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Some Applications of the

Distribution of the Maximum on Independent

Normal Random Variables

Roberto N. Padua Liceo de Cagayan Date Submitted: February 20, 2011

Date Revised: June 24, 2011

ABSTRACT

The paper deals with the distribution of the maximum of n independent normal random variables and some of its applications in the electricity power industry in the area of peak load estimation and in genetic selection for animal breeding. For small n, the difficulty in finding the value of multiple integrals involved in the distribution function of the maximum order statistics (and hence, in the computation of its expected value) is recognized. The paper provides for simple approximations to the mean of the largest order statistics both in the iid and non-identically distributed cases. Large sample asymptotic results for extreme values of normal random variables are often used in reliability theory and of late, used in the analysis of extreme weather changes in relation to climate change. While the large sample results for the iid case have been treated in the past, we focused on the relatively unexplored non-identical but independent case. Results show that : (a) the simple approximations to the mean of the largest order statistic both for the iid and non-iid cases have good MSE properties, and (b) the large sample distribution for a non-identically distributed case still obeys the Type I Gumbel distribution with shifted parameters.

Keywords: largest order statistic, multivariate normal, error function, peak load, genetic selection, Rayleigh Distribution, Gumbel Distribution

INTRODUCTION

The hourly load demand for electricity as noted by an Electric Cooperative roughly follows a normal distribution so that if Xj represents the demand at hour j, then Xj ~ N(μj,σj2). We seek the probability that the peak demand occurs at hour t, that is, we want to evaluate:

(1) ...Pr(Xt>Y)

where Y=max{Xi},i≠t

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distribution N(μ1-μ2, σ12 +σ22) by a table look up or actual numerical integration. When there are n competing hours, then one must resolve Equation (1) which is the focus of the present paper.

The problem of finding the distribution of Y is, by itself, nothing new. In fact, the classical approach would be to consider:

...Pr(Y≤y) = Pr(X1≤y‚ X2≤y‚ … xn≤y) = πPr(Xi≤y) = πφi(y) (2)

where Φi(.) is the cumulative distribution function of a normal distribution with mean μi and variance σi2. The probability problem given by (1) reduces to:

...Pr(Y≤y)=1-Pr(Xt≤Y|Y)πPr(Y≤Y) (3)

Equation (3) demands a large amount of integration and we shall not approach the problem this way.

A related problem arises out of this basic problem of determining the probability distribution of the maximum of n independent normal random variables. Consider the daily peak loads noted Y1,Y2,...,Yn for a period n. The distribution of each Yi is given by (2) and will be more explicitly stated in the body of this paper. We wish to know the joint distribution of these peak loads f(Y1,Y2,..,Yn). Knowledge of the joint distribution f(.) allows us to ask relevant questions such as, “What is the probability that the daily peak loads do not exceed a capacity limit L?” That is, we seek answer to the probability question:

...Pr(Y1≤L,Y2≤L,...Yn≤L)? (4)

With the passage of the Electricity Power Industry Reform Act (1998), the issues of generation capacity and demand requirements, average and peak load requirements and others had been highlighted because of the unbundling of the electricity charges passed on to the consumers. For instance, distribution utilities have to contend with the issue of determining how much electricity to procure from power generation companies considering both the average demand and the peak load demand in their service areas. A miscalculation on the part of the distribution utilities could mean millions in terms of losses.

Hill (2010) considered a similar problem in relation to the calculation of the probability that a runner wins in an n-player running match. Specifically, he obtained the probability distribution of the minimum of n independent normal random variables. He found that the probability distribution of Y, the minimum of n independent normal random variables obeys a multivariate normal distribution with mean vector μ and covariance matrix S where:

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... μ = (μ1, μ2,...,μn)’ and S = diag(σi2) is a diagonal matrix

with σi2 on the ith diagonal and zeroes elsewhere. (5) or

...���� �√��� ��√��������������������� (6)

The application of Hill’s (2010) results in more practical areas such as the power industry sector is obvious. One might be interested in the downtime power load (instead of the peak power load) for purposes of planning and forecasting of a distribution utility’s daily demand requirements.

