Table of Contents
ISRN Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 945627, 13 pages
Research Article

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

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.Β Kam and G.Β Stavroulakis

Copyright Β© 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.


This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution (𝑍) and its powers of order three and five (𝑍3 and 𝑍5). The procedure is demonstrated for sample sizes of 𝑛≀9. It is shown that 𝑍3 and 𝑍5 have expectations of order statistics that are functions of the expectations for 𝑍 and can be expressed in terms of explicit elementary functions for sample sizes of 𝑛≀5. For sample sizes of 𝑛=6,7 the expectations of the order statistics for 𝑍, 𝑍3, and 𝑍5 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 𝑋. Specifically, the first four L-moments are expressed asπœ†1𝑋=𝐸1∢1ξ€»,πœ†2=12𝐸𝑋2∢2βˆ’π‘‹1∢2ξ€»,πœ†3=13𝐸𝑋3∢3βˆ’2𝑋2∢3+𝑋1∢3ξ€»,πœ†4=14𝐸𝑋4∢4βˆ’3𝑋3∢4+3𝑋2∢4βˆ’π‘‹1∢4ξ€»(1.1)or more generally as πœ†π‘Ÿ=1π‘Ÿπ‘Ÿβˆ’1𝑗=0(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈξ€Ίπ‘‹π‘Ÿβˆ’1π‘Ÿβˆ’π‘—βˆΆπ‘Ÿξ€»,(1.2) where the order statistics 𝑋1βˆΆπ‘›β‰€π‘‹2βˆΆπ‘›β‰€β‹―β‰€π‘‹π‘›βˆΆπ‘› are drawn from the random variable 𝑋. The values of πœ†1 and πœ†2 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 πœπ‘Ÿ=πœ†π‘Ÿ/πœ†2 for π‘Ÿβ‰₯3, and where 𝜏3 and 𝜏4 are the analogs to the conventional measures of skew and kurtosis. In general, L-moment ratios are bounded in the interval βˆ’1<πœπ‘Ÿ<1 as is the index of L-skew (𝜏3) 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 (𝜏4) has the boundary condition for continuous distributions of [7] 5𝜏23βˆ’14<𝜏4<1.(1.3)

Headrick [8] derived classes of standard normal-L-moment-based power method distributions using the polynomial transformation𝑝(𝑍)=π‘šξ“π‘–=1π‘π‘–π‘π‘–βˆ’1,(1.4) where π‘βˆΌπ‘–.𝑖.𝑑.𝑁(0,1). Setting π‘š=4(π‘š=6) gives the third- (fifth-) order class of power method distributions. The shape of 𝑝(𝑍) in (1.4) is contingent on the values of the constant coefficients 𝑐𝑖. For the larger class of nonnormal distributions associated with π‘š=6, the coefficients are computed from the system of equations given in Headrick ([8, Equations (2.8)-(2.13)] for specified values of L-moment ratios (𝜏3,…,6). In general, πœ†1 and πœ†2 are standardized to the unit normal distribution asπœ†1=𝑐1+𝑐3+3𝑐5πœ†=0,2=ξ€·4𝑐2+10𝑐4+43𝑐6ξ€Έ4βˆšπœ‹=1βˆšπœ‹.(1.5)

The pdf and cdf associated with (1.4) are given in parametric form as in [8, Equations (1.3) and (1.4)]𝑓𝑝(𝑧)(𝑝(𝑧))=𝑓(𝑧)=𝑝(𝑧),πœ™(𝑧)π‘ξ…žξ‚Ά,𝐹(𝑧)𝑝(𝑧)(𝑝(𝑧))=𝐹(𝑧)=(𝑝(𝑧),Ξ¦(𝑧)),(1.6) where π‘“βˆΆβ„œβ†¦β„œ2 and πΉβˆΆβ„œβ†¦β„œ2 are the parametric forms of the pdf and cdf with the mappings 𝑧↦(π‘₯,𝑦) and 𝑧↦(π‘₯,𝑣) with π‘₯=𝑝(𝑧),𝑦=πœ™(𝑧)/π‘ξ…ž(𝑧),𝑣=Ξ¦(𝑧), and where πœ™(𝑧) and Ξ¦(𝑧) 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 𝑝𝑑(𝑍)=𝑐2𝑑𝑍2π‘‘βˆ’1⎧βŽͺ⎨βŽͺ⎩,whereif𝑑=1,𝑐2=1,𝑐4=0,𝑐6=0,𝑑=2,then𝑐2=0,𝑐4=2/5,𝑐6=0,𝑑=3,𝑐2=0,𝑐4=0,𝑐6=4/43,(1.7) and thus 𝑝1(𝑍)=𝑍,𝑝2(𝑍)=(2/5)𝑍3 and 𝑝3(𝑍)=(4/43)𝑍5. Note that these power method distributions are symmetric and imply that 𝑐1,3,5=0 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], β€œβ€¦power transformations [such as 𝑝2(𝑍) and 𝑝3(𝑍)] 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.”

