Upper and lower bounds on the variances of linearcombinations of the kth records

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Upper and lower bounds on the variances of linear combinations of the kth records

Paweł Marcin Kozyra & Tomasz Rychlik

To cite this article: Paweł Marcin Kozyra & Tomasz Rychlik (2017): Upper and lower bounds on the variances of linear combinations of the kth records, Statistics, DOI:

10.1080/02331888.2017.1343828

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https://doi.org/10.1080/02331888.2017.1343828

Upper and lower bounds on the variances of linear combinations of the kth records

Paweł Marcin Kozyra and Tomasz Rychlik

Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

ABSTRACT

We describe a method of determining upper bounds on the variances of linear combinations of thekth records values from i.i.d. sequences, expressed in terms of variances of parent distributions. We also present conditions for which the bounds are sharp, and those for which the respective lower ones are equal to zero. A special attention is paid to the case of thekth record spacings, i.e. the differences of consecutivekth record values.

ARTICLE HISTORY Received 18 November 2016 Accepted 8 May 2017 KEYWORDS kth record value; linear combination;kth record spacing; variance; sharp bound

2010 MATHEMATICS SUBJECT CLASSIFICATION 60E15; 62G32

1. Introduction

Let X1, X2,. . . be i.i.d. random variables with common continuous distribution function F. Assume that Xi:n stands for the ith order statistic obtained from the first n observations. For a given k∈N, the kth record times Tn,k and the kth record values Rn,k were defined by Dziubdziela and Kopociński [1] as follows: T_1,k= 1 and Tn+1,k = min{j > Tn,k: X_j:j+k−1> XTn,k:Tn,k+k−1}, n ∈N, and Rn,k= XT_n,k:T_n,k+k−1. Another way of defining kth record times (see, e.g. [2, p. 82]) ˜T1,k= k and

˜T_n+1,k= min{j > ˜Tn,k: Xj> Xj−k:j−1} is more natural, because it starts counting the records from moment k, when k observations appear, and the first kth record R1,k= X1:kcan be defined. Note that

˜T_n,k= Tn,k+ k − 1 for all positive integer n and k. Another convention in the literature (see, e.g. [3, p. 42]) is to define the initial value of the record sequence X_1:kas R_0,k. Here we use the more popular notation with R_1,k= X1:k. It is worth pointing out that in the case of discontinuous parent distribu- tions when the ties are possible more subtle definitions of kth record values are necessary (see, e.g.

[4]). We do not consider the record occurrence times in the paper, and so their precise definition is immaterial here.

Letc = (c1,. . . , cn) ∈Rⁿbe an arbitrary nonzero vector, and R1,k,. . . , Rn,kdenote the first kth records based on an i.i.d. sequence X1, X2,. . . of continuously distributed random variables. Our purpose is to provide bounds for the ratios of variancesVar(_n

i=1ciRi,k)/Var X1 for all possible non-degenerate baseline distribution functions F for which the above variances are finite. Note that finiteness ofVar X1implies the same forVar Rn,k, n∈N, when k≥ 2. For the classic records with k= 1, conditionVar X1 < ∞ does not suffice forVar Rn,1 < ∞, n = 2, 3, . . . (see, e.g. [5]).

Throughout the paper, writingVar Y we tacitly assume that this is finite.

The first paper devoted to evaluation of variances of records was due to Klimczak and Rych- lik [5]. They determined bounds on variances of single kth record values measured in the population

CONTACT T. Rychlik trychlik@impan.pl

Downloaded by [77.113.19.101] at 03:49 08 August 2017

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variance units, using an idea of Papadatos [6] who studied variances of order statistics. The results of Klimczak and Rychlik [5] were specified by Jasiński [7]. Much more results are known on evaluations of the expectations of record values. Nagaraja [8] applied the Schwarz inequality for determining mean–variance bounds on the means of first records. Using the same idea, Grudzień and Szynal [9]

derived non-sharp bounds for the kth record values with k≥ 2. With use of the greatest convex mino- rant method of Moriguti [10], Raqab [11] derived analogous sharp bounds. Raqab [12] and Raqab and Rychlik [13] calculated bounds on expectations of the first and kth records, respectively, in more general scale units based on the pth absolute central moments with p≥ 1. Danielak [14] provided similar results for record increments. Rychlik [15], Danielak and Raqab [16,17], Raqab and Rychlik [18] and Raqab [19] studied evaluations of differences of consecutive and non-adjacent record values coming from various restricted families of parent distributions. Bounds on the expectations of arbitrary linear combinations of the kth records, expressed in the Gini mean difference units of basic population were presented by Kozyra and Rychlik [20].