Finally, a related problem that might be of interest is the probability distribution of the maximum of the maxima of random variables. Let S1 = {x11,

x12,...,x1n}, S2 = {x21,x22,...,x2n},..., Sp={x1p,x2p,...,xpn} be subsets of independent

random variables of equal length n. Let :

… �� � ������������� (7)

…� � ������� � � � �� �� � … � �

We are interested in computing the probability that Pr(T > a). As a practical application of this problem, consider the hourly demand for electricity on a daily basis over a one-month period. The subsets Si consists of the hourly

demand records (n = 24). The maxima Yi represents the peak demand for the

ith day. The random variable T represents the peak of the peak demands over a one month period. Knowing the probability distribution of the Y’s can be used to calculate the desired probability distribution of T.

Mathematical Derivation of the Distribution of the Maxima of n Normal Random Variables

We are interested in evaluating Pr(X0 > Y) where Y = max { Xj } j = 1,2,..,n

with the same assumptions as Hill’s(2010) paper. Following his derivation, we first obtain the probability distribution of Y:

Pr�� � �� � ����� � �� (8)

� ��

�� ��� � �����

��√��)

= � √�����

����� �����

��√�

�� � �

� √�����

����� �����

��√�

�� dx1...dxn

= � ������ � � � �

���������√�����

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Hence, the probability distribution of the maximum of n independent normal random variables is precisely a multivariate normal distribution where:

� � ���� ��� � � ����� � � ������

���� � � ���� ��� � � ���.

Let us write down the density function of X0 as:

g(x) = �

� √���� �

������������

We seek the probability ���� � ��.

������ �� � ���� � ��� � � ��� � ����� � ��� �� � � �������� � ���� ��

= � ��������������� and the outer integral is from (-��� ��. (9)

However, the integral is precisely the integral of an (n+1)-variate normal distribution or the n-variate normal distribution before plus one additional variable. That is,

h(t) = �

���� ����√�������

���������������

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where S is an (n+1) x (n+1) strictly diagonal positive definite matrix whose

diagonal elements are σi2 , i = 0,1,...,n and μ = (μ0, μ1,..., μn). The computation

of the integral , however, is not a trivial task and we shall return to this issue later when we perform our simulation exercises.

Application 1: Given the hourly data for electricity demand in a given locality, we can ask the question of finding the probability that the peak demand is less

than 50MW. If X1,X2,..,X24 are the hourly electricity demands, and Y = max

{Xi}, then we can compute P(Y ≤ 50) using Equation (8). Of course, we need the

hourly data (24-hour data) over at least one month to establish the distribution of the X’s.

Application 2: Our motivation comes from a genetic selection problem in

agricultural research, originally considered by Rawlings (1976) and Hill (1976,

1977), and more recently by Tong (1990, p. 129). To describe the problem briefly, suppose that an agricultural genetic selection project involves n animals, for example pigs, and the top performer is to be selected for breeding.

Let X1; . . . ;Xn be the measurements of a certain biological or physical

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The animal with score X(n) is to be selected. If X1; . . . ;Xn are independent with

mean μ, then the common mean of the observations of offsprings of the selected animal with score X(n) is E(X(n)), and therefore the expected gain in one

generation is E(X(n)) - μ. The assumption of independence will be violated if the

pigs come from the same stock and so we make sure that the animals to be observed come from different stocks. If Y = max{Xi} we can ask the same

probability question as before.

Independent and Identically (IID) Normal Random Variables Case

Often, our interest rests mainly on the mean of the maximum of n random variables. Even in the case where the random variables are normally distributed, the numerical integration required to determine the expected value of the maximum is tedious. It is possible to develop heuristic approximations to the expected value of the maximum of n iid normal random variables.