Figure 1: Graphs of the three standard normal-based power method distributions 𝑝𝑑(𝑍) in (1.7) and their values of L-skew (𝜏3) and L-kurtosis (𝜏4).

The standard normal distribution 𝑝1(𝑍) in (1.7) is the only case of the three distributions considered that is moment determinant. That is, 𝑝2(𝑍) and 𝑝3(𝑍) 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], 𝑝2(𝑍) and 𝑝3(𝑍) are determinant in the context of order statistics moments.

The derivation of the expected values of single order statistics associated with 𝑝1(𝑍) 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 𝑛>5. However, Renner [19] provides a technique for expressing the expected values of order statistics associated with 𝑝1(𝑍) for 𝑛=6,7 based on a single power series.

There is a paucity of research on the expectations of order statistics associated with 𝑝2(𝑍) and 𝑝3(𝑍) 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 𝑝2(𝑍) and 𝑝3(𝑍), which is based on a generalization of Renner’s [19] discussion in the context of 𝑝1(𝑍). 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]: 𝐸𝑝(𝑍)π‘—βˆΆπ‘›ξ€»=𝑛2βˆ’π‘›βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ€œπ‘›βˆ’1π‘—βˆ’1∞0𝑝𝑑[](𝑧)πœ‘(𝑧)1+Ξ¨(𝑧)π‘—βˆ’1[]1βˆ’Ξ¨(𝑧)π‘›βˆ’π‘—βˆ’[]1βˆ’Ξ¨(𝑧)π‘—βˆ’1[]1+Ξ¨(𝑧)π‘›βˆ’π‘—ξ€Έπ‘‘π‘§,(2.1) where 𝑝𝑑(𝑧) is defined as in (1.7) and πœ‘(𝑧)=2πœ™(𝑧) and Ξ¨(𝑧)=2Ξ¦(𝑧)βˆ’1 are the pdf and cdf of the folded unit normal distribution at 𝑧=0. Table 1 gives a summary of some specific expansions of the polynomial in (2.1) for sample sizes of 𝑛=1,…,9, which are applicable to all three distributions related to 𝑝𝑑(𝑧). Inspection of Table 1 indicates that we have in general (a) 𝐸[𝑝(𝑍)π‘—βˆΆπ‘›]=βˆ’πΈ[𝑝(𝑍)𝑛+1βˆ’π‘—βˆΆπ‘›], (b) the median 𝐸[𝑝(𝑍)π‘—βˆΆπ‘›]=βˆ’πΈ[𝑝(𝑍)π‘—βˆΆπ‘›]=0, and (c) the 𝐸[𝑝(𝑍)π‘—βˆΆπ‘›] are linear combinations of the integrals 𝐼2π‘Ÿβˆ’1 for π‘Ÿ=1,2,…, with only odd subscripts appearing as only odd powers of Ξ¨(𝑧) appear in the polynomial expansions associated with (2.1). As such, 𝐼2π‘Ÿβˆ’1 in (2.1) can be expressed as𝐼2π‘Ÿβˆ’1=ξ€œβˆž0𝑝𝑑[](𝑧)πœ‘(𝑧)Ξ¨(𝑧)2π‘Ÿβˆ’1𝑑𝑧.(2.2)

Table 1: General expressions for the expected values of the order statistics for 𝑝𝑑=1,2,3(𝑍) in (1.7) and sample sizes of 𝑛=1,…,9. 𝐼2π‘Ÿβˆ’1 denotes an integral in (2.1) where π‘Ÿ=1,…,4.

