ISRN Applied MathematicsVolume 2012 (2012), Article ID 945627, 13 pagesdoi:10.5402/2012/945627

Research Article

On the Order Statistics of Standard Normal-Based Power Method Distributions

Todd C. Headrick and Mohan D. Pant

Section on Statistics and Measurement, Department of EPSE, Southern Illinois University Carbondale, P.O. Box 4618, 222-J Wham Building, Carbondale, IL 62901-4618, USA

Received 17 January 2012; Accepted 5 March 2012

Academic Editors: T. Y. Kam and G. Stavroulakis

Copyright © 2012 Todd C. Headrick and Mohan D. Pant. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution () and its powers of order three and five ( and ). The procedure is demonstrated for sample sizes of . It is shown that and have expectations of order statistics that are functions of the expectations for and can be expressed in terms of explicit elementary functions for sample sizes of . For sample sizes of the expectations of the order statistics for , , and only require a single remainder term.

1. Introduction

Order statistics have played an important role in the development of techniques associated with estimation [1, 2], hypothesis testing [3, 4], and describing data in the context of L-moments [5, 6]. In terms of the latter, L-moments are based on the expectations of linear combinations of order statistics associated with a random variable . Specifically, the first four L-moments are expressed asor more generally as where the order statistics are drawn from the random variable . The values of and are measures of location and scale and are the arithmetic mean and one-half the coefficient of mean difference (or Gini’s index of spread), respectively. Higher-order L-moments are transformed to dimensionless quantities referred to as L-moment ratios defined as for , and where and are the analogs to the conventional measures of skew and kurtosis. In general, L-moment ratios are bounded in the interval as is the index of L-skew () where a symmetric distribution implies that all L-moment ratios with odd subscripts are zero. Other smaller boundaries can be found for more specific cases. For example, the index of L-kurtosis () has the boundary condition for continuous distributions of [7]

Headrick [8] derived classes of standard normal-L-moment-based power method distributions using the polynomial transformation where . Setting gives the third- (fifth-) order class of power method distributions. The shape of in (1.4) is contingent on the values of the constant coefficients . For the larger class of nonnormal distributions associated with , the coefficients are computed from the system of equations given in Headrick ([8, Equations ] for specified values of L-moment ratios (). In general, and are standardized to the unit normal distribution as

The pdf and cdf associated with (1.4) are given in parametric form as in [8, Equations and ] where and are the parametric forms of the pdf and cdf with the mappings and with , and where and are the standard normal pdf and cdf, respectively. For further details on the distributional properties associated with power method transformations see [9, pages 9–30] and [8] in terms of conventional moment and L-moment theory, respectively.

Of concern in this study are three power method distributions related to (1.4) and (1.5) as and thus and . Note that these power method distributions are symmetric and imply that in (1.4). The graphs of the pdfs associated with the distributions in (1.7) are given in Figure 1 along with their values of L-skew and L-kurtosis. We would point out that the importance of these distributions was noted by Stoyanov [10, page 281], “…power transformations [such as and ] can be considered as functional transformations on random data, usually called Box-Cox transformations. Their importance in the area of statistics and its applications is well known.”

(a)

(b)

(c)

Figure 1: Graphs of the three standard normal-based power method distributions in (1.7) and their values of L-skew () and L-kurtosis ().

The standard normal distribution in (1.7) is the only case of the three distributions considered that is moment determinant. That is, and have the so-called classical problem of moments insofar as their respective cdfs have nonunique solutions (i.e., they are moment indeterminant, see [10–12]). However, as pointed out by Huang [12], and are determinant in the context of order statistics moments.

The derivation of the expected values of single order statistics associated with in terms of explicit elementary functions has been attempted by numerous authors (see [13–17]). As indicated by Johnson et al. [18, pages 93-94] these attempts fail to give explicit expressions in terms of elementary functions for the expected values of order statistics with sample sizes of . However, Renner [19] provides a technique for expressing the expected values of order statistics associated with for based on a single power series.

