since g is a convex function and

where U ⁺= max {U, 0}, it follows from Proposition 5 and Theorem 1 of[2] that as c → 0, where ρ denotes the asymptotic mean of the excess over the stopping boundary, B_c = tZ_t – n*. Moreover,

where

The theorem follows.

Expanding h(y) = 1/y about and substituting in the obtained expansion yields

(7)

where W_t is a random variable lying between and

Lemma 1: Let t be as in (4) and let p > 1. Then,

(i) w.p.1 as c → 0.

(ii) is uniformly integrable;

(iii) is uniformly integrable if m > p.

Proof. Assertion (i) holds by Proposition 2 of Aras and Woodroofe[2]. For the second assertion,

since by the definition of t. Thus, Assertion (ii) holds since

By Doob’s maximal inequality (see[4]). To establish (iii), observe that on {t > m}, by the definition of t; so that

where I_{.} denotes the indicator function. This implies that

by the c_r-inequality (see Loève[6]). Since

and

for some A >0, by Doob’s maximal inequality, Assertion (ii) follows.

Lemma 2: Let p > 1. If m > p, then,

(i) is uniformly integrable and

(ii) is uniformly integrable.

Proof. The first assertion follows by using Assertion (i) of Lemma 1 and Theorem 2 of[3]. The second assertion follows since and

for some constant B > 0, by Doob’s maximal inequality.

Theorem 2: Let be as in (5). Then, as c → 0.

Proof. It follows from (5) that

(8)

for c > 0. Next, by Corollary 1 of Aras and Woodroofe[2],

(9)

as c → 0. Now, let

(10)

and let Z denote a random variable having the Standard Normal distribution. Then,

(11)

as c → 0, by Assertion (i) of Lemma 1, the fact that Q_t → Z in distribution as c → 0 by Anscombe’s theorem (see[1]) and the fact that W_t → 1/θ w.p.1 as c → 0. The uniform integrability of follows from Assertions (ii) of Lemmas 1 and 2 since

Now take the limit as c → 0 in (8) and use (9) and (11) to complete the proof of the theorem.

Lemma 3: Let t be defined by (4). If m > 4, then as c → 0.

Proof: It follows from (7) that

(12)

where W_t is a random variable lying between and µ_Y = Next,

(13)

by (3) and Wald’s lemma. Moreover, it follows from (6) that

say, where denotes the excess over the stopping boundary and Q_t is as in (10). Now, as c → 0, by Anscombe’s theorem and the fact that is uniformly integrable. Also, by Hölder’s inequality,

as c → 0, by Proposition 7 of[2] and Lemmas 1 and 2.Using these results in (13) yields

(14)

For the second term in (12), observe that

(15)

by the fact that W_t → 1/θ w.p.1 as c → 0, Assertions (i) and (ii) of Lemma 1, Anscombe’s theorem and the uniform integrability of . The lemma follows by using (14) and (15) in (12).

3. Conclusions

1- We have proposed a sequential procedure for estimating the shape parameter of the type I Pareto distribution and provided a second-order asymptotic expansion for the regret. The procedure can be used for the data representing the losses for insurance policyholders or the insured values of homes. It specifies how many policyholders or homes should be selected and provides an estimate of θ based on this number.

2- Since the asymptotic regret is positive for small values of the cost of sampling, it would be interesting to find a sequential procedure whose asymptotic regret is negative.