Equation (2.2) may be integrated by parts as 𝐼2π‘Ÿβˆ’1ξ€œ=(2π‘Ÿβˆ’1)∞0π‘žπ‘‘(𝑧)πœ‘(𝑧)2[]Ξ¨(𝑧)2π‘Ÿβˆ’2𝑑𝑧,(2.3) where π‘ž1(𝑧)=1, π‘ž2(𝑧)=(2/5)(𝑧2+2) and π‘ž3(𝑧)=(4/43)(𝑧4+4𝑧2+8), for 𝑝1(𝑧), 𝑝2(𝑧), and 𝑝3(𝑧), respectively. Note that Ξ¨(0)=0 and lim𝑧→+βˆžπœ‘(𝑧)=0. Evaluating (2.3) for π‘Ÿ=1 gives a coefficient of mean difference of 𝐼1=ξ€œβˆž0π‘žπ‘‘(𝑧)πœ‘(𝑧)21𝑑𝑧=βˆšπœ‹(2.4) for all 𝑝𝑑(𝑧) in (1.7), which is consistent with the specification in (1.5) and given in Table 1.

The expression [Ξ¨(𝑧)]2π‘Ÿβˆ’2 in (2.3) can be expressed as []Ξ¨(𝑧)2π‘Ÿβˆ’2=ξ‚€2πœ‹ξ‚π‘Ÿβˆ’1ξ‚Έξ€œπ‘§0ξ‚†βˆ’1exp2𝑒2𝑑𝑒2π‘Ÿβˆ’2(2.5) or analogously as a double integral over β„œ2 as []Ξ¨(𝑧)2π‘Ÿβˆ’2=ξ‚€2πœ‹ξ‚π‘Ÿβˆ’1𝑧0ξ‚†βˆ’1exp2𝑧21+𝑧22𝑑𝑧1𝑑𝑧2ξ‚Ήπ‘Ÿβˆ’1.(2.6) Using (2.6), let 𝑧2=𝑧1tanπœƒ1 and thus 𝑑𝑧2=𝑧1sec2πœƒ1π‘‘πœƒ1. Further, let 𝑧21+𝑧22=𝑧21sec2πœƒ1. 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 []Ξ¨(𝑧)2π‘Ÿβˆ’2=ξ‚€2πœ‹ξ‚π‘Ÿβˆ’1ξ‚Έ2ξ€œ0πœ‹/4ξ€œπ‘§0ξ‚†βˆ’1exp2𝑧21sec2πœƒ1𝑑𝑧1𝑧1sec2πœƒ1π‘‘πœƒ1ξ€Έξ‚Ήπ‘Ÿβˆ’1=ξ‚€4πœ‹ξ‚π‘Ÿβˆ’1ξ‚Έξ€œ0πœ‹/4ξ‚»ξ€œπ‘§0ξ‚†βˆ’1exp2𝑧21sec2πœƒ1𝑧1𝑑𝑧1ξ‚Όsec2πœƒ1π‘‘πœƒ1ξ‚Ήπ‘Ÿβˆ’1.(2.7) Subsequently, setting 𝑧21=𝑀 in (2.7), where 𝑧1𝑑𝑧1=𝑑𝑀/2, gives[]Ξ¨(𝑧)2π‘Ÿβˆ’2=ξ‚€4πœ‹ξ‚π‘Ÿβˆ’1ξƒ¬ξ€œ0πœ‹/4ξƒ―ξ€œπ‘§20ξ‚€βˆ’1exp2𝑀sec2πœƒ1𝑑𝑀2ξƒ°sec2πœƒ1π‘‘πœƒ1ξƒ­π‘Ÿβˆ’1=ξ‚€4πœ‹ξ‚π‘Ÿβˆ’1βŽ‘βŽ’βŽ’βŽ£ξ€œ0πœ‹/4ξƒ―12β‹…ξ€·expβˆ’(1/2)𝑀sec2πœƒ1ξ€Έβˆ’(1/2)sec2πœƒ1𝑧20sec2πœƒ1π‘‘πœƒ1⎀βŽ₯βŽ₯βŽ¦π‘Ÿβˆ’1,(2.8) and hence []Ξ¨(𝑧)2π‘Ÿβˆ’2=ξ‚€4πœ‹ξ‚π‘Ÿβˆ’1ξ‚Έξ€œ0πœ‹/4ξ‚€ξ‚†βˆ’11βˆ’exp2𝑧2sec2πœƒ1ξ€Έξ‚‡ξ‚π‘‘πœƒ1ξ‚Ήπ‘Ÿβˆ’1.(2.9) Expanding (2.9) yields []Ξ¨(𝑧)2π‘Ÿβˆ’2⎧βŽͺ⎨βŽͺ⎩=1+π‘Ÿβˆ’1ξ“π‘˜=1(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚π‘˜ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4ξƒ―βˆ’1exp2𝑧2π‘˜ξ“π‘–=1sec2πœƒπ‘–ξƒ°π‘‘πœƒ1β‹―π‘‘πœƒπ‘˜βŽ«βŽͺ⎬βŽͺ⎭,(2.10) where the subscript 𝑖 runs faster than π‘˜. For example, if π‘Ÿ=4, then (2.10) would appear more specifically as []Ξ¨(𝑧)2π‘Ÿβˆ’2βŽ›βŽœβŽœβŽ1βŽžβŽŸβŽŸβŽ ξ‚€4=1βˆ’π‘Ÿβˆ’1πœ‹ξ‚ξ€œ0πœ‹/4ξ‚†βˆ’1exp2𝑧2sec2πœƒ1ξ‚‡π‘‘πœƒ1+βŽ›βŽœβŽœβŽ2βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚20πœ‹/4ξ‚†βˆ’1exp2𝑧2ξ€·sec2πœƒ1+sec2πœƒ2ξ€Έξ‚‡π‘‘πœƒ1π‘‘πœƒ2βˆ’βŽ›βŽœβŽœβŽ3βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚3ξ€ž0πœ‹/4ξ‚†βˆ’1exp2𝑧2ξ€·sec2πœƒ1+sec2πœƒ2+sec2πœƒ3ξ€Έξ‚‡π‘‘πœƒ1π‘‘πœƒ2π‘‘πœƒ3.(2.11)