The paper is organized as follows. General linear combinations of record values are treated in Section2. Tight upper bounds on variances of the combinations as well as some sufficient assumptions on their attainability are presented. There are also described conditions under which sharp lower variance bounds trivially amount to zero. Special case of record spacings R_m+1,k− Rm,kis thoroughly examined in Section 3 that also contains some numerical results. The proofs of the statements presented in Sections 2 and 3 are postponed to Section 4.

2. Linear combinations of thekth records

For given positive integers n and k, and for a fixed vectorc = (c1,. . . , cn) ∈Rⁿ,n

i=1|ci| > 0, we define function

c,k(u, v) = (1 − v)^k⁻¹ u

⎧⎨

⎩

⎡

⎣ⁿ

j=1

cj− (1 − u)^k

n−1

i=0

⎛

⎝ⁿ

j=i+1

cj

⎞

⎠ [−k ln(1 − u)]ⁱ i!

⎤

⎦

×

n−1

i=0

⎛

⎝ⁿ

j=i+1

cj

⎞

⎠ [−k ln(1 − v)]ⁱ i!

−

1≤i<j≤n

cicj j−i−1

p=0

p q=0

(−1)^q[−k ln(1 − u)]ⁱ^+q[−k ln(1 − v)]^p^−q (i − 1)!q!(p − q)!(p + i)

⎫⎬

⎭ (1)

acting on the triangle 0< u ≤ v < 1. For brevity, parameter n is suppressed in the notation. The diagonal versionc,k(u) of c,k(u, u) for 0 < v = u < 1 has much simpler form

c,k(u) = (1 − u)^k⁻¹ u

⎧⎨

⎩

n−1

i=0

⎛

⎝ⁿ

j=i+1

cj

⎞

⎠

2

[−k ln(1 − u)]ⁱ i!

− (1 − u)^k

⎡

⎣ⁿ⁻¹

i=0

⎛

⎝ⁿ

j=i+1

cj

⎞

⎠ [−k ln(1 − u)]ⁱ i!

⎤

⎦

2⎫

⎬

⎭, (2)

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because due to the identity (p − 1)!(q − 1)!

(p + q − 1)! = 1 0

u^p⁻¹(1 − u)^q⁻¹du

=

q−1 r=0

q− 1 r

(−1)^r 1 0

u^p+r−1du=

q−1

r=0

(q − 1)!(−1)^r r!(q − 1 − r)!(p + r), the last line of Equation (1) for u= v can be rewritten as

n j=2

cj

j−1 i=1

ci j−i−1

p=0

[−k ln(1 − u)]ⁱ^+p (i − 1)!

p q=0

(−1)^q q!(p − q)!(q + i)

=

n j=2

cj

j−1 i=1

ci j−i−1

p=0

[−k ln(1 − u)]ⁱ^+p (i + p)! =

n j=2

cj

j−1 i=1

ci

j−1 p=i

[−k ln(1 − u)]^p p!

=

n j=2

cj j−1

p=1

_p

i=1

ci

[−k ln(1 − u)]^p

p! =

n−1 p=1

⎛

⎝ⁿ

j=p+1

cj

⎞

⎠

_p

i=1

ci

[−k ln(1 − u)]^p

p! .

Theorem 2.1: Suppose that X1, X2,. . . is a sequence of i.i.d. random variables with a common contin- uous distribution function F, say, such thatEX²₁ andER²_n,kare finite for fixed n, k∈N. Then for any non-zeroc ∈Rⁿ, we have

Var(n

i=1ciRi,k)

Var X1 ≤ sup

0<u≤v<1c,k(u, v). (3)

Moreover, if

0<u≤v<1sup c,k(u, v) = sup

0<u<1c,k(u), (4)

then bound (3) is sharp. Precisely, we have the following.

(i) If sup₀_<u<1c,k(u) = c,k(u0) for some 0 < u0 < 1, then the upper bound in Equation (3) is attained in the limit by the sequence of parent distribution functions Fm= u0Fm,a+ (1 − u0)Fm,b, m= 1, 2, . . . , where Fm,adenote the distribution function of the uniform random variable on the interval [a− 1/m, a], and a < b are arbitrary.

(ii) If sup₀_<u<1c,k(u) = limu0c,k(u), then the equality in Equation (3) is attained in the limit by any sequence of distribution functions Fm= umFm,a+ (1 − um)Fm,bas m→ ∞ and um 0, whereas a< b.

(iii) If sup0<u<1c,k(u) = limu1c,k(u), then the upper bound in Equation (3) is attained in the limit by any sequence of distribution functions Fm= umFm,a+ (1 − um)Fm,b as m→ ∞ and um 1, with a < b.

