Page 1

arXiv:1008.2297v1 [cs.IT] 13 Aug 2010

An MGF-based Unified Framework to

Determine the Joint Statistics of Partial Sums

of Ordered Random Variables

Sung Sik Nam, Member, IEEE, Mohamed-Slim Alouini, Fellow Member, IEEE,

and Hong-Chuan Yang, Senior Member, IEEE.

Abstract

Order statistics find applications in various areas of communications and signal processing. In this

paper, we introduce an unified analytical framework to determine the joint statistics of partial sums of

ordered random variables (RVs). With the proposed approach, we can systematically derive the joint

statistics of any partial sums of ordered statistics, in terms of the moment generating function (MGF)

and the probability density function (PDF). Our MGF-based approach applies not only when all the K

ordered RVs are involved but also when only the Ks(Ks< K) best RVs are considered. In addition,

we present the closed-form expressions for the exponential RV special case. These results apply to the

⋆This work was supported in part by Qatar Telecom (Qtel). This is an extended version of a paper which was presented

in Proc. of IEEE International Conference on Wireless Communications and Signal Processing (WCSP 2009), Nanjing, China,

November 2009. S. S. Nam was with Department of Electrical and Computer Engineering, Texas A&M University at College

Station, Texas, USA. He is now with Department of Electronic Engineering in Hanyang University, Seoul, Korea. M.-S. Alouini

was with the Electrical and Computer Engineering Program at Texas A&M University at Qatar, Doha, Qatar. He is now with

Electrical Engineering Program, KAUST, Thuwal, Saudi Arabia. H. -C. Yang is with Department of Electrical and Computer

Engineering, University of Victoria, BC V8W 3P6, Canada. They can be reached by E-mail at <ssnam11@tamu.edu,

slim.alouini@kaust.edu.sa, hyang@ece.uvic.ca>.

August 19, 2010DRAFT

Page 2

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS1

performance analysis of various wireless communication systems over fading channels.

Index Terms

Joint PDF, Moment generating function (MGF), Order statistics, Probability density function (PDF),

Rayleigh fading.

I. INTRODUCTION

The subject of order statistics deals with the properties and distributions of the ordered

random variables (RVs) and their functions. It has found applications in many areas of statistical

theory and practice [1], with examples including life-testing, quality control, signal and image

processing [2], [3]. Recently, order statistics makes a growing number of appearance in the

analysis and design of wireless communication systems (see for example [4]–[14]). For example,

diversity techniques have been used for over the past fifty years to mitigate the effects of fading on

wireless communication systems. These techniques improve the performance of wireless systems

over fading channels by generating and combining multiple replicas of the same information

bearing signal at the receiver. The analysis of low-complexity selection combining schemes,

which select the best replica, requires some basic results of order statistics, i.e. the distribution

functions of the largest one among several random variables.

More recently, the design and analysis of adaptive diversity combining techniques and mul-

tiuser scheduling strategies call for some further results on order statistics [5], [6]. In particular,

the joint statistics of partial sums of ordered RVs are often necessary for the accurate character-

ization of system performance [7], [14]. The major difficulty in obtaining the statistics of partial

sums of ordered RVs resides in the fact that even if the original unordered RVs are independently

distributed, their ordered versions are necessarily dependent due to the inequality relations

August 19, 2010DRAFT

Page 3

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS2

among them. Recently, the co-author has applied a successive conditioning approach to convert

dependent ordered random variables to independent unordered ones [5], [6]. That approach,

however, requires some case-specific manipulations, which may not always be generalizable.

In this paper, we present an unified analytical framework to determine the joint statistics of

partial sums of ordered RVs using a moment generating functions (MGF) based approach. More

specifically, we extend the result in [15]–[17], which only derive the joint MGF of the selected

individual order statistics and the sum of the remaining ones, and systematically solve for the

joint statistics of arbitrary partial sums of ordered RVs. The main advantage of the proposed

MGF-based unified framework is that it applies not only to the cases when all the K ordered

RVs are considered but also to those cases when only the Ks(Ks< K) best RVs are involved.

After considering several illustrative examples, we focus on the exponential RV special case and

derive the closed-form expressions of the joint statistics. These statistical results can apply to

the performance analysis of various wireless communication systems over generalized fading

channels.