Let X(n) = max {Xi}. Clearly X(n) > E(Xi) = μ for all i. Suppose that:

X(n) = μ + ε where ε ~ F(ε) , and (11)

F(ε) = 1 – exp(-��

���) , 0 < ε < ∞

The distribution F(.) is called a Rayleigh Distribution used in extreme value analysis and is a special case of the Weibull distribution with α = √2 b and β = 2.. The Rayleigh distribution has been used to model the distance covered per unit time of a particle whose (x,y) coordinates are each independent standard normal random variables. Model 11 says that the maximum order statistic X(n)

exceeds the mean of the component normal random variables by a random amount ε whose expected value is :

E(ε) = √2 b Γ(1 + ½) (12)

The expected value of X(n) can be obtained from (11):

E(X(n)) = μ + E(ε). (13)

In other words, if we can model the extreme value statistic X(n) as a sum of the

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Procedurally, if x1,x2,..,xn are iid N(μ,σ2) are random samples, then we first take the maximum likelihood estimators of μ and σ2 corrected for bias:

� � = �

�∑xi (14)

s2 =

���∑ (xi – � �)2

Put εi = �xi – � �� for i = 1, 2, ...,n, which we now assume obeys the Rayleigh distribution with mean equal to (13). The maximum likelihood estimator of the parameter b of the Rayleigh distribution is given by:

b = ���� ∑ ��

� (15)

Our estimate for E(X(n)) = μ(n) is thus:

����� = � � + 2√����� ∑ ��� �� = � � + √2���∑ ��� ��≈ � � + 2.51 sd(x) (16)

Or heuristically, ����� = � � + 3 sd(x) which is intuitively appealing. We

verify this error model by using simulation in the next section.

Independent but not Identically Distributed Normal Random Variables

The situation when the random variables Xi are not identically distributed but independent normal random variables with μi = E(Xi) and σi2 = var(Xi) is more complicated but also more useful in practice. In the succeeding discussions, we agree on the following notation:

�� = �

� ∑ �� = mean of the means

μ(n) = max { μ1, μ2, ..., μn} = maximum of the means X(n) = max { X1,X2,...,Xn} = maximum order statistic Dn = μ(n) - ��

We create a data model similar to (11). We start with the obvious inequality:

μ(n) - �� ≥ 0 with equality only if μi = μ for all i. (17) Procedurally, if x1,x2,..,xn are iid N(μ,σ2) are random samples, then we

first take the maximum likelihood estimators of μ and σ2 corrected for bias:

� � = �

�∑xi (14)

s2 =

���∑ (xi – � �)2

Put εi = �xi – � �� for i = 1, 2, ...,n, which we now assume obeys the

Rayleigh distribution with mean equal to (13). The maximum likelihood estimator of the parameter b of the Rayleigh distribution is given by:

b = ���� ∑ ��

� (15)

Our estimate for E(X(n)) = μ(n) is thus:

����� = � � + 2√����� ∑ ��� �� = � � + √2���∑ ��� �� ≈ � � + 2.51 sd(x) (16)

Or heuristically, ����� = � � + 3 sd(x) which is intuitively appealing. We

verify this error model by using simulation in the next section.

Independent but not Identically Distributed Normal Random Variables

The situation when the random variables Xi are not identically distributed

but independent normal random variables with μi = E(Xi) and σi2 = var(Xi) is

more complicated but also more useful in practice. In the succeeding discussions, we agree on the following notation:

�� = �

� ∑ �� = mean of the means

μ(n) = max { μ1, μ2, ..., μn} = maximum of the means

X(n) = max { X1,X2,...,Xn} = maximum order statistic

Dn = μ(n) - ��

We create a data model similar to (11). We start with the obvious inequality: μ(n) - �� ≥ 0 with equality only if μi = μ for all i. (17)

To see this, consider:

μ(n) = �

�∑ ��� ��� > ��

�∑ ��� � = �� since μ(n) > μi for all i. (18) It follows that Dn ≥ 0.

We now claim that :

X(n) = �� + εi where εi = �xi - �����, i = 1,2,..,n. (19)

and : εi ~ F(ε) is a Rayleigh distribution function.

In practice, the underlying means are unknown and so we replace (19) by its sample counterpart:

X(n) = ���+ �xi - ����. (20)

The second term on the right is assumed to obey a Rayleigh distribution with the mean given by Equation (12). However, the MLE of b is now:

�� = ���� ∑ ���� ��������� . (21)

Equation (21) is no longer approximately equal to √2 sd(x). We can argue heuristically to obtain a sense of the magnitude of (21) in relation to the case when the xi’s are properly centered around their means.

