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 asλ1=E[X1:1],λ2=12E[X2:2-X1:2],λ3=13E[X3:3-2X2:3+X1:3],λ4=14E[X4:4-3X3:4+3X2:4-X1:4]or more generally as
λr=1r∑j=0r-1(-1)j(r-1j)E[Xr-j:r],
where the order statistics X1:n≤X2:n≤⋯≤Xn:n are drawn from the random variable X. The values of λ1 and λ2 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 τr=λr/λ2 for r≥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 -1<τr<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]
5τ32-14<τ4<1.

Headrick [8] derived classes of standard normal-L-moment-based power method distributions using the polynomial transformationp(Z)=∑i=1mciZi-1,
where Z~i.i.d.N(0,1). Setting m=4(m=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 m=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, λ1 and λ2 are standardized to the unit normal distribution asλ1=c1+c3+3c5=0,λ2=(4c2+10c4+43c6)4π=1π.

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))=f̅(z)=(p(z),ϕ(z)p′(z)),Fp(z)(p(z))=F̅(z)=(p(z),Φ(z)),
where f̅:ℜ↦ℜ2 and F̅:ℜ↦ℜ2 are the parametric forms of the pdf and cdf with the mappings z↦(x,y) and z↦(x,v) with x=p(z),y=ϕ(z)/p′(z),v=Φ(z), and where ϕ(z) and Φ(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
pt(Z)=c2tZ2t-1,whereif{t=1,c2=1,c4=0,c6=0,t=2,thenc2=0,c4=2/5,c6=0,t=3,c2=0,c4=0,c6=4/43,
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], “…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.”

Graphs of the three standard normal-based power method distributions pt(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)j:n]=n2-n(n-1j-1)∫0∞pt(z)φ(z)([1+Ψ(z)]j-1[1-Ψ(z)]n-j-[1-Ψ(z)]j-1[1+Ψ(z)]n-j)dz,
where pt(z) is defined as in (1.7) and φ(z)=2ϕ(z) and Ψ(z)=2Φ(z)-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 pt(z). Inspection of Table 1 indicates that we have in general (a) E[p(Z)j:n]=-E[p(Z)n+1-j:n], (b) the median E[p(Z)j:n]=-E[p(Z)j:n]=0, and (c) the E[p(Z)j:n] are linear combinations of the integrals I2r-1 for r=1,2,…, with only odd subscripts appearing as only odd powers of Ψ(z) appear in the polynomial expansions associated with (2.1). As such, I2r-1 in (2.1) can be expressed asI2r-1=∫0∞pt(z)φ(z)[Ψ(z)]2r-1dz.

General expressions for the expected values of the order statistics for pt=1,2,3(Z) in (1.7) and sample sizes of n=1,…,9. I2r-1 denotes an integral in (2.1) where r=1,…,4.

Equation (2.2) may be integrated by parts as
I2r-1=(2r-1)∫0∞qt(z)φ(z)2[Ψ(z)]2r-2dz,
where q1(z)=1, q2(z)=(2/5)(z2+2) and q3(z)=(4/43)(z4+4z2+8), for p1(z), p2(z), and p3(z), respectively. Note that Ψ(0)=0 and limz→+∞φ(z)=0. Evaluating (2.3) for r=1 gives a coefficient of mean difference of
I1=∫0∞qt(z)φ(z)2dz=1π
for all pt(z) in (1.7), which is consistent with the specification in (1.5) and given in Table 1.

