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On the asymptotic and approximate distributions of the product of an inverse Wishartmatrix and a Gaussian vector

Kotsiuba, Igor; Franko, Ivan; Mazur, Stepan

2015

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Citation for published version (APA):Kotsiuba, I., Franko, I., & Mazur, S. (2015). On the asymptotic and approximate distributions of the product of aninverse Wishart matrix and a Gaussian vector. (Working Papers in Statistics; No. 11). Department of Statistics,Lund university.

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On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian vector

IGOR KOTSIUBA, IVAN FRANKO NATIONAL UNIVERSITY OF LVIV

STEPAN MAZUR, LUND UNIVERSITY

Working Papers in StatisticsNo 2015:11Department of StatisticsSchool of Economics and ManagementLund University

ON THE ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS OF

THE PRODUCT OF AN INVERSE WISHART MATRIX AND A

GAUSSIAN VECTOR

I. KOTSIUBA AND S. MAZUR

Abstract. In this paper we study the distribution of the product of an inverse

Wishart random matrix and a Gaussian random vector. We derive its asymptotic

distribution as well as its approximate density function formula which is based on theGaussian integral and the third order Taylor expansion. Furthermore, we compare

obtained asymptotic and approximate density functions with the exact density whichis obtained by Bodnar and Okhrin (2011). A good performance of obtained results

is documented in the numerical study.

1. Introduction

The multivariate normal distribution is one of the most important and very usefuldistribution in multivariate statistical analysis. If we have sample of size n from the k-variate normal distribution then the sample unbiased estimators of the mean vector andthe covariance matrix have k-variate normal and k-variate Wishart distributions, respec-tively, and they are independently distributed (see Section 3 of Mardia et al. (1980)).

Recently, several papers study the investigation of properties of the estimators formean vector and covariance matrix. For instance, Stein (1956) and Jorion (1986) discussimprovement techniques for the mean estimator. Ledoit and Wolf (2004), Bodnar andGupta (2010), Bai and Shi (2011), Cai et al. (2011), Cai and Yuan (2012), Cai andZhou (2012), and Bodnar et al. (2014a, 2015a) improve the estimation techniques forthe (inverse) covariance matrices. Papers by von Rosen (1988), Styan (1989), Dıaz-Garcıa et al. (1997), Bodnar and Okhrin (2008), Drton et al. (2008), and Bodnar etal. (2015b) develop the distributional properties of the Wishart matrices, the inverseWishart matrices and related quantities.

However, functions that depend on the product of the (inverse) Wishart random ma-trix and the Gaussian random vector are not comprehensively investigated in literature.Bodnar and Okhrin (2011), Bodnar et al. (2013, 2014b) derived the exact distributionsof the product of the (inverse) Wishart random matrix and the Gaussian random vector,which have integral representations. We note that these products have direct applicationsin the portfolio theory and in the discriminant analysis. For example, the elements of thesample estimator of the discriminant function are expressed as products of the inverseWishart random matrix and the Gaussian random vector, and the weights of the tan-gency portfolio are estimated by the same product. In this paper we extend the resultsobtained by Bodnar and Okhrin (2011) by providing the asymptotic and approximatedensity functions of this product.

Received 02/03/2015.2010 Mathematics Subject Classification. 62E17, 62E20.Key words and phrases. Wishart distribution, multivariate normal distribution, asymptotic distribu-

tion, integral approximation.The second author appreciate the financial support of the Swedish Research Council Grant Dnr: 2013-5180 and Riksbankens Jubileumsfond Grant Dnr: P13-1024:1.

1

2 I. KOTSIUBA AND S. MAZUR

The rest of the paper is structured as follows. The main results are presented inSection 2, where the asymptotic distribution of the product of an inverse Wishart randommatrix and a Gaussian random vector is derived as Theorem 2.1. Its approximate densityfunction which is based on the third order Taylor series approximation is obtained inTheorem 2.2. The numerical performance of the obtained results is outlined in details inSection 3, while Section 4 summarizes the paper.

2. Main Results

In this section we consider the product of the inverse Wishart random matrix and anormally distributed random vector. In particular, we derive its asymptotic and approx-imative density functions.