Let Y = (x1-μ1)2 + (x2 – μ2)2 +...+ (xn – μn)2, and Z =(x1-μ(n))2 + (x2 – μ(n))2 +...+ (xn – μ(n))2. If we take expectations:

E(Y) = σ2 + σ2 + ... + σ2 = nσ2 , (22) assuming equal variances. Next consider one term of the quantity Z:

E(xi – μ(n) )2 = E(xi – μi + μi – μ(n) )2 = E(xi-μi)2 (23) + 2E(xi – μi)(μi – μ(n)) + E(μi – μ(n))2

= σ2 + 0 + θ12 , since the second term above is zero and θ12 = (μi – μ(n))2

It follows that E(Y) = nσ2 ≤ E(Z) = nσ2 + ∑ � �� �

� . Equation (21) is greater than Equation (15), and so we expect a greater additive factor to the expected value of the means than in the IID case. In fact, the inequality provides us an insight on the magnitude of the difference since:

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To see this, consider:

μ(n) = �

�∑ ������ > ��

�∑ ���� = �� since μ(n) > μi for all i. (18) It follows that Dn ≥ 0.

We now claim that :

X(n) = �� + εi where εi = �xi - �����, i = 1,2,..,n. (19)

and : εi ~ F(ε) is a Rayleigh distribution function.

In practice, the underlying means are unknown and so we replace (19) by its sample counterpart:

X(n) = ���+ �xi - �����.

(20)

The second term on the right is assumed to obey a Rayleigh distribution with the mean given by Equation (12). However, the MLE of b is now:

�� = ���� ∑ ����� �������� . (21)

Equation (21) is no longer approximately equal to √2 sd(x). We can argue heuristically to obtain a sense of the magnitude of (21) in relation to the case when the xi’s are properly centered around their means.

Let Y = (x1-μ1)2 + (x2 – μ2)2 +...+ (xn – μn)2, and Z =(x1-μ(n))2 + (x2 – μ(n))2

+...+ (xn – μ(n))2. If we take expectations:

E(Y) = σ2 + σ2 + ... + σ2 = nσ2 , (22)

assuming equal variances. Next consider one term of the quantity Z:

E(xi – μ(n) )2 = E(xi – μi + μi – μ(n) )2 = E(xi-μi)2 (23)

+ 2E(xi – μi)(μi – μ(n)) + E(μi – μ(n))2

= σ2 + 0 + θ12 , since the second term above is zero

and θ12 = (μi – μ(n))2

It follows that E(Y) = nσ2 ≤ E(Z) = nσ2 + ∑ �

�� �

� . Equation (21) is greater than Equation (15), and so we expect a greater additive factor to the expected value of the means than in the IID case. In fact, the inequality provides us an insight on the magnitude of the difference since:

E(Z) – E(Y) = ∑ ��� ��. (24)

To see this, consider:

μ(n) = �

�∑ ��� ��� > ��

�∑ ��� � = �� since μ(n) > μi for all i. (18)

It follows that Dn≥ 0.

We now claim that :

X(n) = �� + εi where εi = �xi - �����, i = 1,2,..,n. (19)

and : εi ~ F(ε) is a Rayleigh distribution function.

In practice, the underlying means are unknown and so we replace (19) by its sample counterpart:

X(n) = ���+ �xi - �����.

(20)

The second term on the right is assumed to obey a Rayleigh distribution with the mean given by Equation (12). However, the MLE of b is now:

�� = ���� ∑ ���� ��������� . (21)

Equation (21) is no longer approximately equal to √2 sd(x). We can argue heuristically to obtain a sense of the magnitude of (21) in relation to the case when the xi’s are properly centered around their means.