Substituting (2.10) into (2.3) and initially integrating with respect to 𝑧 (Lichtenstein, [21]) yields βˆšπœ‹ξ€œβˆž0π‘žπ‘‘(𝑧)πœ‘(𝑧)2ξƒ―βˆ’1exp2𝑧2π‘˜ξ“π‘–=1sec2πœƒπ‘–ξƒ°π‘‘π‘§=𝑔𝑑sec2πœƒπ‘–ξ€Έ,(2.12) where the specific forms of 𝑔𝑑(sec2πœƒπ‘–), which are associated with 𝑝𝑑(𝑧), are 𝑔1ξ€·sec2πœƒπ‘–ξ€Έ=√2ξ‚€βˆ‘2+π‘˜π‘–=1sec2πœƒπ‘–ξ‚1/2,𝑔2ξ€·sec2πœƒπ‘–ξ€Έ=2√2ξ‚€βˆ‘5+2π‘˜π‘–=1sec2πœƒπ‘–ξ‚5ξ‚€βˆ‘2+π‘˜π‘–=1sec2πœƒπ‘–ξ‚3/2,𝑔3ξ€·sec2πœƒπ‘–ξ€Έ=4√2ξ‚΅ξ‚€βˆ‘3+42+π‘˜π‘–=1sec2πœƒπ‘–ξ‚ξ‚€βˆ‘+82+π‘˜π‘–=1sec2πœƒπ‘–ξ‚2ξ‚Άξ‚€βˆ‘432+π‘˜π‘–=1sec2πœƒπ‘–ξ‚5/2.(2.13)