Theorem 2.2: Under assumptions of Theorem 2.1, if either c1 = 0 or k ≥ 2, then the trivial bound Var(_n

i=1ciRi,k) Var X1 ≥ 0

is optimal. If the former(latter) condition holds, then the zero bound is attained for the sequence of baseline distributions described in Theorem 2.1(ii) (Theorem 2.1(iii), respectively).

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In the case of the single nth value of the kth records withc = en= (0, . . . , 0, 1), function (1) simplifies to

en,k(u, v) = 1− (1 − u)^kn−1

i=0 [−k ln(1−u)]ⁱ i!

u (1 − v)^k−1

n−1

i=0

[−k ln(1 − v)]ⁱ

i! ,

and then Equations (3) and (4) hold (cf [5]). When k= 1, then clearly

0<u<1sup en,k(u) = lim

u1en,k(u) = +∞.

Klimczak and Rychlik [5] showed that for any k≥ 2 function en,k is maximized at some inner point of open interval(0, 1). Under the restriction 2 ≤ k ≤ max{2, n((n + 4)/(3n + 4))}, Jasiński [7]

proved that the function is first increasing, and then decreasing, and has a unique local maximum.

Since

u0limen,k(u) = 0, n ≥ 2,

ulim1en,k(u) = 0, k ≥ 2, the trivial bound

Var(_n

i=1ciRi,k) Var X1 ≥ 0 is sharp for all k and n except for n= k = 1.

3. Thekth record spacings

In this section, we thoroughly analyse variances of the kth record spacings. To simplify the notation, we writeem+1−em,kandem+1−em,kasm,kandm,k, respectively. The former has the representation

m,k(u, v) = [−k ln(1 − u)]^m(1 − v)^k−1 um!

1−(1 − u)^k[−k ln(1 − v)]^m m!

, (5)

and the latter satisfiesm,k(u) = m,k(u, u).

Theorem 3.1: Let X1, X2,. . . be i.i.d. with a continuous distribution function, and assume thatEX₁²<

∞ andER²_m+1,k < ∞. Then

Var(Rm+1,k− Rm,k) Var X1 ≤ sup

0<u<1m,k(u), (6)

and the bound is sharp. In particular, the following holds.

(i) If k= 1 and m ≥ 1, then

Var(Rm+1,1− Rm,1) Var X1 ≤ lim

u1m,1(u) = +∞, (7)

and this upper bound is attained by the sequence of baseline distributions described in Theorem 2.1(iii).

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(ii) If m= 1 and k ≥ 2, then

Var(R2,k− R1,k) Var X1 ≤ lim

u01,k(u) = k,

and the bound is attained by the sequence of distributions described in Theorem 2.1(ii).

(iii) If either k= 2 ≤ m or k ≥ 3 with 2 ≤ m ≤ (2/3)k, then Var(Rm+1,k− Rm,k)

Var X1 ≤ m,k(u0),

where u0= u0(m, k) is the unique solution to _m,k (u) = 0, and the equality is attained by the sequence of parent distributions described in Theorem 2.1(i).

(iv) If finally k≥ 3 with m > (2/3)k, then

Var(Rm+1,k− Rm,k)

Var X1 ≤ m,k(u0),

where 0< u0< 1 is the global maximum point of m,kover(0, 1), and attainability conditions are presented in Theorem 2.1(i).

Remark 3.1: Relation (7) is not surprising in view of the fact that one can construct parent distri- butions such thatVar Rm,1< ∞ =Var R_m+1,1 for every m∈N(cf. [5,8]). It is surprising, though, that arbitrary large values of variance ratio are possible for so simple parent distributions with very restricted supports, defined in Theorem 2.1.

Remark 3.2: We can check that the derivative _m,k (u), 0 < u < 1, has for arbitrary k, m ≥ 3 at most five zeros, and, in consequence,m,k(u) itself has three local maxima at most. The proof is presented in Section4. On the other hand, many numerical examples show that for various k, m≥ 3 function

m,k(u) is merely increasing-decreasing in (0, 1), and has a single maximum there. However, we are not able to prove the claim formally. Below we present exemplary graphs ofm,kfor m= 3, . . . , 12 with k= 5,10 and m = 3,6,12,21,33,48,66,87,111,138 with k = 37 (see Figures 1–3). The following table contains the global maxima of these functions accompanied by respective arguments attaining the extremes.

Table1depicts the upper bounds on ratiosVar(Rm+1,k− Rm,k)/Var X1 for k= 5,10 with m = 3,. . . , 12 and k = 37 with m = 3,6,12,21,33,48,66,87,111,138, and arguments u0for which respective

Figure 1.Graphs ofm,k(u) for k = 5 and m = 3, . . . , 12.

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Figure 2.Graphs ofm,k(u) for k = 10 and m = 3, . . . , 12.