The remainder of this paper is organized as follows. In section II, we summarize the main idea

behind the proposed unified analytical framework, including the general idea and some special

considerations. We then introduce some common functions and useful relations in section III,

which will help make the results in later sections more compact. In section IV and V, we

present some selected examples on the derivation of joint PDF based on our proposed approach.

Following this, we show in section VI some closed form expressions for the selected examples

presented in previous sections under i.i.d. Rayleigh fading conditions. Finally, we discuss some

useful applications of these results in section VII.

August 19, 2010 DRAFT

Page 4

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS3

II. THE MAIN IDEA

Let ∞ ≥ γ1:K≥ γ2:K≥ γ3:K··· ≥ γK:K≥ 0 be the order statistics obtained by arranging

K nonnegative i.i.d. RVs, {γi}K

i=1, in decreasing order of magnitude. The objective is to derive

the joint PDF of their partial sums involving either all K or the first Ks(Ks< K) ordered RVs,

e.g. the joint PDF of γm:Kand

K ?

n=1

n?=m

γn:Kor the joint PDF of

m ?

n=1γn:Kand

Ks

?

n=m+1γn:K.

A. General steps

The proposed analytical framework adopts a general two-step approach: i) obtain the analytical

expressions of the joint MGF of partial sums (not necessarily the partial sums of interest as will be

seen later), ii) apply inverse Laplace transform to derive the joint PDF of partial sums (additional

integration may be required to obtain the desired joint PDF). To facilitate the inverse Laplace

transform calculation, the joint MGF from step i) should be made as compact as possible. An

observation made in [15]–[17] involving the interchange of multiple integrals of ordered RVs

becomes useful in the following analysis. Suppose for example that we need to evaluate a multiple

integral over the range γa≥ γ1≥ γ2≥ γ3≥ γ4≥ γb. More specifically, let

I=

?γa

γb

dγ1

?γ1

γb

dγ2

?γ2

γb

dγ3

?γ3

γb

dγ4 p(γ1,γ2,γ3,γ4).

(1)

It can be shown by interchanging the order of integration, while ensuring each pair of limits is

chosen to be as tight as possible, the multiple integral in (1) can be rewritten into the following

August 19, 2010DRAFT

Page 5

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS4

equivalent representations,

I =

?γa

?γa

?γa

?γa

?γa

γb

dγ4

?γa

?γ2

?γa

?γ1

?γa

γ4

dγ3

?γa

?γ3

?γ3

?γ1

?γ1

γ3

dγ2

?γa

?γa

?γ1

?γ1

?γ1

γ2

dγ1 p(γ1,γ2,γ3,γ4)

=

γb

dγ2

γb

dγ3

γb

dγ4

γ2

dγ1 p(γ1,γ2,γ3,γ4)

=

γb

dγ3

γ3

dγ1

γb

dγ4

γ3

dγ2 p(γ1,γ2,γ3,γ4)

=

γb

dγ1

γb

dγ4

γ4

dγ3

γ3

dγ2 p(γ1,γ2,γ3,γ4)

=

γb

dγ4

γ4

dγ1

γ4

dγ3

γ3

dγ2 p(γ1,γ2,γ3,γ4).

(2)

The general rule is that the integration limits should be selected as tight as possible using the

remaining variables. For example, in the first equation of (2), the variables are integrated in the

order of γ1, γ2, γ3, and γ4. Based on the given inequality condition γa≥ γ1≥ γ2≥ γ3≥ γ4≥ γb,

the integration limit of γ1should be from γ2to γabecause γ2is the tightest among the remaining

RVs. Similarly, the integration limit γ3is from γ4to γabecause γ1and γ2were already integrated

out.

After obtaining the joint MGF in a compact form, we can derive joint PDF of selected

partial sum through inverse Laplace transform. For most cases of our interest, the joint MGF

involves basic functions, for which the inverse Laplace transform can be calculated analytically.

In the worst case, we may rely on the Bromwich contour integral. In most of the case, the

result involves a single one-dimensional contour integration, which can be easily and accurately

evaluated numerically with the help of integral tables [18], [19] or using standard mathematical

packages such as Mathematica and Matlab.