Let A ∼ Wk(n,Σ), i.e. the random matrix A has the k-dimensional Wishart distri-bution with n degrees of freedom and covariance matrix Σ, which is positive definite.We consider the case when k < n, i.e. A is a non-singular random matrix. Also, letz ∼ Nk(µ, λΣ), it means that the random vector z has the k-dimensional normal dis-tribution with mean vector µ and covariance matrix λΣ, where λ > 0 is a constant.Furthermore, let L is the p × k non-zero matrix of constants of rank(L) = p < k, thesymbol 0 denotes the suitable vector of zeros, and Ik stands for the k×k identity matrix.Throughout the paper it is assumed that A and z are independently distributed. Thenit holds that the distribution of LA−1z|(z = z∗) equals to the distribution of LA−1z∗.It is noted that

LA−1z∗d= z∗TΣ−1z∗

LA−1z∗

z∗TA−1z∗z∗TA−1z∗

z∗TΣ−1z∗, (1)

where the symbold= denotes the equality in distribution.

Using Theorem 3.2.12 of Muirhead (1982) we obtain that

z∗TΣ−1z∗

z∗TA−1z∗∼ χ2

n−k+1

and is independently distributed of z∗. Furthermore, it holds that z∗TA−1z∗ is in-dependent of LA−1z∗/z∗TA−1z∗ for given z∗ (see Theorem 3 of Bodnar and Okhrin(2008)). Consequently, it is independent of z∗TΣ−1z∗LA−1z∗/z∗TA−1z∗ and respec-tively of zTΣ−1zLA−1z/zTA−1z. Using the proof of Theorem 1 of Bodnar and Schmid(2008) it follows that

z∗TΣ−1z∗LA−1z∗

z∗TA−1z∗∼ tp

(n− k + 2; LΣ−1z∗,

1

n− k + 2z∗TΣ−1z∗LRz∗LT

), (2)

where Rz∗ = Σ−1 −Σ−1z∗z∗TΣ−1/z∗TΣ−1z∗. The symbol tp(d; b; B) stands for the p-dimensional multivariate t-distribution with d degrees of freedom, the location parameterb, and the dispersion matrix B.

Let ξ = z∗TΣ−1z∗/z∗TA−1z∗. Then ξ and z are independent and ξ ∼ χ2n−k+1. As a

result, we obtain the following stochastic representation of LA−1z which is given by

LA−1zd= ξ−1

LΣ−1z +

√zTΣ−1z

n− k + 2(LRzL

T )1/2t0

, (3)

where Rz = Σ−1 − Σ−1zzTΣ−1/zTΣ−1z; ξ ∼ χ2n−k+1, z ∼ Nk(µ, λΣ), and t0 ∼

tp(n− k + 2,0p, Ip). Moreover, ξ, z, and t0 are independently distributed.

Next, we consider the asymptotic distribution of nLA−1z. It is remarkable thatusually λ = 1/n (see Section 3 of Muirhead (1982)). Then

E(nLA−1z) =n

n− k − 1LΣ−1µ. (4)

ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS 3

Also, it is easy to see that

limn→∞

ξ

n= 1 and lim

n→∞z = µ. (5)

Thus, from (3), (4), (5), and Theorem 6.15 of Arnold (1990), we get the asymptoticdistribution of nLA−1z which is presented in the following theorem.

Theorem 2.1. Let matrix A ∼ Wk(n,Σ) and vector z ∼ Nk(µ, λΣ) with Σ positivedefinite and λ = 1/n. Furthermore, let A and z be independent and L be a p× k matrixof constants of rank(L) = p < k. Then the asymptotic distribution of nLA−1z, i.e.n→∞, is given by

√n(nLA−1z− LΣ−1µ

) d→ Np(0, Σ), (6)

where Σ = (1 + µTΣ−1µ)LΣ−1LT − LΣ−1µµTΣ−1LT .

From Theorem 2.1 we get that the asymptotic distribution of the random vectornLA−1z has a p-variate normal distribution.

In many applications the sample size n is not large. For this reason, we need anadjustment like a finite-sample approximation of the density. Bodnar and Okhrin (2011)derived an exact density function of LA−1z. However, density function is expressed asa four-dimensional integral of the generalized function. Hence, the aim of this paper isto derive a simpler approximative density function of the random vector LA−1z.

Let Q = Σ−1/2LT(LΣ−1LT

)−1LΣ−1/2, P = Σ−1/2LT

(LΣ−1LT

)−1/2, S1 = r1r

T1 ,

S2 = r3rT3 , r1 = Σ−1/2µ, r2 = PT r1, r3 = (Ik −Q)r1, r4 = Qr1, r5 = −(p − 1)/ar1 +

1/br3, a = rT1 r1, b = rT1 (Ik −Q)r1, c = rT1 Qr1. It is noted that Q is a k × k singular

and projection matrix with Q = PPT .In the next theorem we suggest the approximate density function of LA−1z which is

based on the Gaussian integral and the third order Taylor series expansion.