Figure 3.Graphs ofm,k(u) for k = 37 and m = 3,6,12,21,33,48,66,87,111,138.

Table 1.Upper bounds on variances ofkth record spacingsVar(Rm+1,k− Rm,k)/Var X1fork = 5,10 with m = 3, . . . , 12 and k = 37 withm = 3,6,12,21,33,48,66,87,111,138.

k 5 10 37

m u0(m, k) m,k(u0) u0(m, k) m,k(u0) m u0(m, k) m,k(u0)

3 0.41750 0.70681 0.18959 0.99836 3 0.048661 2.9832

4 0.57701 0.63879 0.28428 0.73769 6 0.12726 1.1307

5 0.68530 0.64205 0.36766 0.61334 12 0.26465 0.51284

6 0.76273 0.68360 0.44044 0.54564 21 0.42999 0.32282

7 0.81971 0.75473 0.50410 0.50723 33 0.59350 0.26689

8 0.86231 0.85410 0.56000 0.48632 48 0.73326 0.27486

9 0.89444 0.98378 0.60924 0.47707 66 0.83891 0.33927

10 0.91884 1.1481 0.65270 0.47630 87 0.91044 0.48782

11 0.93745 1.3532 0.69114 0.48219 111 0.95415 0.79993

12 0.95170 1.6073 0.72517 0.49367 138 0.97839 1.4726

functionsm,kattain their maxima. Table2contains analogous results for small k= 2,3,4 and m = 2,. . . , 8. The numerical examples show that for fixed k, the maximum arguments u0 increase as m increases. For small k, the bound values increase as well then. For greater k, the bounds first decrease and then increase.

Now we show a direct application of general Theorem 2.1 for establishing possibly non-sharp upper bounds on variances of second spacings of kth records R_m+2,k− Rm,k. In this case, function

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Table 2.Upper bounds on_Var(Rm+1,k− Rm,k)/Var X1fork = 2,3,4 and m = 2, . . . , 8.

k 2 3 4

m u0(m, k) m,k(u0) u0(m, k) m,k(u0) u0(m, k) m,k(u0)

2 0.86934 1.06896 0.44816 0.79186 0.26639 0.88481

3 0.95642 1.72309 0.75606 0.81215 0.55201 0.69942

4 0.98432 3.01160 0.86434 0.99074 0.70792 0.70797

5 0.99419 5.45217 0.92156 1.29382 0.80154 0.78566

6 0.99782 10.0581 0.95377 1.75316 0.86280 0.91637

7 0.99918 18.7747 0.97243 2.42904 0.90421 1.10242

8 0.99969 35.3359 0.98344 3.41419 0.93267 1.35393

Table 3.Upper bounds on_Var(Rm+2,k− Rm,k)/Var X1fork = 2,3 with m = 2, . . . , 8.

k 2 3

m u0(m, 2) v0(m, 2) c,2(u0,v0) u0(m, 3) v0(m, 3) c,3(u0,v0)

2 0.96071 0.97163 0.62903 0.25054 0.25054 0.61965

3 0.98266 0.98803 1.3188 0.46736 0.46736 0.43117

4 0.99259 0.99501 2.6491 0.93446 0.94172 0.56797

5 0.99689 0.99794 5.2213 0.95858 0.96340 0.84949

6 0.99871 0.99916 10.191 0.97399 0.97711 1.2623

7 0.99947 0.99966 19.789 0.98371 0.98572 1.8695

8 0.99979 0.99986 38.321 0.98982 0.99110 2.7636

(1) takes on the following form:

em+2−em,k(u, v) =(1 − v)^k⁻¹ u

−(1 − u)^k

[−k ln(1 − u)]^m⁻²

(m − 2)! +[−k ln(1 − u)]^m⁻¹ (m − 1)!

×

[−k ln(1 − v)]^m−2

(m − 2)! +[−k ln(1 − v)]^m−1 (m − 1)!

+[−k ln(1 − u)]^m−2 (m − 2)!

+ [−k ln(1 − u)]^m−2[−k ln(1 − v)]

(m − 3)!(m − 1) −[−k ln(1 − u)]^m−1 (m − 3)!(m − 1)

.

Respective suprema over the triangle 0< u ≤ v < 1 provide bounds on the ratiosVar(Rm+2,k− Rm,k)/Var X1. In Table3we collected numerical values of the bounds for k= 2,3 and m = 2, . . . , 8.

They are accompanied by the pairs of arguments(u0,v0) = (u0(m, k), v0(m, k)) providing the max- imal values. In most the cases u0< v0which means that the bounds are not optimal. E.g., for k= 2 and m= 4 the function em+2−em,2(u) = em+2−em,2(u, u) attains its maximum approximately equal to 0.58104 at 0.96273, whereasem+2−em,2(u, v) attains its maximal value 0.62903 at the interior triangle point(0.96071, 0.97163). Observe that for k = 2 and m = 2,3, we have u0 = v0, and respective bounds are sharp.