The expression [Ψ(z)]2r-2 in (2.3) can be expressed as
[Ψ(z)]2r-2=(2π)r-1[∫0zexp{-12u2}du]2r-2
or analogously as a double integral over ℜ2 as
[Ψ(z)]2r-2=(2π)r-1[∬0zexp{-12(z12+z22)}dz1dz2]r-1.
Using (2.6), let z2=z1tanθ1 and thus dz2=z1sec2θ1dθ1. Further, let z12+z22=z12sec2θ1. 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
[Ψ(z)]2r-2=(2π)r-1[2∫0π/4∫0zexp{-12(z12sec2θ1)}dz1(z1sec2θ1dθ1)]r-1=(4π)r-1[∫0π/4{∫0zexp{-12(z12sec2θ1)}z1dz1}sec2θ1dθ1]r-1.
Subsequently, setting z12=w in (2.7), where z1dz1=dw/2, gives[Ψ(z)]2r-2=(4π)r-1[∫0π/4{∫0z2exp(-12wsec2θ1)dw2}sec2θ1dθ1]r-1=(4π)r-1[∫0π/4{12⋅exp(-(1/2)wsec2θ1)-(1/2)sec2θ1}0z2sec2θ1dθ1]r-1,
and hence
[Ψ(z)]2r-2=(4π)r-1[∫0π/4(1-exp{-12(z2sec2θ1)})dθ1]r-1.
Expanding (2.9) yields
[Ψ(z)]2r-2=1+{∑k=1r-1(-1)k(r-1k)(4π)k∫0π/4⋯∫0π/4exp{-12z2∑i=1ksec2θi}dθ1⋯dθk},
where the subscript i runs faster than k. For example, if r=4, then (2.10) would appear more specifically as
[Ψ(z)]2r-2=1-(r-11)(4π)∫0π/4exp{-12z2sec2θ1}dθ1+(r-12)(4π)2∬0π/4exp{-12z2(sec2θ1+sec2θ2)}dθ1dθ2-(r-13)(4π)3∭0π/4exp{-12z2(sec2θ1+sec2θ2+sec2θ3)}dθ1dθ2dθ3.

Substituting (2.10) into (2.3) and initially integrating with respect to z (Lichtenstein, [21]) yields
π∫0∞qt(z)φ(z)2exp{-12z2∑i=1ksec2θi}dz=gt(sec2θi),
where the specific forms of gt(sec2θi), which are associated with pt(z), are
g1(sec2θi)=2(2+∑i=1ksec2θi)1/2,g2(sec2θi)=22(5+2∑i=1ksec2θi)5(2+∑i=1ksec2θi)3/2,g3(sec2θi)=42(3+4(2+∑i=1ksec2θi)+8(2+∑i=1ksec2θi)2)43(2+∑i=1ksec2θi)5/2.

Equations (2.13) can be more conveniently expressed as
gt(sec2θi)=g1(sec2θi)-ht(sec2θi),
where the specific forms of ht(sec2θi) are
h1(sec2θi)=0,h2(sec2θi)=2(∑i=1ksec2θi)5(2+∑i=1ksec2θi)3/2,h3(sec2θi)=2(11∑i=1ksec4θi+28∑i=1ksec2θi+22∑i<jsec2θisec2θj)43(2+∑i=1ksec2θi)5/2
and where ∑i<j in (2.17) indicates summing over all k(k-1)/2 pairwise combinations. Hence, the integral in (2.3) can be expressed as
I2r-1=2r-1π(1+{∑k=1r-1(-1)k(r-1k)(4π)k∫0π/4⋯∫0π/4gt(sec2θi)dθ1⋯dθk}),
and subsequently substituting (2.14) into (2.18) gives
I2r-1=2r-1π(1+{∑k=1r-1(-1)k(r-1k)(4π)k∫0π/4⋯∫0π/4(g1(sec2θi)-ht(sec2θi))dθ1⋯dθk}).

The integral associated with g1(sec2θi) in (2.19) cannot be expressed in terms of explicit elementary functions for k>1, which also implies r>2 and sample sizes of n>5 in Table 1. As such, we will consider the approximating function g1*(sec2θi) as
g1*(sec2θi)=(2k/2)∏i=1k1(2+sec2θi)1/2,
where
∫0π/4⋯∫0π/4g1(sec2θi)dθ1⋯dθk=∫0π/4⋯∫0π/4g1*(sec2θi)dθ1⋯dθk={tan-1(1/2),k=1,0,k⟶∞.