Theorem 2.2. Let matrix A ∼ Wk(n,Σ) and vector z ∼ Nk(µ, λΣ) with Σ positivedefinite. Furthermore, let A and z be independent and L be a p× k matrix of constantsof rank(L) = p < k. Then the approximate density function of LA−1z is given by

fLA−1z(y) ≈ |LΣ−1LT |−1/2

a(p−1)/2b1/2πp/2

Γ(p+n−k+2

2

)Γ(n−k+2

2

)×∫ +∞

0

f−(p+n−k+2)/22 (sy,0)

[1 +

λ

2tr[g2(sy)]

]×spfχ2

n−k+1(s)ds, (7)

4 I. KOTSIUBA AND S. MAZUR

where y = s(LΣ−1LT )−1/2y, f2(sy,0) = 1 + 1a

(yT y − 2yT r2 + c+ (c−yT r2)2

b

),

ω(sy) = −(f2(sy,0)− 1)r1 + r4 −Py +c− yrT2

b(2r4 −Py)− (c− yr2)2

b2r3,

g2(sy) =p− 1

a

(2

aS1 − Ik

)+

1

b

(2

bS2 − (Ik −Q)

)+ r5

(rT5 +

2(p+ n− k + 2)

af2(sy,0)ωT (sy)

)+

p+ n− k + 2

a2f22 (sy,0)

{p+ n− k + 4

a2ω(sy)ωT (sy)

+ f2(sy,0)

[a(f2(sy,0)− 1)

(Ik −

4

aS1

)− 4r1 [ω(sy) + (f2(sy,0)− 1)r1]

T

]− aQ +

a(c− yT r2)

b2

{b(2r4 −Py)

[2

brT3 +

2rT4 − yTPT

c− yT r2

]+ 2bQ

− 2r3(2rT4 − yTPT )− (c− yT r2)(Ik −Q)− 4

[2r4 −Py − c− yT r2

br3

]rT3

}}.

The symbol fχ2n−k+1

(·) denotes the density of the χ2-distributed random variable with

(n− k + 1) degrees of freedom.

Proof. From (2) the unconditional density function of zTΣ−1zLA−1z/zTA−1z is givenby

fzT Σ−1z LA−1z

zT A−1z

(y) =

∫Rk

fz∗T Σ−1z∗ LA−1z∗

z∗A−1z∗(y|z = z∗)fz(z∗) dz∗

= λ−k/2|Σ|−1/2

(2π)k/2

Γ(p+n−k+2

2

)πp/2Γ

(n−k+2

2

)×∫Rk

exp

(− (z∗ − µ)TΣ−1(z∗ − µ)

2λ

)|LRz∗LT |−1/2

(z∗TΣ−1z∗)p/2

×(

1 +(y − LΣ−1z∗)T (LRz∗LT )−1(y − LΣ−1z∗)

z∗TΣ−1z∗

)−(p+n−k+2)/2

dz∗.

From Theorem 18.2.8 of Harville(1997) it follows that(LRz∗LT

)−1=

(LΣ−1LT

)−1+

(LΣ−1LT

)−1LΣ−1z∗z∗TΣ−1LT

(LΣ−1LT

)−1

z∗TΣ−1z∗ − z∗TΣ−1LT(LΣ−1LT

)−1LΣ−1z∗

,

∣∣LRz∗LT∣∣ =

∣∣LΣ−1LT∣∣ z∗TΣ−1z∗ − z∗TΣ−1LT

(LΣ−1LT

)−1LΣ−1z∗

z∗TΣ−1z∗.