Theorem 3.2: Under assumptions of Theorem 3.1, with the exception of case m = k = 1, bound Var(Rm+1,k− Rm,k)

Var X1 ≥ 0 is the best possible. In particular,

(i) if m≥ 2, and k ≥ 1, then

u0limm,k(u) = 0,

and the zero bound is attained under conditions of Theorem 2.1(ii),

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(ii) if m≥ 1, and k ≥ 2, then

ulim1m,k(u) = 0,

and the zero bound is attained under conditions of Theorem 2.1(iii).

Remark 3.3: Calculating the sharp lower bounds for the case k = m = 1 is an open problem. We have

0<u<1inf 1,1(u) = 1,1(0.37977) = 0.88514,

which is the lowest possible value attained by the sequences of continuous distributions tending weakly to some two-point ones, as in Theorem 2.1(i). However, when we consider the family of Weibull baseline distribution functions F_α(x) = 1 − exp(−x^α), x > 0, with shape parameter α > 0 (the scale parameter does not matter here), then we obtain

Var_α(R2,1− R1,1)

Var_αX1 = V(α)

=(1 + 1/α) + (2 + 2/α) − 2(1 + 1/α)^(2+2/α)_(2+1/α) − [(2 + 1/α)−(1 + 1/α)]²

(1 + 2/α) − ²(1 + 1/α)

≥ V(4.88090) = 0.57492

(cf. [3, p. 55]). This shows in particular that

0<u≤v<1inf 1,1(u, v) < inf0<u<11,1(u).

4. Proofs

Proof of Theorem 2.1: Our first aim is to determine the joint distribution function of two kth record values Rm,k and Rn,k with 1≤ m < n. We start from analysing first records in the standard expo- nential sequence X₁^∗, X₂^∗,. . .. It is well known that the record spacings R^∗1,1, R^∗_2,1− R^∗1,1,. . . are i.i.d.

standard exponential, and so R^∗_m,1 and R^∗_n,1− R^∗m,1, 1≤ m < n, are independent and have Erlang (gamma) distributions with unit scale parameter and shape parameters m and n−m, respectively. It easily follows that R^∗_m,1and R^∗_n,1= R^∗m,1+ R^∗n,1− R^∗m,1have the joint density function

g_m,n,1^∗ (x, y) = x^m−1(y − x)^n−m−1e^−y

(m − 1)!(n − m − 1)! 0< x < y.

(cf [3, p. 11]). When x≥ y, we obtain the marginal distribution function of the latter variable

G^∗_m,n,1(x, y) =P(R^∗m,1≤ x, R^∗n,1≤ y) =P(R^∗n,1≤ y)

= G^∗n,1(y) = 1 − e^−y

n−1

i=0

yⁱ

i!, y> 0.

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Otherwise

G^∗_m,n,1(x, y) =

_x

0

s^m−1 (m − 1)!

_y

s

(t − s)^n−m−1

(n − m − 1)!e^−tdt ds

= x 0

s^m⁻¹ (m − 1)!

y−s 0

vⁿ^−m−1

(n − m − 1)!e^−v−sdv ds

=

_x

0

s^m−1 (m − 1)!e^−s

1− e^s−y

n−m−1

i=0

(y − s)ⁱ i!

ds

= 1 − e^−x

m−1

i=0

xⁱ i! − e^−y

n−m−1

i=0

_x

0

s^m−1(y − s)ⁱ (m − 1)!i! ds

= G^∗m,1(x) − e^−y

n−m−1

i=0

i j=0

(−1)^jx^m^+jyⁱ^−j (m − 1)!j!(i − j)!(m + j).

An important classic result of the record theory asserts that the sequence of the kth records based on an i.i.d. sequence with a continuous baseline distribution function F has the distribution identical with that of an i.i.d. sequence of first records based on distribution function 1− (1 − F)^k(see, e.g.

[2, Theorem 22.6]). When F is standard exponential in particular, the kth records R^∗_1,k, R^∗_2,k,. . . have the distribution of the first records from the sequence X₁^∗/k, X₂^∗/k, . . ., i.e. identical with those of R^∗_1,1/k, R^∗2,1/k, . . .. Accordingly, their finite-dimensional versions are simple scale transformations of their counterparts for the first records, and so

G^∗_n,k(x) = G^∗n,1(kx), G^∗_m,n,k(x, y) = G^∗_m,n,1(kx, ky).