B. Special cases

The general steps can be directly applied when all K ordered RVs are considered and the RVs

in the partial sums are continuous. When these conditions do not hold, we need to apply some

August 19, 2010DRAFT

Page 6

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS5

extra steps in the analysis in order to obtain a valid joint MGF. Specifically, when only the best

Ks(Ks< K) ordered RVs are involved in the partial sums, we should consider the Ksth order

statistics γKs:Kseparately. Without this separation, we cannot find the valid integration limit in

calculating the joint MGF. As an illustration, the example in Fig. 1 considers 3-dimensional joint

PDF of the partial sums of three group of RVs which are {γ1:K,γ2:K,γ3:K}, {γ4:K,γ5:K,γ6:K},

and {γ7:K,γ8:K}, with K = 10 and Ks= 8. Following the proposed approach, we will derive

the 4-dimensional joint MGF in step i) by considering γ8:K separately, i.e. the joint MGF of

the following four groups of RVs {γ1:K,γ2:K,γ3:K}, {γ4:K,γ5:K,γ6:K}, {γ7:K}, and {γ8:K}.

After obtaining the corresponding 4-dimensional joint MGF, we need to perform another finite

integration to obtain the desired 3-dimensional joint PDF.

When the RVs involved in one partial sum is not continuous, i.e. separated by the other RVs,

we need to divide these RVs into smaller sums. The example in Fig. 2 illustrate this process. If

we consider 3-dimensional joint PDF of {γ1:K, γ2:K, γ5:K, γ6:K}, {γ3:K,γ4:K}, and {γ7:K,γ8:K},

then there are three partial sum of RVs {γ1:K, γ2:K, γ5:K, γ6:K}, {γ3:K,γ4:K}, and {γ7:K,γ8:K}.

Note that the first group is split by the second group of RVs (as such discontinuous). More

specifically, the second group, {γ3:K,γ4:K}, split the original group, {γ1:K, γ2:K, γ5:K, γ6:K},

into two groups as {γ1:K,γ2:K}, and {γ5:K,γ6:K}. Therefore, we also consider this split group

{γ1:K,γ2:K} and {γ5:K,γ6:K} as two smaller groups. As a result, we will derive 5-dimensional

joint MGF of {γ1:K,γ2:K}, {γ3:K,γ4:K}, {γ5:K,γ6:K}, {γ7:K}, and {γ8:K} in step i). Similarly

to the first example, after the joint PDF of the new substituted partial sums are derived with

inverse Laplace transform in step ii), we can transform it to a lower dimensional desired joint

PDF with finite integration.

The proposed approach is summarized in the flowchart given in Fig. 3, where we consider

August 19, 2010 DRAFT

Page 7

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS6

three different case separately. In the following sections, we present several examples to illustrate

the proposed analytical framework. Our focus is on how to obtain a compact expression of the

joint MGFs, which can be greatly simplified with the application of the following function and

relations.

III. COMMON FUNCTIONS AND USEFUL RELATIONS

In this section, we introduce some common functions and their properties. These results will

be used to simplify the derivation of joint MGFs in later sections.

A. Common Functions

i) A mixture of a CDF and an MGF c(γ,λ):

c(γ,λ) =

?γ

0

dx p(x)exp(λx),

(3)

where p(x) denotes the PDF of the RV of interest. Note that c(γ,0) = c(γ) is the CDF

and c(∞,λ) leads to the MGF. Here, the variable γ is real, while λ can be complex.

ii) A mixture of an exceedance distribution function (EDF) and an MGF, e(γ,λ):

e(γ,λ) =

?∞

γ

dx p(x)exp(λx).

(4)

Note that e(γ,0) = e(γ) is the EDF while e(0,λ) gives the MGF.

iii) An Interval MGF µ(γa,γb,λ):

µ(γa,γb,λ) =

?γb

γa

dx p(x)exp(λx).

(5)

Note that µ(0,∞,λ) gives the MGF.

August 19, 2010DRAFT

Page 8

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS7

Note that the functions defined in (3), (4) and (5) are related as follows

c(γ,λ)=

e(0,λ) − e(γ,λ)

(6)

=

c(∞,λ) − e(γ,λ)

e(γ,λ)=

c(∞,λ) − c(γ,λ)

(7)

=

e(0,λ) − c(γ,λ)

µ(γa,γb,λ)=

c(γb,λ) − c(γa,λ)

(8)

=

e(γa,λ) − e(γb,λ).