Using the transformation t = Σ−1/2(z∗−µ) in the last integral with Jacobian |Σ|1/2 weget

fzT Σ−1z LA−1z

zT A−1z

(y) = λ−k/2|LΣ−1LT |−1/2

(2π)k/2

Γ(p+n−k+2

2

)πp/2Γ

(n−k+2

2

)×∫Rk

exp

(−tT t

2λ

) [(t + r1)T (t + r1)

]−(p−1)/2

[(t + r1)T (Ik −Q)(t + r1)]1/2

×{

1 +1

(t + r1)T (t + r1)

[yT y − 2yTPT (t + r1) + (t + r1)TQ(t + r1)

+((t + r1)TQ(t + r1)− yTPT (t + r1))2

(t + r1)T (Ik −Q)(t + r1)

]}−(p+n−k+2)/2

dt,

ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS 5

where r1 = Σ−1/2µ, y = (LΣ−1LT )−1/2y, P = Σ−1/2LT (LΣ−1LT )−1/2, and Q =

PPT .Let

f1(t) =

[(t + r1)T (t + r1)

]−(p−1)/2

[(t + r1)T (Ik −Q)(t + r1)]1/2

and f2(y, t) = 1 +1

(t + r1)T (t + r1)f2(y, t)

with

f2(y, t) = yT y − 2yTPT (t + r1) + (t + r1)TQ(t + r1)

+

[(t + r1)TQ(t + r1)− yTPT (t + r1)

]2(t + r1)T (Ik −Q)(t + r1)

.

Next, we approximate the function f1(t)f−(p+n−k+2)/22 (y, t) using a Taylor series ex-

pansion of the third order at t = 0. Notice, that the integrals of summands in Taylorseries expansion, which correspond to the odd derivatives at point t = 0 are all zero asthe moments of the odd order from the multivariate normal distribution with zero meanvector. As a result, we get the next approximation

f1(t)f−(p+n−k+2)/22 (y, t) = g0(y) +

1

2tT g2(y)t + o(||t||4), (8)

where

g0(y) = f1(0)f−(p+n−k+2)/22 (y,0),

g2(y) =∂2[f1(t)f

−(p+n−k+2)/22 (y, t)

]∂t∂tT

∣∣∣∣∣∣t=0

= f−(p+n−k+2)/22 (y,0)

∂2f1(t)

∂t∂tT

∣∣∣∣t=0

+ 2∂f1(t)

∂t

∣∣∣∣t=0

∂f−(p+n−k+2)/22 (y, t)

∂tT

∣∣∣∣∣t=0

+ f1(0)∂2f

−(p+n−k+2)/22 (y, t)

∂t∂tT

∣∣∣∣∣t=0

.

It is easy to see that∂f1(t)

∂t=ϕ1(t)

ϕ2(t)

with

ϕ1(t) = −(p− 1)[(t + r1)T (t + r1)

]−1(t + r1)

−[(t + r1)T (Ik −Q)(t + r1)]−1(Ik −Q)(t + r1),

ϕ2(t) =[(t + r1)T (t + r1)

](p−1)/2[(t + r1)T (Ik −Q)(t + r1)]1/2.

Then the second order derivative ∂2f1(t)∂t∂tT

can be computed as

∂2f1(t)

∂t∂tT=ϕ2(t)∂ϕ1(t)

∂tT− ϕ1(t)∂ϕ2(t)

∂tT

ϕ22(t)

,

where

∂ϕ1(t)

∂tT=

p− 1

(t + r1)T (t + r1)

[2

(t + r1)(t + r1)T

(t + r1)T (t + r1)− Ik

]+

1

(t + r1)T (Ik −Q)(t + r1)

×[2

(Ik −Q)(t + r1)(t + r1)T (Ik −Q)

(t + r1)T (Ik −Q)(t + r1)− (Ik −Q)

],

∂ϕ2(t)

∂tT=

[(t + r1)T (t + r1)

](p−1)/2 [(t + r1)T (Ik −Q)(t + r1)

]1/2×{

(t + r1)T (Ik −Q)

(t + r1)T (Ik −Q)(t + r1)+

p− 1

(t + r1)T (t + r1)(t + r1)T

}.

6 I. KOTSIUBA AND S. MAZUR

Similarly, the first order partial derivative of f−(p+n−k+2)/22 (y, t) is determined by the

following formula

∂f−(p+n−k+2)/22 (y, t)

∂t= −(p+ n− k + 2)

f−(p+n−k+4)/22 (y, t)

(t + r1)T (t + r1)

×

[− f2(y, t)

(t + r1)T (t + r1)(t + r1) +

1

2

∂f2(y, t)

∂t

], (9)

where

∂f2(y, t)

∂t= −2Py + 2Q(t + r1) + 2

ψ1(y, t)− ψ2(y, t)

ψ3(t),

ψ1(y, t) =[(t + r1)T (Ik −Q)(t + r1)

] [(t + r1)TQ(t + r1)− yTPT (t + r1)

]× [2Q(t + r1)−Py] ,

ψ2(y, t) =[(t + r1)TQ(t + r1)− yTPT (t + r1)

]2(Ik −Q)(t + r1),

ψ3(t) =[(t + r1)T (Ik −Q)(t + r1)

]2.