It is obvious that for any strictly increasing function ξ :R+→R, random variables ξ(X^∗1), ξ(X2^∗), . . . are i.i.d. with common distribution function Fξ(x) = 1 − exp(−ξ⁻¹(x)), whereas ξ(R^∗_1,k), ξ(R^∗_2,k), . . . form the respective sequence of the kth records. In particular, 1 − e^−X¹^∗, 1− e^−X²^∗,. . . are standard uniform, and one- and two-dimensional distribution functions of respective records are

Gn,k(x) =P(1 − e^−R^∗^n,k ≤ x) = G^∗_n,k(− ln(1 − x)) = G^∗n,1(−k ln(1 − x))

= 1 − (1 − x)^k

n−1

i=0

[−k ln(1 − x)]ⁱ

i! , (8)

Gm,n,k(x, y) = G^∗m,n,1(−k ln(1 − x), −k ln(1 − y))

=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1− (1 − x)^k

m−1

i=0

[−k ln(1 − x)]ⁱ

i! − (1 − y)^k

×

n−m−1

i=0

i j=0

(−1)^j[−k ln(1 − x)]^m^+j[−k ln(1 − y)]ⁱ^−j

(m − 1)!j!(i − j)!(m + j) , x< y,

1− (1 − y)^k

n−1

i=0

[−k ln(1 − y)]ⁱ

i! , x≥ y,

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when x and y are in(0, 1). If ξ(x) = F⁻¹(1 − e^−x) for some (not necessarily absolutely) continuous distribution F, then Xi= F⁻¹(1 − e^−Xⁱ^∗), i = 1, 2, . . ., are i.i.d. F-distributed, and the respective kth

(11)

records have the following one- and two-dimensional marginals Fn,k(x) = Gn,k(F(x)), Fm,n,k(x, y) = Gm,n,k(F(x), F(y)), respectively.

In the sequel, we use the Hoeffding [21] formula for the covariance Cov(X, Y) =

R²[H(x, y) − F(x)G(y)] dx dy (10) of random variables X and Y with joint distribution function H and marginals F and G, respec- tively (for a simple proof, see [22]). Note that either of conditions F(x) = 0 and G(y) = 0 implies H(x, y) = 0. Similarly, from F(x) = 1 and G(y) = 1 follows that H(x, y) = G(y) and H(x, y) = F(x), respectively. Therefore, we can rewrite Equation(10) as

Cov(X, Y) =

0<F(x),G(y)<1[H(x, y) − F(x)G(y)] dx dy. (11) Using Equation (10), we also obtain

Var X=Cov(X, X) =

R²[F(min{x, y}) − F(x)F(y)] dx dy

= 2

0<F(x)≤F(y)<1F(x)[1 − F(y)] dx dy. (12)

Noting that the supports of record values are contained in the supports of the original variables, and using Equations (8), (9), (11) and (12), we conclude

Var

_n

i=1

ciRi,k

=

n i=1

c²_iVar(Ri,k) + 2

1≤i<j≤n

cicjCov(Ri,k, Rj,k)

= 2

n i=1

c²_i

0<F(x)≤F(y)<1[Gi,k(F(x)) − Gi,k(F(x))Gi,k(F(y))] dx dy

+ 2

1≤i<j≤n

cicj

0<F(x),F(y)<1[Gi,j,k(F(x), F(y)) − Gi,k(F(x))Gj,k(F(y))] dx dy

= 2

n i=1

c²_i

0<F(x)≤F(y)<1[Gi,k(F(x)) − Gi,k(F(x))Gi,k(F(y)] dx dy

+ 2

1≤i<j≤n

cicj

0<F(x)≤F(y)<1

⎧⎨

⎩Gi,k(F(x)) − [1 − F(y])^k

j−i−1

p=0

p q=0

(−1)^q

× [−k ln(1 − F(x))]^i+q[−k ln(1 − F(y))]^p−q

(i − 1)!q!(p − q)!(i + q) − Gi,k(F(x))Gj,k(F(y))

dx dy

+ 2

1≤i<j≤n

cicj

0<F(x)≤F(y)<1[Gj,k(F(x)) − Gj,k(F(x))Gi,k(F(y))] dx dy

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= 2

n i=1

c²_i

0<F(x)≤F(y)<1Gi,k(F(x)) dx dy

+ 2

1≤i=j≤n

cicj

0<F(x)≤F(y)<1Gi,k(F(x)) dx dy

− 2

1≤i<j≤n

cicj

0<F(x)≤F(y)<1[1− F(y])^k

×

j−i−1

p=0

p q=0

(−1)^q[−k ln(1 − F(x))]^i+q[−k ln(1 − F(y))]^p−q (i − 1)!q!(p − q)!(i + q)

− 2

n i=1

c²_i

0<F(x)≤F(y)<1Gi,k(F(x))Gi,k(F(y)) dx dy

− 2

1≤i=j≤n

cicj

0<F(x)≤F(y)<1Gi,k(F(x))Gj,k(F(y)) dx dy

= 2

0<F(x)≤F(y)<1

⎧⎨

⎩

_n

i=1

ciGi,k(F(x))

⎡

⎣ⁿ

j=1

cj[1− Gj,k(F(y))]

⎤

⎦ −

1≤i<j≤n

cicj

× [1 − F(y])^k

j−i−1

p=0

p q=0

(−1)^q[−k ln(1 − F(x))]ⁱ^+q[−k ln(1 − F(y))]^p^−q (i − 1)!q!(p − q)!(i + q)

⎫⎬

⎭dx dy.