Equations (2.13) can be more conveniently expressed as 𝑔𝑑sec2πœƒπ‘–ξ€Έ=𝑔1ξ€·sec2πœƒπ‘–ξ€Έβˆ’β„Žπ‘‘ξ€·sec2πœƒπ‘–ξ€Έ,(2.14) where the specific forms of β„Žπ‘‘(sec2πœƒπ‘–) are β„Ž1ξ€·sec2πœƒπ‘–ξ€Έβ„Ž=0,(2.15)2ξ€·sec2πœƒπ‘–ξ€Έ=√2ξ‚€βˆ‘π‘˜π‘–=1sec2πœƒπ‘–ξ‚5ξ‚€βˆ‘2+π‘˜π‘–=1sec2πœƒπ‘–ξ‚3/2β„Ž,(2.16)3ξ€·sec2πœƒπ‘–ξ€Έ=√2ξ‚€βˆ‘11π‘˜π‘–=1sec4πœƒπ‘–βˆ‘+28π‘˜π‘–=1sec2πœƒπ‘–βˆ‘+22𝑖<𝑗sec2πœƒπ‘–sec2πœƒπ‘—ξ‚ξ‚€βˆ‘432+π‘˜π‘–=1sec2πœƒπ‘–ξ‚5/2(2.17) and where βˆ‘π‘–<𝑗 in (2.17) indicates summing over all π‘˜(π‘˜βˆ’1)/2 pairwise combinations. Hence, the integral in (2.3) can be expressed as 𝐼2π‘Ÿβˆ’1=2π‘Ÿβˆ’1βˆšπœ‹βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺ⎩1+π‘Ÿβˆ’1ξ“π‘˜=1(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚π‘˜ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4𝑔𝑑sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜βŽ«βŽͺ⎬βŽͺ⎭⎞⎟⎟⎠,(2.18) and subsequently substituting (2.14) into (2.18) gives 𝐼2π‘Ÿβˆ’1=2π‘Ÿβˆ’1βˆšπœ‹βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺ⎩1+π‘Ÿβˆ’1ξ“π‘˜=1(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚π‘˜ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4𝑔1ξ€·sec2πœƒπ‘–ξ€Έβˆ’β„Žπ‘‘ξ€·sec2πœƒπ‘–ξ€Έξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜βŽ«βŽͺ⎬βŽͺ⎭⎞⎟⎟⎠.(2.19)

The integral associated with 𝑔1(sec2πœƒπ‘–) in (2.19) cannot be expressed in terms of explicit elementary functions for π‘˜>1, which also implies π‘Ÿ>2 and sample sizes of 𝑛>5 in Table 1. As such, we will consider the approximating function π‘”βˆ—1(sec2πœƒπ‘–) as π‘”βˆ—1ξ€·sec2πœƒπ‘–ξ€Έ=ξ€·2π‘˜/2ξ€Έπ‘˜ξ‘π‘–=11ξ€·2+sec2πœƒπ‘–ξ€Έ1/2,(2.20) where ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4𝑔1ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜=ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4π‘”βˆ—1ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜=ξƒ―tanβˆ’1ξ‚€βˆš1/2,π‘˜=1,0,π‘˜βŸΆβˆž.(2.21)

Thus, for finite π‘˜>1 we have ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4𝑔1ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜=ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4π‘”βˆ—1ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜+πœ€π‘˜=tanβˆ’11√2ξƒͺξƒͺπ‘˜+πœ€π‘˜,(2.22) where πœ€π‘˜ is the remainder term required for π‘˜>1 and where πœ€1=0 for π‘Ÿ=1,2 and 𝑛≀5. Thus, using (2.22), (2.19) can be expressed as 𝐼2π‘Ÿβˆ’1=2π‘Ÿβˆ’1βˆšπœ‹βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺ⎩1+π‘Ÿβˆ’1ξ“π‘˜=1(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚π‘˜Γ—βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽξƒ©tanβˆ’11√2ξƒͺξƒͺπ‘˜+πœ€π‘˜βŽžβŽŸβŽŸβŽ βˆ’ξ€œ0πœ‹/4β‹―ξ€œ0πœ‹/4β„Žπ‘‘ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘˜βŽžβŽŸβŽŸβŽ βŽ«βŽͺ⎬βŽͺ⎭⎞⎟⎟⎠.(2.23)

The remainder terms πœ€π‘˜>1 in (2.23) can be solved by using (2.3), (2.15), (2.23), and the error function Erf [22], where Erf would replace Ξ¦(𝑧) in (2.3) where Ξ¨(𝑧)=2Ξ¦(𝑧)βˆ’1. More specifically, Table 2 gives the values of πœ€π‘˜ for π‘˜=1,…12,25, and 50 with 40-digit precision. Inspection of Table 2 indicates that the (positive) remainder term achieves a maximum at πœ€4 and thereafter tends to zero as π‘˜ increases (i.e., πœ€π‘˜β†’0 for π‘˜>4).