B. Simplifying Relationship

i) Integral Im:

Based on the derivation given Appendix II, the integral Imdefined as:

Im=

?γm−1:K

?γm+1:K

0

dγm:Kp(γm:K)exp(λγm:K)

?γm:K

0

dγm+1:Kp(γm+1:K)exp(λγm+1:K)

×

0

dγm+2:Kp(γm+2:K)exp(λγm+2:K)···

?γK−1:K

0

dγK:Kp(γK:K)exp(λγK:K),

(9)

can be expressed in terms of the function c(γ,λ) as

Im=

1

(K − m + 1)![c(γm−1:K,λ)](K−m+1).

(10)

ii) Integral I′

m:

Following the similar derivation as given in Appendix II, the integral I′

mdefined as

I′

m=

?∞

γm+1:K

?∞

dγm:Kp(γm:K)exp(λγm:K)

?∞

γm:K

dγm−1:Kp(γm−1:K)exp(λγm−1:K)

×

γm−1:K

dγm−2:Kp(γm−2:K)exp(λγm−2:K)···

?∞

γ2:K

dγ1:Kp(γ1:K)exp(λγ1:K),

(11)

can be expressed in terms of the function e(γ,λ) as

I′

m=

1

m![e(γm+1:K,λ)]m.

(12)

August 19, 2010 DRAFT

Page 9

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS8

iii) Integral I′′

a,b:

Based on the derivation given in Appendix III, the integral I′′

a,bdefined as

I′′

a,b=

?γa:K

?γa:K

γb:K

dγb−1:Kp(γb−1:K)exp(λγb−1:K)

?γa:K

γb−1:K

dγb−2:Kp(γb−2:K)exp(λγb−2:K)

×

γb−2:K

dγb−3:Kp(γb−3:K)exp(λγb−3:K)···

?γa:K

γa+2:K

dγa+1:Kp(γa+1:K)exp(λγa+1:K),

(13)

can be expressed in terms of the function µ(·,·) as

I′′

a,b=

1

(b − a − 1)![µ(γb:K,γa:K,λ)](b−a−1)

for b > a.

(14)

IV. SAMPLE CASES WHEN ALL K ORDERED RVS ARE CONSIDERED

Assume the original RVs {γi} are i.i.d. with a common arbitrary PDF p(γ), the K-dimensional

joint PDF of {γi:K}K

i=1is simply given by [1]

p(γ1:K,γ2:K,··· ,γK:K) = F ·

K

?

i=1

p(γi:K)

for γ1:K≥ γ2:K≥ γ3:K··· ≥ γK:K,

(15)

where F = K!.

Theorem 4.1: (PDF of

K ?

n=1γn:Kamong K ordered RVs)

Let Z1=

K ?

n=1γn:Kfor convenience. We can derive the PDF of Z = [Z1] as

pZ(z1)=L−1

S1

?

[c(∞,−S1)]K?

,

(16)

where L−1

S1{·} denotes the inverse Laplace transform with respect to S1.

Proof: The MGF of Z is given by the expectation

MGFZ(λ1) = E{exp(λ1Z1)}= F ·

∞

?

0

dγ1:Kp(γ1:K)exp(λ1γ1:K)

γ1:K

?

0

dγ2:Kp(γ2:K)exp(λ1γ2:K)

×··· ×

γK−1:K

?

0

dγK:Kp(γK:K)exp(λ1γK:K),

(17)

where E{·} denotes the expectation operator. By applying (10), we can obtain the MGF of

Z1=?K

m=1γm:Kas

MGFZ(λ1)=[c(∞,λ1)]K.

(18)

August 19, 2010DRAFT

Page 10

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS9

Therefore, we can derive the PDF of Z1=?K

as

m=1γm:Kby applying the inverse Laplace transform

pZ(z1)=L−1

S1{MGFZ(−S1)}

=L−1

S1

?

[c(∞,−S1)]K?

.

(19)

Theorem 4.2: (Joint PDF of γm:Kand

K ?

n=1

n?=m

γn:K)

Let Z1= γm:Kand Z2=

K ?

n=1

n?=m

γn:Kfor convenience. We can obtain the 2-dimensional joint PDF

of Z = [Z1,Z2] as

pZ(z1,z2) = p

γm:K,

K

?

n=1

n?=m

γn:K

(z1,z2)

=

K!