Using (9) we can calculate the second partial derivative of f−(p+n−k+2)/22 (y, t) which

has the following representation

∂2f−(p+n−k+2)/22 (y, t)

∂t∂tT=

(p+ n− k + 2)

[(t + r1)T (t + r1)]2 f−(p+n−k+6)/22 (y, t) {(p+ n− k + 4)

×

[− f2(y, t)

(t + r1)T (t + r1)(t + r1) +

1

2

∂f2(y, t)

∂t

]

×

[− f2(y, t)

(t + r1)T (t + r1)(t + r1) +

1

2

∂f2(y, t)

∂t

]T

+f2(y, t)

[f2(y, t)

(Ik −

4(t + r1)(t + r1)T

(t + r1)T (t + r1)

)+2(t + r1)

∂f2(y, t)

∂tT− (t + r1)T (t + r1)

2

∂2f2(y, t)

∂t∂tT

]},

ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS 7

where

∂2f2(y, t)

∂t∂tT= 2Q + 2

ψ3(t)(∂ψ1(y,t)∂tT

− ∂ψ2(y,t)∂tT

)− (ψ1(y, t)− ψ2(y, t))∂ψ3(t)

∂tT

ψ23(t)

,

∂ψ1(y, t)

∂tT=

[(t + r1)T (Ik −Q)(t + r1)

] [(t + r1)TQ(t + r1)− yTPT (t + r1)

]×{

2Q + [2Q(t + r1)−Py]

[2(t + r1)T (Ik −Q)

(t + r1)T (Ik −Q)(t + r1)

+2(t + r1)TQ− yTPT

(t + r1)TQ(t + r1)− yTPT (t + r1)

]},

∂ψ2(y, t)

∂tT=

[(t + r1)TQ(t + r1)− yTPT (t + r1)

]2×

[2(Ik −Q)(t + r1)

[2(t + r1)TQ− yTPT

](t + r1)TQ(t + r1)− yTPT (t + r1)

+ (Ik −Q)

],

∂ψ3(t)

∂tT= 4

[(t + r1)T (Ik −Q)(t + r1)

](t + r1)T (Ik −Q).

By taking all partial derivatives at t = 0 and putting them in (8), we obtain the Taylor

series expansion of the function f1(t)f−(p+n−k+2)/22 (y, t). Then, the approximate density

function for zTΣ−1zLA−1z/zTA−1z is given by

fzT Σ−1z LA−1z

zT A−1z

(y) ≈ λ−k/2|LΣ−1LT |−1/2

(2π)k/2

Γ(p+n−k+2

2

)πp/2Γ

(n−k+2

2

)×

∫Rk

[g0(y) +

1

2tT g2(y)t

]exp

(−tT t

2λ

)dt.

Using Harville (1997, p. 321), the last integral leads to

fzT Σ−1z LA−1z

zT A−1z

(y) ≈ |LΣ−1LT |−1/2

πp/2

Γ(p+n−k+2

2

)Γ(n−k+2

2

) [g0(y) +

λ

2tr[g2(y)]

].

Finally, applying the facts that zTΣ−1zLA−1z/zTA−1z and zTA−1z/zTΣ−1z areindependently distributed, and zTΣ−1z/zTA−1z ∼ χ2

n−k+1 we obtain that

fLA−1z(y) =

∫ +∞

0

fzT Σ−1z LA−1z

zT A−1z

(ys)fχ2n−k+1

(s)spds.

Summarizing the above, we get the statement of the theorem. �

3. Numerical study

In this section we compare the area under the approximative density which is de-rived in Theorem 2.2, with the target value of unity. Moreover, the asymptotic and theapproximate densities obtained in Section 2 are compared with the exact one which isderived by Bodnar and Okhrin (2011, Theorem 1). The comparison is done for λ = 1/n,µ = (1, ..., 1)T , and Σ = Ik.

In the first numerical study, we investigate the performance of the approximate densityfunction which is obtained in Theorem 2.2. For this reason the area under the approx-imative density function is compared with one. This comparison allows us to concludehow many information is ignored by deleting the moments of order higher than two whichare positive. If the deviation from 1 is very big, then we should include the moments ofhigher order in the Taylor series expansion.