Let Y = (x1-μ1)2 + (x2 – μ2)2 +...+ (xn – μn)2, and Z =(x1-μ(n))2 + (x2 – μ(n))2

+...+ (xn – μ(n))2. If we take expectations:

E(Y) = σ2 + σ2 + ... + σ2 = nσ2 , (22)

assuming equal variances. Next consider one term of the quantity Z:

E(xi – μ(n) )2 = E(xi – μi + μi – μ(n) )2 = E(xi-μi)2 (23)

+ 2E(xi – μi)(μi – μ(n)) + E(μi – μ(n))2

= σ2 + 0 + θ12 , since the second term above is zero

and θ12 = (μi – μ(n))2

It follows that E(Y) = nσ2 E(Z) = nσ2 + ∑ � �� �

� . Equation (21) is greater

than Equation (15), and so we expect a greater additive factor to the expected value of the means than in the IID case. In fact, the inequality provides us an insight on the magnitude of the difference since:

E(Z) – E(Y) = ∑ ��� ��. (24)

To see this, consider:

μ(n) = �

�∑ ��� ��� > ��

�∑ ��� � = �� since μ(n) > μi for all i. (18) It follows that Dn ≥ 0.

We now claim that :

X(n) = �� + εi where εi = �xi - �����, i = 1,2,..,n. (19)

and : εi ~ F(ε) is a Rayleigh distribution function.

In practice, the underlying means are unknown and so we replace (19) by its sample counterpart:

X(n) = ���+ �xi - ����. (20)

The second term on the right is assumed to obey a Rayleigh distribution with the mean given by Equation (12). However, the MLE of b is now:

�� = ���� ∑ ���� ��������� . (21)

Equation (21) is no longer approximately equal to √2 sd(x). We can argue heuristically to obtain a sense of the magnitude of (21) in relation to the case when the xi’s are properly centered around their means.

Let Y = (x1-μ1)2 + (x2 – μ2)2 +...+ (xn – μn)2, and Z =(x1-μ(n))2 + (x2 – μ(n))2 +...+ (xn – μ(n))2. If we take expectations:

E(Y) = σ2 + σ2 + ... + σ2 = nσ2 , (22) assuming equal variances. Next consider one term of the quantity Z:

E(xi – μ(n) )2 = E(xi – μi + μi – μ(n) )2 = E(xi-μi)2 (23) + 2E(xi – μi)(μi – μ(n)) + E(μi – μ(n))2

= σ2 + 0 + θ12 , since the second term above is zero and θ12 = (μi – μ(n))2

It follows that E(Y) = nσ2 ≤ E(Z) = nσ2 + ∑ � �� �

� . Equation (21) is greater than Equation (15), and so we expect a greater additive factor to the expected value of the means than in the IID case. In fact, the inequality provides us an insight on the magnitude of the difference since:

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There is a rough approximation on the value of E(X(n)) provided by

Hamza(2008) and we borrow his theorem below:

Theorem (Hamza). Let X1,X2,..,Xn be independent random variables

with Mi = E(Xi), then:

ܯഥ≤ E(X(n)) ≤ܯഥ + ௡ିଵ ௡ M(n).

where:

ܯഥ = average of the Mi’s

M(n) = max{Mi}.

In relation to the present problem, it may often be more useful for the electric distribution utility to have an idea of the magnitude of the peak demand (on the average). Thus, Hamza’(s) (2008) theorem will be most useful in providing such information. Similarly, in the genetic selection problem of application 2, we can assume that the means differ across the animals and that we chose the maximum observed Xi as the animal to be used for breeding.

Then, again , Hamza’s(2008) results will apply. Note that the Theorem does not require that the random variables be normal. It applies to all independent random variables, and so, is quite general.

Meanwhile for sufficiently large n, we can consider some asymptotic results to simplify the calculations.

Asymptotic Results for the IID Case

We wish to show that for large n, the asymptotic distribution of the largest order statistic X(n) is a Gumbel Type I distribution. The Fisher–Tippet–

Gnedenko theorem (also the Fisher–Tippet theorem or the extreme value theorem) is a general result regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization converges in distribution to one of three possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem (or convergence to types theorem) is given to Gnedenko (1948); previous versions were stated by Fisher and Tippett in 1928 and Fréchet in 1927. The role of the extremal types theorem for maxima is similar to that of the central limit theorem for averages.