Table 2: Computed values of the remainder term πœ€π‘˜ associated with (2.23). The values were computed with 40-digit precision.

We would note that the approach taken here to determine πœ€2 is analogous to Renner’s [19] approach of developing a power series for this value. That is, the remainder term πœ€2 in Table 2 is also the value approximated in [19] for 𝑝1(𝑍). Further, we would note that extending the approach in [19] for computing the remainder terms for π‘˜>2 would become computationally burdensome.

To demonstrate (2.23) more specifically, if π‘Ÿ=4 and 𝑑=2 in (1.7), then the integral 𝐼7 associated with 𝑝2(𝑍) would appear as𝐼7=2π‘Ÿβˆ’1βˆšπœ‹βŽ§βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽ1βŽžβŽŸβŽŸβŽ ξ‚€41βˆ’π‘Ÿβˆ’1πœ‹ξ‚ξƒ©ξƒ©tanβˆ’11√2βˆ’ξ€œξƒͺξƒͺ0πœ‹/4β„Ž2ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1ξƒͺ+βŽ›βŽœβŽœβŽ2βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚2βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽξƒ©tanβˆ’11√2ξƒͺξƒͺ2+πœ€2βŽžβŽŸβŽŸβŽ βˆ’ξ€0πœ‹/4β„Ž2ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1π‘‘πœƒ2βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ3βŽžβŽŸβŽŸβŽ ξ‚€4π‘Ÿβˆ’1πœ‹ξ‚3βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽξƒ©tanβˆ’11√2ξƒͺξƒͺ3+πœ€3βŽžβŽŸβŽŸβŽ βˆ’ξ€ž0πœ‹/4β„Ž2ξ€·sec2πœƒπ‘–ξ€Έπ‘‘πœƒ1π‘‘πœƒ2π‘‘πœƒ3⎞⎟⎟⎠⎫βŽͺ⎬βŽͺ⎭.(2.24)

3. Evaluations

Tables 3–5 give evaluations for the expected values of the order statistics for 𝑝1(𝑍),𝑝2(𝑍), and 𝑝3(𝑍) in (1.7), which are based on (2.23) and the general formulae given in Table 1 for sample sizes of 𝑛=4,5. Inspection of Tables 4 and 5 indicates that the expected values for 𝑝2(𝑍) and 𝑝3(𝑍) are all expressed in terms of elementary functions and are also functions of the expectations associated with 𝑝1(𝑍) in Table 3.

Table 3: Expected values of order statistics for 𝑝1(𝑍)=𝑍 for 𝑛=4,5.
Table 4: Expected values of order statistics for 𝑝2(𝑍)=(2/5)𝑍3 for 𝑛=4,5.
Table 5: Expected values of order statistics for 𝑝3(𝑍)=(4/43)𝑍5 for 𝑛=4,5.

Presented in Tables 6, 7 and 8 are the evaluations for all three distributions in (1.7) for samples of sizes 𝑛=6,7 where the expectations of the order statistics for 𝑝1(𝑍), 𝑝2(𝑍), and 𝑝3(𝑍) 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 𝑝1(𝑍) for sample sizes of 𝑛=8 and 𝑛=9, 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 𝑝1(𝑍)=𝑍 for 𝑛=6,7.
Table 7: Expected values of order statistics for 𝑝2(𝑍)=(2/5)𝑍3 for 𝑛=6,7.
Table 8: Expected values of order statistics for 𝑝3(𝑍)=(4/43)𝑍5 for 𝑛=6,7.
Table 9: Expected values of order statistics for 𝑝1(𝑍)=𝑍 for 𝑛=8.
Table 10: Expected values of order statistics for 𝑝1(𝑍)=𝑍 for 𝑛=9.


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