Since

n i=1

ci[1− Gi,k(F(x))] = [1 − F(x)]^k

n−1

i=0

⎛

⎝ⁿ

j=i+1

cj

⎞

⎠ [−k ln(1 − F(x)]ⁱ i!

we have

Var

_n

i=1

ciRi,k

= 2

0<F(x)≤F(y)<1c,k(F(x), F(y))F(x)[1 − F(y)] dx dy

≤ sup

0<u≤v<1c,k(u, v)Var X1. (13)

Suppose now that Equation (4) holds.

(i) Assume first that the supremum ofc,k(u) is attained at some 0 < u0< 1. If u0is the unique value of F(x), different from 0 and 1, then equality holds in Equation (13). The condition is satisfied by the distribution functions

F(x) =

⎧⎪

⎨

⎪⎩

0, x< a, u0, a≤ x < b, 1, x> b

(13)

for arbitrary a< b. Mixtures of uniform distribution functions Fm= u0Fm,a+ (1 − u0)Fm,btend to F(x) for all x ∈R, and functionc,k(u, v)u(1 − v) is continuous. Therefore, as m → ∞, we get

Varm(n

i=1ciRi,k) VarmX1 = 2

a−1/m<x<y<bc,k(Fm(x), Fm(y))Fm(x)(1 − Fm(y)) dx dy 2

a−1/m<x<y<bFm(x)(1 − Fm(y)) dx dy

→ 2

a<x<y<bc,k(F(x), F(y))F(x)(1 − F(y)) dx dy 2

a<x<y<bF(x)(1 − F(y)) dx dy

= c,k(u0) = sup

0<u<1c,k(u).

(ii) If sup0<u<1c,k(u) = limu0c,k(u), then by the previous statement and continuity of c,k, for the sequence of mixtures defined in Theorem 2.1(ii) yields

Varm

_n

i=1ciRi,k

VarmX1 → lim_u0c,k(u).

The proof of statement (iii) is similar.

Proof of Theorem 2.2: We first calculate the right limit of Equation (2) at 0. Denote the respec- tive expression in the curly brackets byχc,k(u). Since factor (1 − u)^k−1 is immaterial here, and lim_u0χc,k(u) = 0, with use the l’Hospital rule, we obtain

u0limc,k(u) = lim_u0χ_c,k (u) = k

⎛

⎝ⁿ

j=2

cj

⎞

⎠

2

+ k

⎛

⎝ⁿ

j=1

cj

⎞

⎠

2

− 2k

⎛

⎝ⁿ

j=1

cj

⎞

⎠

⎛

⎝ⁿ

j=2

cj

⎞

⎠

= k

⎛

⎝ⁿ

j=1

cj−

n j=2

cj

⎞

⎠

2

= kc²₁.

Hence c1 = 0 implies that limu0c,k(u) = 0, and using the sequence of baseline distributions of Theorem 2.1(ii), we attain zero for the variance ratios in the limit. If k≥ 2, then limu1c,k(u) = 0, because each expression(1 − u)^p[−k ln(1 − u)]^qfor p≥ 1 and q ≥ 0 tends to 0 as u approaches 1.

If k= 1, then c,ktends to+∞ at 1, because it behaves asymptotically as a combination of functions [−k ln(1 − u)]ⁱ, i= 0, . . . , n − 1, and the coefficient c²n/(n − 1)! of the fastest increasing term [−k ln(1 − u)]ⁿ⁻¹ is clearly positive. In conclusion, applying construction of Theorem 2.1(iii) for

k≥ 2, we obtain zero limit.

In the proof of Theorem 3.1, we use three lemmas.

Lemma 4.1: Let

fn,M(x) = e^x− Mxⁿ n!

be defined on the non-negative half-axis for fixed real M and non-negative integer n. Then f0,M(0) = 1 − M and fn,M(0) = 1 for n ∈N. Also, lim_x∞fn,M(x) = +∞ for every n ∈N0 and M∈R. Moreover, we have the following.

(i) Function f0,Mis strictly increasing everywhere.