Let Y = max {Xi} of a sequence of independent and identically distributed

standard normal random variables Xi. Let Φ(x) denote the cumulative

distribution function of a standard normal random variable x. The cumulative distribution function of the maximum order statistic Y is:

Fn(y) = [Φ(y)]n , -∞ < y < ∞ as before (25)

There is a rough approximation on the value of E(X(n)) provided by

Hamza(2008) and we borrow his theorem below:

Theorem (Hamza). Let X1,X2,..,Xn be independent random variables

with Mi = E(Xi), then:

ܯഥ≤ E(X(n)) ≤ܯഥ + ௡ିଵ ௡ M(n).

where:

ܯഥ = average of the Mi’s

M(n) = max{Mi}.

In relation to the present problem, it may often be more useful for the electric distribution utility to have an idea of the magnitude of the peak demand (on the average). Thus, Hamza’(s) (2008) theorem will be most useful in providing such information. Similarly, in the genetic selection problem of application 2, we can assume that the means differ across the animals and that we chose the maximum observed Xi as the animal to be used for breeding.

Then, again , Hamza’s(2008) results will apply. Note that the Theorem does not require that the random variables be normal. It applies to all independent random variables, and so, is quite general.

Meanwhile for sufficiently large n, we can consider some asymptotic results to simplify the calculations.

Asymptotic Results for the IID Case

We wish to show that for large n, the asymptotic distribution of the largest order statistic X(n) is a Gumbel Type I distribution. The Fisher–Tippet–

Gnedenko theorem (also the Fisher–Tippet theorem or the extreme value theorem) is a general result regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization converges in distribution to one of three possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem (or convergence to types theorem) is given to Gnedenko (1948); previous versions were stated by Fisher and Tippett in 1928 and Fréchet in 1927. The role of the extremal types theorem for maxima is similar to that of the central limit theorem for averages.

Let Y = max {Xi} of a sequence of independent and identically distributed

standard normal random variables Xi. Let Φ(x) denote the cumulative

distribution function of a standard normal random variable x. The cumulative distribution function of the maximum order statistic Y is:

Fn(y) = [Φ(y)]n , -∞ < y < ∞ as before (25)

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Clearly, lim Fn(y) = 1 or 0 depending on whether Φ(y) = 1 or 0 as n →∞. In order to obtain a non-degenerate limiting distribution, it is necessary to transform Y by applying a linear transformation with coefficients which depend on the sample size n but not on y. This process is similar to the standardization process in statistics. Let Yn’ = anY + bn where an and bn are coefficients depending on n but not on y. Suppose first that the limiting distribution G(y) exists. That is:

lim Fn(y) = G(y) exists for properly transformed Y.

If we increase the sample size to nN where N > 0, then the largest of the nN values X1,X2,...,XnN is also equal to the largest of the values Xj-1(n+1), Xj-1(n+2),...,Xjn, for j=1,2,...,N. It follows that the limiting distribution G(.) obeys:

[G(y)]N = G(aN y + bN) (26)

Equation (26) is called the stability postulate and was first discovered by Frechet (1927). Now, take aN = 1. Equation (26) now becomes:

[G(y)]N = G(y + bN) (27)

We iterate Equation (27) for larger samples NM:

[G(y)]NM = G((x + bNM) = G(y + bN)M = G(y + bN + bM) (28)

We infer that : bNM = bN + bM . This equation tells that bN must be some type of logarithmic function. In particular, bN = σ log N where σ is a constant. We plug this value into Equation (27) to get:

[G(y)]N = G(y + σ log N ) (29)

Take the logarithm of both sides of (29) to get : N{- log G(y)} = - log G(y + σN) where the negative sign emerges from the fact that G(y) ≤ 1. We take the logarithm once again (sometimes called the law of iterated logarithms):

log N + log {- log G(y)} = log{- log G(y + σN)} (30)

Let h(y) = log {- log G(y)}. Equation (30) becomes:

log N + h(y) = h(σ(௬

ఙ + log N) or h(y ) = h(σ( ௬

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Put σ(௬

ఙ + log N) = 0 (or find h(0) on the right hand side. This means that

ఙ = - log N. Substituting back to Equation (31), we obtain:

h(y) = h(0) - ௬

ఙ, , since h(y) decreases as y increases. (32)