Thus, for finite k>1 we have
∫0π/4⋯∫0π/4g1(sec2θi)dθ1⋯dθk=∫0π/4⋯∫0π/4g1*(sec2θi)dθ1⋯dθk+εk=(tan-1(12))k+εk,
where εk is the remainder term required for k>1 and where ε1=0 for r=1,2 and n≤5. Thus, using (2.22), (2.19) can be expressed as
I2r-1=2r-1π({1+∑k=1r-1(-1)k(r-1k)(4π)k×(((tan-1(12))k+εk)-∫0π/4⋯∫0π/4ht(sec2θi)dθ1⋯dθk)}).

The remainder terms εk>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 Φ(z) in (2.3) where Ψ(z)=2Φ(z)-1. More specifically, Table 2 gives the values of εk for k=1,…12,25, and 50 with 40-digit precision. Inspection of Table 2 indicates that the (positive) remainder term achieves a maximum at ε4 and thereafter tends to zero as k increases (i.e., εk→0 for k>4).

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

Sample size (n)

Integral

Remainder term

1,…,5

I1, I3

ε1=0.0

6,7

I5

ε2=0.03140698829552010270731937950881276500595

8,9

I7

ε3=0.05156068650031409787170392919312656858246

10,11

I9

ε4=0.05900198710355817149868423817928465212298

12,13

I11

ε5=0.05808975458203638968882522593413660371348

14,15

I13

ε6=0.05274763616761422221709626523935998463539

16,17

I15

ε7=0.04559236574104643530748593758544745949676

18,19

I17

ε8=0.03815223895234453779274127861572423887877

20,21

I19

ε9=0.03122205691467168489718556870682270636055

22,23

I21

ε10=0.02514855254614865670209122288596241803047

24,25

I23

ε11=0.02002429921405354560405588075438666460570

26,27

I25

ε12=0.01580928681263632398753707685232879723154

⋮

⋮

⋮

52,53

I51

ε25=0.00057455597453332805073409074487236584232

⋮

⋮

⋮

102,103

I101

ε50=0.00000099193614769461065745252616987082859

We would note that the approach taken here to determine ε2 is analogous to Renner’s [19] approach of developing a power series for this value. That is, the remainder term ε2 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 k>2 would become computationally burdensome.

To demonstrate (2.23) more specifically, if r=4 and t=2 in (1.7), then the integral I7 associated with p2(Z) would appear asI7=2r-1π{1-(r-11)(4π)((tan-1(12))-∫0π/4h2(sec2θi)dθ1)+(r-12)(4π)2(((tan-1(12))2+ε2)-∬0π/4h2(sec2θi)dθ1dθ2)-(r-13)(4π)3(((tan-1(12))3+ε3)-∭0π/4h2(sec2θi)dθ1dθ2dθ3)}.

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.

Expected values of order statistics for p1(Z)=Z for n=4,5.

E[p1(Z)3:4]=-3π+18tan-1(1/2)π3/2=0.29701138…

E[p1(Z)4:4]=3π-6tan-1(1/2)π3/2=1.02937537…

E[p1(Z)3:5]=0

E[p1(Z)4:5]=-5π+30tan-1(1/2)π3/2=0.49501897…

E[p1(Z)5:5]=5π-15tan-1(1/2)π3/2=1.16296447…

Expected values of order statistics for p2(Z)=(2/5)Z3 for n=4,5.

E[p1(Z)3:4]=-325π3/2+E[p1(Z)3:4]=0.14462665…

E[p2(Z)4:4]=25π3/2+E[p1(Z)4:4]=1.08017028…

E[p2(Z)3:5]=0

E[p2(Z)4:5]=-2π3/2+E[p1(Z)4:5]=0.24104442…

E[p2(Z)5:5]=12π3/2+E[p1(Z)5:5]=1.28995174…

Expected values of order statistics for p3(Z)=(4/43)Z5 for n=4,5.

E[p3(Z)3:4]=-77432π3/2+E[p1(Z)3:4]=0.069615569…

E[p3(Z)4:4]=771292π3/2+E[p1(Z)4:4]=1.10517397…

E[p3(Z)3:5]=0

E[p3(Z)4:5]=-3851292π3/2+E[p1(Z)4:5]=0.11602594…

E[p3(Z)5:5]=3852582π3/2+E[p1(Z)5:5]=1.35246098…

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.

Expected values of order statistics for p1(Z)=Z for n=6,7.

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