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Volume 20, Issue 2, June 2021, Pages 289 - 303

On Upper k-Record Values from the Generalized Linear Exponential Distribution

Authors
M. Alam, M. A. Khan*, R. U. Khan
Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, 202 002, India
*Corresponding author. Email: khanazam2808@gmail.com
Corresponding Author
M. A. Khan
Received 12 June 2020, Accepted 18 January 2021, Available Online 1 March 2021.
DOI
10.2991/jsta.d.210216.001How to use a DOI?
Keywords
Record values; Single moments; Product moments; Recurrence relations; Characterization
Abstract

In this paper, we derive the exact expressions as well as recurrence relations for single and product moment of generalized upper record values from the four-parameter generalized linear exponential distribution. Further, we characterize the given distribution through conditional expectation and recurrence relations.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The exponential, Rayleigh, Weibull, and the linear exponential distribution (LED) are widely used to model lifetime data. These distribution have several desirable properties which are frequently used in analyzing lifetime data. In particular, the generalized linear exponential distribution (GLED) having linearly increasing failure rates is very useful for modeling lifetime data. The exponential and Rayleigh distributions are special cases of LED. Since the LED does not provide a reasonable parametric fit for bathtub shape (decreasing, nonlinear increasing, nonmonotone failure rates). Therefore, a lot of work is devoted to GLED and propose some new distribution which are more flexible. So, in this continuation Mahmoud and Alam [1] introduced the four-parameter GLED, which defines the different distributional properties of GLED. Lee and Tsai [2] modify the moment of the GLED in a different way, which is more robust. The GLED shows the increasing, decreasing, and bathtub-shaped hazard rate functions which are quite useful in reliability and biological studies.

A random variable X is said to have a GLED with four parameters θ ,α>0 and β ,λ≥0, if its probability density function (pdf) is of the form

f(x)=α(β+θx)(θ2x2+βx−λ)α−1exp{−(θ2x2+βx−λ)α}I(Ψ,∞)(x), x>0,(1.1)
where
I(Ψ,∞)(x)={1x>Ψ0otherwise 
is the indicator function, and
Ψ=−β+β2+2θλθ
and the distribution function (df) is
F(x)=[1−exp{−(θ2x2+βx−λ)α}]I(Ψ,∞)(x), x>0.(1.2)

It is easy to see that

(θ2x2+βx−λ)f(x)=α(β+θx)[−lnF¯(x)][F¯(x)],(1.3)
where
F¯(x)=1−F(x).

The concept of record values was introduced by Chandler [3]. An observation is called a record if its value is greater (or less) than all the previous observations. Record values are used in a wide variety of practical situations, such as industrial stress testing, meteorological analysis, hydrology, seismology, oil, mining surveys, sports, and athletic events. For a survey on important results in this area one may refer to Ahsanullah [4], Kamps [5], Arnold et al. [6] and Ahsanullah and Nevzorov [7]. Dziubdziela and Kopcoiński [8] have generalized the concept of record values of Chandler [3] by random variables of a more generalized nature and called them the k-th record values. Later, Minimol and Thomas ([9], p. 487) called the record values defined by Dziubdziela and Kopcoiński [8] also as the generalized record values, since the r-th member of the sequence of the classical record values is also known as the r-th record value. By setting k=1, we obtain ordinary record statistics.

Let {Xn, n≥1} be a sequence of independent and identical distributed (iid) continuous random variables with df F(x) and pdf f(x). Then for a fixed positive integer k≥1, the sequence of k-th upper record times {Un(k),n≥1} is defined as Nevzorov [10]:

U1(k)=k
and for n≥1,
Un+1(k)=min{j:j>Un(k),Xj>XUn(k)−k+1:Un(k)}.

The sequence {Yn(k),n≥1}, where Yn(k)=XUn(k):Un(k)+k−1 is called the sequence of generalized upper record values (k-th upper record values) of {Xn,n≥1}. Note that for k = 1, we have Yn(1)=XUn, n≥1, which are the record values of {Xn,n≥1} as defined in Ahsanullah [4].

The pdf of Yn(k) and the joint pdf of Ym(k) and Yn(k) are given by (Dziubdziela and Kopcoiński [8], Grudzień [11])

fYn(k)(x)=kn(n−1)![−lnF¯(x)]n−1[F¯(x)]k−1f(x), n≥1,(1.4)
fYm(k),Yn(k)(x,y)=kn(m−1)!(n−m−1)![−lnF¯(x)]m−1f(x)F¯(x)×  [lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y), x<y,1≤m<n,(1.5)
and the conditional pdf of Yn(k) given Ym(k)=x, is
fYn(k)|Ym(k)(y|x)=kn−m(n−m−1)![lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)F¯(x)]k−1f(y)F¯(x),  x<y.(1.6)

For some recent developments on generalized record values with special reference to those arising from exponential, Gumbel, Pareto, generalized Pareto, Burr, Weibull, Gompertz, Makeham, modified-Weibull, exponential-Weibull, additive Weibull, and Kumaraswamy log-logistic distributions, see Grudzień and Szynal [12,13], Pawlas and Szynal [14–15,16], Minimol and Thomas [9,17], Khan and Khan [18], Khan et al. [19,20], and Singh et al. [21], respectively. In this paper, we mainly focus on the study of generalized upper record values arising from the GLED and discussed exact explicit expressions as well as several recurrence relations satisfied by single and product moments. In addition, conditional expectations and recurrence relations of generalized record values are used to characterize this distribution.