Thus, -logG(y) = exp(h(y)) = exp (h(0) - ௬

ఙ) = exp( - (

௬ିఙ௛ሺ଴ሻ

ఙ ). Put μ = σh(0)

and we have:

– log G(y) = exp (-(௬ିఓሻ

ఙ ). (33)

The last step is easy to see:

G(y) = exp (-exp (-(௬ିఓሻ

ఙ )) or G(y) = exp (-exp (-(

௬ିఈሻ

ఉ )). (34)

The asymptotic mean and variance of Y can now be obtained from G(y). It is an easy exercise to find the moments of this distribution:

The mean, variance, skewness, and kurtosis are

where is the Euler-Mascheroni constant and is Apéry's constant.

Asymptotic Results for the Independent but not Identically Distributed Case (INID)

Let X1,X2,...,Xn be independent normal random variables with means E(Xi) = μi and variances given by var(Xi) = σi2. As before, we let Y = max{Xi}. The distribution of Y is given by:

Fn*(y) = π[Fi(y)], i = 1 2,...,n (35)

(11)

without bound. In practice, it is the second interpretation that appears to be reasonable and implementable. Hence, our asymptotic analysis will follow the second interpretation.

To this end, let:

�� = �

� ∑ �� = arithmetic average of the means which is non-stochastic,

��

���� = �

� ∑ ��� = arithmetic average of the variances, also non-stochastic

� � = �

√� �∑ ��� = the square root of the arithmetic average of the

variances.

We consider:

Fn*(����

�� ) = π[Fi( ����

�� )], i = 1 2,...,n.

If samples of size M were obtained from each d.f., then:

Fn*(����

�� ) = π[Fi( ����

�� )]M, i = 1 2,...,n., M = 1,2,3,...→∞ (36)

Assume , as before , that the limiting distribution of (36) exists and is G(.):

π[Fi(����

�� )]M → G( ����

�� ) as M →∞. (37)

Applying the same stability postulate as before, we have:

G(����

�� �M = G( ����

�� + bM) (38)

For which we conclude that bM = θ log M. Hence:

log M + log (-log(G(����

�� ) ) = log(-log(G( ����

�� + θ log M )). (39)

Equation (39) implies that if h(����

�� ) = log (-log(G( ����

�� ) ), then:

h(����

�� ) = h(0) - ����

�� � = -( ����

�� � - ��

�� �h(0)) (40)

without bound. In practice, it is the second interpretation that appears to be reasonable and implementable. Hence, our asymptotic analysis will follow the second interpretation.

To this end, let:

�� = �

� ∑ �� = arithmetic average of the means which is non-stochastic,

��

���� = �

� ∑ ��� = arithmetic average of the variances, also non-stochastic

� � = �

√� �∑ ��� = the square root of the arithmetic average of the

variances.

We consider:

Fn*(����

�� ) = π[Fi( ����

�� )], i = 1 2,...,n.

If samples of size M were obtained from each d.f., then:

Fn*(����

�� ) = π[Fi( ����

�� )]M, i = 1 2,...,n., M = 1,2,3,...→∞ (36)

Assume , as before , that the limiting distribution of (36) exists and is G(.):

π[Fi(����

�� )]M → G( ����

�� ) as M →∞. (37)

Applying the same stability postulate as before, we have:

G(����

�� �M = G(

����

�� + bM) (38)

For which we conclude that bM = θ log M. Hence:

log M + log (-log(G(����

�� ) ) = log(-log(G(

����

�� + θ log M )). (39)

Equation (39) implies that if h(����

�� ) = log (-log(G(

����

�� ) ), then:

h(����

�� ) = h(0) -

���� �� � = -(

���� �� � -

��

(12)

It follows that:

G(yሻ = exp (-exp(-ሺ௬ିఓഥ Ȃఙഥ ఏഥ ௛ሺ଴ሻሻ) (41)

which we recognize as a Type I Gumbel distribution with α = ߤ + ߪh(0) and β

= ߪഥߠ. Here h(0)= log(-log (G(0)).