8 I. KOTSIUBA AND S. MAZUR

HHHHHn

k2 4 6 8 10

30 0.999974 1.000000 1.000000 1.000000 1.000000

60 1.000000 0.999917 0.999943 0.999972 0.999989

90 1.000000 0.999999 0.999993 0.999946 0.999475

120 1.000000 0.999999 0.999902 0.999898 0.999494

Table 1. Area under the approximate density function which is givenin Theorem 2.2 for n ∈ {30, 60, 90, 120}, k ∈ {2, 4, 6, 8, 10}, p = 1,l = (1, 0 . . . , 0)T .

HHHHHn

k11 12 13 14 15

30 0.999766 0.999678 0.999753 0.999958 0.999968

60 0.999662 0.999568 0.999893 0.9997898 0.999734

90 0.999659 0.999779 0.999753 0.999582 0.9987399

120 0.999677 0.999738 0.999431 0.999578 0.998789

Table 2. Area under the approximate density function which is givenin Theorem 2.2 for n ∈ {30, 60, 90, 120}, k ∈ {11, 12, 13, 14, 15}, p = 10,L = (Ip, 0p,k−p).

The results of the first simulation study are presented in Tables 1 and 2 for a severalvalues of k with p = {1, 10}, and L = (Ip, 0p,k−p). From Tables 1 and 2 we observe thatour approximate density works quite good for different values p.

In the second numerical study, we compare the asymptotic and the approximate den-sity functions with the exact one. The results are given for k = {2, 4}, p = 1, andlT = (1, 0, ..., 0). We note that in this case the exact density is given as the two-dimensional integral (see Theorem 1 b) of Bodnar and Okhrin (2011)). Also, it is re-markable that all integrals can be easily evaluated using any mathematical software,e.g., Mathematica. Here, in Figure 1, the Taylor series approximation is shown by theshort-dashed line, the asymptotic density by the long-dashed line, and the exact one bythe solid line. We observe that the approximative density function coincides with theexact one. Furthermore, we observe that they are slightly skewed to the right. Also, wesee that the asymptotic density leads very closed to the exact one and the approximatedensities.

4. Summary

In this paper, the product of a random matrix which includes an inverse Wishartdistributed and a normally distributed vector is studied. We derived its asymptoticdistribution. Moreover, we obtained the formula of an approximative density of linearfunctions of the product which is based on the Gaussian integral and the third orderTaylor expansion. In the numerical study we documented the good performance of theapproximate and asymptotic densities.

ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS 9

0.01 0.02 0.03 0.04x

20

40

60

80

f (x)

Asymptotics

Taylor series

Exact

0.004 0.006 0.008 0.010 0.012 0.014 0.016x

50

100

150

200

250

f (x)

Asymptotics

Taylor series

Exact

a) k = 2, n = 60, l = (1, 0)T b) k = 2, n = 120, l = (1, 0)T

0.01 0.02 0.03 0.04x

10

20

30

40

50

60

70

f (x)

Asymptotics

Taylor series

Exact

0.005 0.010 0.015x

50

100

150

200

f (x)

Asymptotics

Taylor series

Exact

c) k = 4, n = 60, l = (1, 0, 0, 0)T d) k = 4, n = 120, l = (1, 0, 0, 0)T

Figure 1. The exact density and its two approximations as given inSection 2 with k ∈ {2, 4}, n ∈ {60, 120}, and l ∈ {(1, 0)T , (1, 0, 0, 0)T }.

5. Acknowledgements

The authors are thankful to Professor Yuliya Mishura and two anonymous Reviewersfor careful reading of the paper and for their suggestions which have improved an earlierversion of this paper. We also thank Khrystyna Kyrylych for her comments used in thepreparation of the revised version of the paper.

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Department of Probability Theory and Statistics, Ivan Franko National University ofLviv, Universytetska 1, 79000, Lviv, Ukraine

E-mail address: kotsiuba@hotmial.com

Department of Statistics, Lund University, PO Box 743, SE-22007 Lund, Sweden

E-mail address: stepan.mazur@stat.lu.se

Working Papers in Statistics 2015LUND UNIVERSITYSCHOOL OF ECONOMICS AND MANAGEMENTDepartment of StatisticsBox 743220 07 Lund, Sweden

http://journals.lub.lu.se/stat