2. RELATIONS FOR SINGLE MOMENTS

In this section, we derive the exact expressions for single moments of generalized upper record values and recurrence relations in the following theorems:

Theorem 2.1.

For the distribution given in (1.1). Fix a positive integer k≥1, for n≥1, (β2+2θλ)>0 and j=0,1,…

E(Yn(k))j=1(n−1)!∑u=0j∑v=0∞(ju)(j−u2v)(−β)u 2v(β2+2θ λ)j−u2−vθj−vkvαγ{vα+n,k(β2+2θλ2θ)α}+∑u=0j∑v=0∞(ju)(j−u2v)(−β)u 2j−u2−v(β2+2θλ)vθj+u2+vkj−u2α−vαΓ{j−u2α−vα+n,k(β2+2θλ2θ)α},(2.1)
where Γ(s,t)=∫t∞xs−1 e−x dx and γ(s,t)=∫0txs−1 e−x dx are the upper and lower incomplete gamma functions, respectively.

Proof.

In the view of (1.4), we have

E(Yn(k))j=kn(n−1)!∫−∞∞xj−lnF¯(x)n−1F¯(x)k−1f(x)dx,  n≥1 .

Now using (1.3), we get

E(Yn(k))j=αkn(n−1)!∫∞Ψxj(β+θx)(θ2x2+βx−λ)−lnF¯(x)nF¯(x)kdx=αkn(n−1)!∫∞Ψxj(β+θx)(θ2x2+βx−λ)[(θ2x2+βx−λ)α]n[e−(θ2x2+βx−λ)α]kdx,(2.2)
after substituting t=(θ2x2+βx−λ)α in (2.2), we get
E(Yn(k))j=kn(n−1)!∫0∞[−β+2θt1/α+(β2+2θλ)θ]jtn−1e−ktdt.(2.3)

On using binomial expansion in (2.3) and we have

E(Yn(k))j=kn(n−1)!∑u=0j(ju)(−β)uθj∫∞02θt1/α+(β2+2θλ)j−u2tn−1e−ktdt.=kn(n−1)!∑u=0j(ju)(−β)uθj∫∞0[1+β2+2θλ2θt1/α]j−u2(2θt1/α)j−u2tn−1e−ktdt.=kn(n−1)!∑u=0j(ju)(−β)uθj∫∞0[1+1ψt1/α]j−u2(2θt1/α)j−u2tn−1e−ktdt.(2.4)
where ψ=2θβ2+2θλ, by the expression |β2+2θλ2θt1/α|<1 only if (β2+2θλ2θ)α<  t < ∞. We can verify that |ψ t1/α|<1 if 0<t<(1ψ)α and |1ψt1/α|<1 if (1ψ)α<t<∞, thus the integral in (2.4) should be separated into two parts as:
EYn(k)j=kn(n−1)!∑u=0j(ju)(−β)uθj[∫​​0(1ψ)α(1+1ψt1/α)j−u2(2θt1/α)j−u2× tn−1e−ktdt+∫​​(1ψ)α∞[1+1ψt1/α]j−u2(2θt1/α)j−u2tn−1e−ktdt=kn(n−1)! ∑u=0j(ju)(−β)uθj(β2+2θλ)j−u2[∫​​0(1ψ)α1+ψt1/αj−u2× tn−1e−ktdt+∫​​(1ψ)α∞[1+1ψt1/α]j−u2(ψt1/α)j−u2tn−1e−ktdt.(2.5)

On using the binomial expansion in (2.5), we get:

E(Yn(k))j=kn(n−1)!∑u=0j(ju)(−β)uθj(β2+2θλ)j−u2[∑v=0∞(j−u2v)ψv×∫0(1ψ)αtvα+n−1e−ktdt+∑v=0∞(j−u2v)ψj−u2−v∫∞(1ψ)αtj−u2α−vα+n−1e−ktdt.=kn(n−1)!∑u=0j(ju)(−β)uθj(β2+2θλ)j−u2[∑v=0∞(j−u2v)(2θβ2+2θλ)v×∫0(β2+2θλ2θ)αtvα+n−1e−ktdt+∑v=0∞(j−u2v)(2θβ2+2θλ)j−u2−v∫∞(β2+2θλ2θ)αtj−u2α−vα+n−1e−ktdt,(2.6)
using the incomplete gamma function in (2.6), we get the result as given in (2.1).