(K−1)!p(z1)L−1

S2

?

[c(z1,−S2)](K−1)?

?

for m = 1, z1≥

1

K−1z2

K!

(K−m)!(m−1)!p(z1)L−1

S2

[c(z1,−S2)](K−m)[e(z1,−S2)](m−1)?

for m ≥ 2.

(20)

Proof: The second order MGF of Z = [Z1,Z2] is given by the expectation

MGFZ(λ1,λ2) = E {exp(λ1Z1+ λ2Z2)}

= F ·

∞

?

0

dγ1:Kp(γ1:K)exp(λ2γ1:K)

γ1:K

?

0

dγ2:Kp(γ2:K)exp(λ2γ2:K)···

γm−2:K

?

0

dγm−1:Kp(γm−1:K)exp(λ2γm−1:K)

×

γm−1:K

?

0

dγm:Kp(γm:K)exp(λ1γm:K)

×

γm:K

?

0

dγm+1:Kp(γm+1:K)exp(λ2γm+1:K)···

γK−1:K

?

0

dγK:Kp(γK:K)exp(λ2γK:K).

(21)

We show in Appendix IV that by applying (10), (2) and (12), we can obtain the second order

MGF of Z1= γm:Kand Z2=

K ?

n=1

n?=m

γn:Kas

MGFZ(λ1,λ2)

=

F

(K − m)!(m − 1)!

∞

?

0

dγm:Kp(γm:K)exp(λ1γm:K)[c(γm:K,λ2)](K−m)[e(γm:K,λ2)](m−1).

(22)

August 19, 2010DRAFT

Page 11

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS10

With the MGF expression given in (22) at hand, we are now in the position to derive the

2-dimensional joint PDF of Z1= γm:Kand Z2=

K ?

n=1

n?=m

γn:K. Letting λ1= −S1and λ2= −S2,

we can derive the 2-dimensional joint PDF by applying the inverse Laplace transform as

pZ(z1,z2) = L−1

S1,S2{MGFZ(−S1,−S2)}

=

K!

(K − m)!(m − 1)!

∞

?

0

dγm:K

?

p(γm:K)L−1

S1{exp(−S1γm:K)}

×L−1

S2

?

[c(γm:K,−S2)](K−m)[e(γm:K,−S2)](m−1)??

.

(23)

Based on the inverse Laplace transform properties in Appendix I,

L−1

S1{exp(−S1γm:K)} = δ (z1− γm:K).

(24)

Therefore, substituting (24) in (23) we can obtain the desired 2-dimensional joint PDF of Z1=

γm:Kand Z2=

K ?

n=1

n?=m

γn:K.

Theorem 4.3: (Joint PDF of

m ?

n=1γn:Kand

K ?

K ?

n=m+1γn:K)

Let Z1=

m ?

n=1γn:Kand Z2=

n=m+1γn:Kfor convenience, then we can derive the 2-dimensional

joint PDF of Z = [Z1,Z2] as

pZ(z1,z2) = p m

?

n=1γn:K,

K

?

n=m+1

γn:K

(z1,z2)

= L−1

S1,S2{MGFZ(−S1,−S2)}

=

K!

(K − m)!(m − 1)!

∞

?

0

dγm:K

?

p(γm:K)L−1

S1

?

exp(−S1γm:K)[e(γm:K,−S1)](m−1)?

×L−1

S2

?

[c(γm:K,−S2)](K−m)??

for z1≥

m

K − mz2.

(25)

Proof: The second order MGF of Z = [Z1,Z2] is given by the expectation

MGFZ(λ1,λ2) = E {exp(λ1Z1+ λ2Z2)}

= F

∞

?

0

dγ1:Kp(γ1:K)exp(λ1γ1:K)···

γm−1:K

?

0

dγm:Kp(γm:K)exp(λ1γm:K)

(26)

×

γm:K

?

0

dγm+1:Kp(γm+1:K)exp(λ2γm+1:K)···

γK−1:K

?

0

dγK:Kp(γK:K)exp(λ2γK:K).

(27)

August 19, 2010DRAFT

Page 12

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS11

We show in Appendix V that by applying (10) and (2) and then (12), we can obtain the second

order MGF of Z as

MGFZ(λ1,λ2)

=

K!

(K − m)!(m − 1)!

∞

?