(ii) If M≤ 1, then f1,M also increases, and otherwise there exists x0> 0 such that f1,M decreases on (0, x0) and increases on (x0,∞).

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(iii) Let n≥ 2. If either M ≤ 1 or M > 1 and minx>0f_n−1,M(x) ≥ 0, then fn,Mis increasing. If M> 1 and min_x>0f_n−1,M(x) < 0, then there exist 0 < x1< x2< ∞ such that fn,Mincreases on(0, x1), decreases on(x1, x2), and finally increases on (x2,∞).

Proof: Calculating the left- and right-end values is immediate. Also, claim (i) is trivial since f_0,M (x) = e^x> 0, x > 0. Note that f_n+1,M = fn,Mfor all n∈N.

(ii) We have f_1,M (x) = e^x− M. If M ≤ 1, then f1,Mis strictly increasing and positive. Otherwise, it is decreasing on(0, x0) with x0= x0(1) = ln M, and increasing elsewhere.

(iii) Suppose first that n= 2. If M ≤ 1, then f2,M = f1,M> 0, and, due to (ii), f2,M is increasing from 1 to∞. If M > 1 and minx>0f1,M(x) = f1,M(ln M) ≥ 0, then by the latter statement of (ii), f2,M is increasing as well. For M> 1 with f1,M(ln M) < 0, by the same argument, there exist 0< x1= x1(2) < x0(1) < x2= x2(2) such that f2,M (xi) = f1,M(xi) = 0, i = 1,2, and, consequently, f2,Mis increasing on(0, x1) ∪ (x2,∞), and decreasing on (x1, x2). It means that conditions of (iii) are satisfied for n= 2.

Assume now that (iii) holds for some n≥ 2. We conclude the same for n+1. If M ≤ 1, then f_n+1,M (x) = fn,M(x) increases onR+from 1 to∞, and so does fn+1,M. If M> 1 and minx>0fn,M(x) = fn,M(x0(n)) ≥ 0, then f_n+1,M (x) is positive (except for possibly at x0(n) when fn,M(x0(n)) = 0), and so f_n+1,M is increasing function. If finally M> 1 and fn,M(x0(n)) < 0, then there are 0 < x1= x1(n + 1) < x0(n) < x2 = x2(n + 1) < ∞ such that fn+1,mis increasing, decreasing, and increasing

in(0, x1), (x1, x2), and (x2,∞), respectively.

Lemma 4.2: Let f and g be polynomials, r(x) = g(x)/f (x), n ∈N, and

h(x) = f (x)e^x− g(x)xⁿ

n!, x≥ 0.

(i) If lim_x→∞f(x) = ∞, xf = max{x ∈R: f(x) = 0} ≥ n, g(xf) > 0, and r(x) < 0 for x > xf, then there exists xh> xf such that h is negative on(xf, xh), and positive on (xh,∞).

Moreover, function h(x) is increasing on (xh,∞) under additional assumption that f(x) > 0 for x> xh.

(ii) If f(0) > 0, 0 < xf = min{x ∈R: f(x) = 0} ≤ n, g(xf) > 0, and r(x) > 0 for 0 < x < xf, then there exists 0< xh< xf such that h is positive on(0, xh), and negative on (xh, xf).

Proof: (i) Note that h(xf) = −g(xf)(xⁿ_f/n!) < 0 and limx→∞h(x) = ∞. Hence the function has some zeros in(xf,∞). Let xhdenote the smallest of them. Obviously, h is negative on(xf, xh).

Define nowϕ(x) = h(x)/f (x) = e^x− r(x)(xⁿ/n!), x > xf, andψ(x) = e^x− M(xⁿ/n!) with M = r(xh). Note that f (xh) > 0, and ψ(xh) = ϕ(xh) = h(xh)/f (xh) = 0. By assumption, r(x) < r(xh) = M, x> xh> xf, and so

ϕ(x) = e^x− r(x)xⁿ

n! > e^x− Mxⁿ

n! = ψ(x), x > xh. Observe further that

ψ(xh) = e^x^h− M xⁿ_h⁻¹

(n − 1)! > 0 = ψ(xh) = e^x^h− Mx_hⁿ n!

iff M< M(xh/n). If M ≤ 1, then by Lemma 4.1(ii) and (iii) ψ increases from 1 at 0 to ∞ at ∞, which contradictsψ(xh) = 0. Therefore condition ψ(xh) > 0 is equivalent with xh> n which is true due to inequalities n≤ xf < xh. By Lemma 4.1 again, relations 0= ψ(xh) < ψ(xh) imply that ψ(x) > 0 for all x> xh, and so isϕ(x) = h(x)/f (x). By positivity of f on (xf,∞), h(x) > 0 for x > xhas well.