There is a paucity of research on the expectations of order statistics associated with and in the context of explicit elementary functions. Thus, what follows in Section 2 is the development of an approach for determining the expected values of the order statistics for and , which is based on a generalization of Renner’s [19] discussion in the context of . In Section 3, some specific evaluations of the generalization are provided to demonstrate the methodology.

2. Methodology

The expected values of the order statistics associated with (1.7) can be determined based on the following expression [20, page 34]: where is defined as in (1.7) and and are the pdf and cdf of the folded unit normal distribution at . Table 1 gives a summary of some specific expansions of the polynomial in (2.1) for sample sizes of , which are applicable to all three distributions related to . Inspection of Table 1 indicates that we have in general (a) , (b) the median , and (c) the are linear combinations of the integrals for , with only odd subscripts appearing as only odd powers of appear in the polynomial expansions associated with (2.1). As such, in (2.1) can be expressed as

Table 1: General expressions for the expected values of the order statistics for in (1.7) and sample sizes of . denotes an integral in (2.1) where .

Sample size ()
Expected value

1

2

3

4

5

6

7

8

9

Equation (2.2) may be integrated by parts as where , and , for , , and , respectively. Note that and Evaluating (2.3) for gives a coefficient of mean difference of for all in (1.7), which is consistent with the specification in (1.5) and given in Table 1.

The expression in (2.3) can be expressed as or analogously as a double integral over as Using (2.6), let and thus . Further, let . As such, the region of integration will be reduced to one-half of the area of the original rectangle associated with (2.6). Thus, we have Subsequently, setting in (2.7), where , gives and hence Expanding (2.9) yields where the subscript runs faster than . For example, if , then (2.10) would appear more specifically as

Substituting (2.10) into (2.3) and initially integrating with respect to (Lichtenstein, [21]) yields where the specific forms of , which are associated with , are

Equations (2.13) can be more conveniently expressed as where the specific forms of are and where in (2.17) indicates summing over all pairwise combinations. Hence, the integral in (2.3) can be expressed as and subsequently substituting (2.14) into (2.18) gives

The integral associated with in (2.19) cannot be expressed in terms of explicit elementary functions for , which also implies and sample sizes of in Table 1. As such, we will consider the approximating function as where

Thus, for finite we have where is the remainder term required for and where for and . Thus, using (2.22), (2.19) can be expressed as

The remainder terms in (2.23) can be solved by using (2.3), (2.15), (2.23), and the error function Erf [22], where Erf would replace in (2.3) where . More specifically, Table 2 gives the values of for , and 50 with 40-digit precision. Inspection of Table 2 indicates that the (positive) remainder term achieves a maximum at and thereafter tends to zero as increases (i.e., for ).

Table 2: Computed values of the remainder term associated with (2.23). The values were computed with 40-digit precision.

Sample size ()
Integral
Remainder term

,

We would note that the approach taken here to determine is analogous to Renner’s [19] approach of developing a power series for this value. That is, the remainder term in Table 2 is also the value approximated in [19] for . Further, we would note that extending the approach in [19] for computing the remainder terms for would become computationally burdensome.

To demonstrate (2.23) more specifically, if and in (1.7), then the integral associated with would appear as

3. Evaluations

Tables 3–5 give evaluations for the expected values of the order statistics for , and in (1.7), which are based on (2.23) and the general formulae given in Table 1 for sample sizes of . Inspection of Tables 4 and 5 indicates that the expected values for and are all expressed in terms of elementary functions and are also functions of the expectations associated with in Table 3.

Table 3: Expected values of order statistics for for .

Table 4: Expected values of order statistics for for .

Table 5: Expected values of order statistics for for .

Presented in Tables 6, 7 and 8 are the evaluations for all three distributions in (1.7) for samples of sizes where the expectations of the order statistics for , , and are all expressed in terms of explicit elementary functions and a single remainder term. Tables 9 and 10 give the expected values of the order statistics associated with the standard normal distribution for sample sizes of and , respectively. We would also note that Mathematica [22] software is available from the authors for implementing the methodology.

Table 6: Expected values of order statistics for for .

Table 7: Expected values of order statistics for for .

Table 8: Expected values of order statistics for for .

Table 9: Expected values of order statistics for for .

Table 10: Expected values of order statistics for for .

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