0

dγm:Kp(γm:K)exp(λ1γm:K)[c(γm:K,λ2)](K−m)[e(γm:K,λ1)](m−1).

(28)

Again, letting λ1= −S1and λ2= −S2, we can obtain the desired 2-dimensional joint PDF

of Z1=

m ?

n=1γn:Kand Z2=

K ?

n=m+1γn:Kby applying the inverse Laplace transform.

V. SAMPLE CASES WHEN ONLY KsORDERED RVS ARE CONSIDERED

Let us now consider the cases where only the best Ks(≤ K) ordered RVs are involved.

Assuming the original {γi} are i.i.d. RVs with a common arbitrary PDF p(γ) and CDF P (γ),

the Ks-dimensional joint PDF of {γi:K}Ks

i=1is simply given by [1]

p(γ1:K,γ2:K,··· ,γKs:K) = F ·

Ks

?

i=1

p(γi:K)[P (γKs:K)]K−Ks,

(29)

where F = Ks!?K

Theorem 5.1: (PDF of

Ks

?=

K!

(K−Ks)!.

Ks

?

n=1γn:K, Ks≥ 2)

Let Z′=

Ks

?

n=1γn:Kfor convenience, then we can derive the PDF of Z′as

pZ′ (x) = p?Ks

n=1γn:K(x) =

?

x

Ks

0

pZ(x − z2,z2)dz2

for Ks≥ 2,

(30)

where

pZ(z1,z2) =

F

(Ks− 1)!p(z2)[c(z2)](K−Ks)L−1

S1

?

[e(z2,−S1)](Ks−1)?

.

(31)

Proof: We only need to consider γKs:K separately in this case. Let Z1=

Ks−1

?

n=1

γn:K and

Z2= γKs:K. The target second order MGF of Z = [Z1,Z2] is given by the expectation in

MGFZ(λ1,λ2)=E{exp(λ1z1+ λ2z2)}

=

F

?∞

?

0

dγ1:Kp(γ1:K)exp(λ1γ1:K)···

?γKs−2:K

0

dγKs−1:Kp(γKs−1:K)exp(λ1γKs−1:K)

×

Ks−1:K

0

dγKs:Kp(γKs:K)exp(λ2γKs:K)[c(γKs:K)]K−Ks.

(32)

August 19, 2010DRAFT

Page 13

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS12

By simply applying (2) and then (12) to (32), we can obtain the second order MGF result as

MGFZ(λ1,λ2) =

F

(Ks− 1)!

?∞

0

dγKs:Kp(γKs:K)exp(λ2γKs:K)[c(γKs:K)]K−Ks[e(γKs:K,λ1)]Ks−1.

(33)

Again, letting λ1= −S1and λ2= −S2, we can obtain the 2-dimensional joint PDF of Z1=

Ks−1

?

Z1+ Z2, we can obtain the target PDF of Z′with the following finite integration

n=1

γn:Kand Z2= γKs:Kby applying the inverse Laplace transform. Finally, noting that Z′=

pZ′(x) =

?

x

Ks

0

pZ(x − z2,z2)dz2.

(34)

Theorem 5.2: (Joint PDF of γm:Kand

Ks

?

n=1

n?=m

γn:K)

Let X = γm:Kand Y =

Ks

?

n=1

n?=m

γn:K, then the joint PDF of Z = [X,Y ] can be obtained as

pZ(x,y) = p

γm:K,

Ks

?

n=1

n?=m

γn:K(x,y)

=

?(Ks−2)x

?x

?y

(Ks−2

Ks−1)yp

γ1:K,

Ks−1

?

n=2

γn:K,γKs:K(x,z2,y − z2)dz2,m = 1,

0

?y−(Ks−m)z4

(m−1)x

pm−1

?

n=1

γn:K,γm:K,

Ks−1

?

n=m+1

γn:K,γKs:K

(z1,x,y−z1−z4,z4)dz1dz4,

1 < m < Ks− 1,

(Ks−2)xpKs−2

?

n=1

γn:K,γKs−1:K,γKs:K(z1,x,y − z1)dz1,m = Ks− 1,

p

γKs:K,

Ks−1

?

n=1

γn:K(x,y),m = Ks,

(35)

August 19, 2010DRAFT

Page 14

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS13

or equivalently

p

γm:K,

Ks

?