LITERATURE CITED

Arellano-Valle, R.B., Genton, M.G. (2005). On fundamental skew distributions. J. Multivariate Anal. 96, 93–116.

Arellano-Valle, R.B., Genton, M.G. (2007). On the exact distribution of linear combinations of order statistics from dependent randomvariables. J. Multivariate Anal. in press.

Arellano-Valle, R.B., Branco, M.D., Genton, M.G. (2006). A unified view on skewed distributions arising from selections. Canad. J.Statist. 34, 581– 601.

Crocetta, C., Loperfido, N. (2005). The exact sampling distribution of L-statistics. Metron 63, 1–11.

David, H.A. (1981). Order Statistics. Wiley, New York.

Fang, K.-T., Kotz, S., Ng, K.-W. (1990). Symmetric multivariate and related distributions. Monographs on Statistics and AppliedProbability, vol. 36. Chapman & Hall, Ltd., London.

Genton, M.G. (2004). Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Edited volume, Chapman & Hall/CRC Press, London, Boca Raton, FL.

Genz, A. (1992). Numerical computation of multivariate Normal probabilities.

J. Comput. Graph Statist. 1, 141–149.

Genz, A., Bretz, F. (2002). Methods for the computation of multivariate t-probabilities. J. Comput. Graph. Statist. 11, 950–971.

Ghosh, M. (1972). Asymptotic properties of linear functions of order statistics for m-dependent random variables. Calcutta Statist. Assoc.Bull. 21, 181–192.

It follows that:

G(yሻ = exp (-exp(-ሺ௬ିఓഥ Ȃఙഥ ఏഥ ௛ሺ଴ሻሻ) (41)

which we recognize as a Type I Gumbel distribution with α = ߤ + ߪh(0) and β

= ߪഥߠ. Here h(0)= log(-log (G(0)).

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combinations of order statistics from dependent randomvariables. J. Multivariate Anal. in press.

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It follows that:

G(yሻ = exp (-exp(-ሺ௬ିఓഥ Ȃఙഥ ఏഥ ௛ሺ଴ሻሻ) (41)

which we recognize as a Type I Gumbel distribution with α = ߤ + ߪh(0) and β

= ߪഥߠ. Here h(0)= log(-log (G(0)).

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Arellano-Valle, R.B., Genton, M.G. (2005). On fundamental skew

distributions. J. Multivariate Anal. 96, 93–116.

Arellano-Valle, R.B., Genton, M.G. (2007). On the exact distribution of linear

combinations of order statistics from dependent randomvariables. J. Multivariate Anal. in press.

Arellano-Valle, R.B., Branco, M.D., Genton, M.G. (2006). A unified view on

skewed distributions arising from selections. Canad. J.Statist. 34, 581– 601.

Crocetta, C., Loperfido, N. (2005). The exact sampling distribution of

L-statistics. Metron 63, 1–11.

David, H.A. (1981). Order Statistics. Wiley, New York.

Fang, K.-T., Kotz, S., Ng, K.-W. (1990). Symmetric multivariate and related

distributions. Monographs on Statistics and AppliedProbability, vol. 36. Chapman & Hall, Ltd., London.

Genton, M.G. (2004). Skew-Elliptical Distributions and Their Applications: A

Journey Beyond Normality. Edited volume, Chapman & Hall/CRC Press, London, Boca Raton, FL.

Genz, A. (1992). Numerical computation of multivariate Normal probabilities.

J. Comput. Graph Statist. 1, 141–149.

Genz, A., Bretz, F. (2002). Methods for the computation of multivariate

t-probabilities. J. Comput. Graph. Statist. 11, 950–971.

Ghosh, M. (1972). Asymptotic properties of linear functions of order statistics

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Genz, A & F Bretz. (2002). Methods for the computation of multivariate t-probabilities. J. Comput. Graph. Statist., 11:950–971.

Genz, A. (1992). Numerical computation of multivariate Normal probabilities. J. Comput. Graph Statist., 1:141–149.

Ghosh, M. (1972). Asymptotic properties of linear functions of order statistics for m-dependent random variables. Calcutta Statist. Assoc.Bull., 21:181–192.

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References

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