n=1

n?=m

γn:K(x,y)

=

?(Ks−2)x

?x

?x

(Ks−2

Ks−1)yp

γ1:K,

Ks−1

?

n=2

γn:K,γKs:K(x,z2,y − z2)dz2,m = 1,

0

?(Ks−m−1)x

(Ks−m−1)z4pm−1

?

n=1

γn:K,γm:K,

Ks−1

?

n=m+1

γn:K,γKs:K

(y−z3−z4,x,z3,z4)dz3dz4,

1 < m < Ks− 1,

0pKs−2

?

n=1

γn:K,γKs−1:K,γKs:K(y − z3,x,z3)dz3,m = Ks− 1,

p

γKs:K,

Ks−1

?

n=1

γn:K

(x,y),m = Ks.

(36)

Proof: To derive the joint PDF of γm:Kand

Ks

?

n=1

n?=m

γn:K, we need to consider four cases i)

m = 1, ii) 1 < m < Ks− 1, iii) m = Ks− 1 and iv) m = Ksseparately based on our unified

framework. While for case iv), we can start with the second order MGF of γm:Kand

Ks

?

n=1

n?=m

γn:K

directly, we should consider substituted groups instead of the original groups for cases i), ii),

and iii). More specifically, for cases i) and iii), we need to consider γKs:Kseparately as shown

in Fig. 4(a) and 4(c) whereas, for case ii), as one of original groups is split by γm:K, we should

consider substituted groups for the split group instead of original groups as shown in Fig. 4(b).

As a result, we will start by obtaining a four order MGF for case ii) and a three order MGF for

case i) and case iii). In all these cases, the higher dimensional joint PDF can then be used to

find the desired 2-dimensional joint PDF of interest by transformation.

Applying the results in (2), (10), (12) and (14), we derive in Appendix VI the following joint

MGF for different cases

August 19, 2010DRAFT

Page 15

S.S. NAM et al.: AN UNIFIED FRAMEWORK ... ON ORDERED STATISTICS14

a. m = 1

Let Z1= γ1:K, Z2=

Ks−1

?

n=2

γn:K, and Z3= γKs:K, then

MGFZ(λ1,λ2,λ3)

= F

∞

?

0

dγKs:Kp(γKs:K)exp(λ3γKs:K)[c(γKs:K)](K−Ks)

×

∞

?

γKs:K

dγ1:Kp(γ1:K)exp(λ1γ1:K)

1

(Ks− 2)![µ(γKs:K,γ1:K,λ2)](Ks−2).

(37)

b. 1 < m < Ks− 1

Let Z1=

m−1

?

n=1γn:K, Z2= γm:K, Z3=

Ks−1

?

n=m+1γn:K, and Z4= γKs:K, then

MGFZ(λ1,λ2,λ3,λ4)

=

F

(Ks− m − 1)!(m − 1)!

∞

?

0

dγKs:Kp(γKs:K)exp(λ4γKs:K)[c(γKs:K)](K−Ks)

×

∞

?

γKs:K

dγm:Kp(γm:K)exp(λ2γm:K)[e(γm:K,λ1)](m−1)[µ(γKs:K,γm:K,λ3)](Ks−m−1).

(38)

c. m = Ks− 1

Let Z1=

Ks−2

?

MGFZ(λ1,λ2,λ3)

n=1

γn:K, Z2= γKs−1:Kand Z3= γKs:K, then

= F

∞

?

0

∞

?

dγKs:Kp(γKs:K)exp(λ3γKs:K)[c(γKs:K)](K−Ks)

×

γKs:K

dγKs−1:Kp(γKs−1:K)exp(λ2γKs−1:K)

1

(Ks− 2)![e(γKs−1:K,λ1)](Ks−2).

(39)

d. m = Ks

Let Z1= γKs:Kand Z2=

Ks−1

?

=

n=1

γn:K, then

MGFZ(λ1,λ2)

F

∞

?

0

dγKs:Kp(γKs:K)exp(λ1γKs:K)[c(γKs:K)](K−Ks)

×

1

(Ks− 1)![e(γKs:K,λ2)](Ks−1).

(40)

Starting from the MGF expressions given above, we apply inverse Laplace transforms in Ap-

pendix VI in order to derive the following joint PDFs

August 19, 2010DRAFT