Corollary 2.1.

The j-th moment of the k-th record values from GLED has another form:

E(Yn(k))j=1(n−1)![(β2+2θ λ)j2θjγ{n,k(β2+2θλ2θ)α}+2j2θj2kj2αΓ{j2α+n,k(β2+2θλ2θ)α}+∑u=1j∑v=1∞(ju)(j−u2v)(−β)u2v(β2+2θλ)j−u2−vθj−vkvαγ{vα+n,k(β2+2θλ2θ)α}+∑u=1j∑v=1∞(ju)(j−u2v)(−β)u2j−u2−v(β2+2θλ)vθj+u2+vkj−u2α−vαΓ{j−u2α−vα+n,k(β2+2θλ2θ)α}.(2.7)

Proof.

On expanding (2.1), at u=0 and v=0. Hence, the result.

Remark 2.1.

  1. Setting j = 0 in (2.1), then

    RHS=1(n−1)![γ{n,k(β2+2θλ2θ)α}+Γ{n,k(β2+2θλ2θ)α}],
    by using the formula [Γ(s,t)+γ(s,t)]=Γ(s)=∫0∞ts−1 e−t dt, we get
    RHS=1(n−1)!Γ(n)=1=LHS as Γ(n)=(n−1)!

  2. Setting k = 1 and n = 1 in (2.7), we get the exact expression for the j-th ordinary moment of GLED as obtained by Lee and Tsai [2].

  3. Setting β=λ=0 in (2.7), we get the exact expression for single moments of generalized upper records from Weibull distribution

    E(Yn(k))j=1(n−1)![2j2θj2kj2αΓ{j2α+n, 0}]=(2k1/αθ)jΓ(j2α+n)(n−1)!,(2.8)
    as obtained by Kamps [5] and at k = 1, we get the result as given in Nagaraja [22].

  4. Setting α=12 in (2.8), we get the exact expression for single moments of generalized upper records from the exponential distribution

    E(Yn(k))j=1(k θ2)jΓ(j+n)(n−1)!,
    as obtained by Kamps [5] and at k = 1 in Ahsanullah [23].

  5. Setting α=1 in (2.8), we get the exact expression for single moments of generalized upper records from Rayleigh distribution

    E(Yn(k))j=(2k θ)j Γ(j2+n)(n−1)!,
    at k = 1 we get the similar result as given in Ahsanullah and Shakil [24].

  6. Setting λ=0 and α=1 in (2.1), we get the exact expression for single moments of generalized upper records from the LED

    E(Yn(k))j=1(n−1)![∑u=0j∑v=0∞(ju)(j−u2v)(−1)u2vβj−2vθj−vkvγ{v+n,k(β22θ)}+∑u=0j∑v=0∞(ju)(j−u2v)(−1)u2j−u2−vβu+2vθj+u2+vkj−u2−vΓ{j−u2−v+n,k(β22θ)}.

  7. Setting β=0 in (2.7), we get the exact expression for single moments of generalized upper records from the generalized Rayleigh distribution

    E(Yn(k))j=1(n−1)![(2θλ)j2θjγ{n,kλα}+2j2θj2 kj2αΓ{j2α+n,kλα}]      =(2θ)j 1(n−1)![λj2 γ{n, kλα}+1kj2αΓ{j2α+n, k λα}].

Theorem 2.2.

For the distribution given in (1.1), fix a positive integer k≥1, for n≥1 and j=0,1,…

E(Yn(k))j=nαθ(j+2)λ[(1+j+22nα)E(Yn(k))j+2−E(Yn+1(k))j+2]+nαβ(j+1)λ[(1+j+1nα)E(Yn(k))j+1−E(Yn+1(k))j+1].(2.9)

Proof

From (1.4) for n≥1 and j=0,1,…, we have

θ2[E(Yn(k))j+2]+β[E(Yn(k))j+1]−λ[E(Yn(k))j]=kn(n−1)!×∫Ψ∞(θ2x2+βx−λ)xj[−lnF¯(x)]n−1(F¯(x))k−1f(x)dx,(2.10)

on using (1.3) in (2.10), we get

θ2[E(Yn(k))j+2]+β[E(Yn(k))j+1]−λ[E(Yn(k))j]
=kn(n−1)!∫Ψ∞α(β+θx)xj−lnF¯(x)nF¯(x)kdx=αβkn(n−1)!∫Ψ∞xj−lnF¯(x)nF¯(x)kdx+αθkn(n−1)!∫Ψ∞xj+1−lnF¯(x)nF¯(x)kdx.(2.11)

Now applying Lemma 2.1 stated in Khan et al. [19] to (2.11), we have

θ2[E(Yn(k))j+2]+β[E(Yn(k))j+1]−λ[E(Yn(k))j]
=nαθj+2[E(Yn+1(k))j+2−E(Yn(k))j+2]+nαβj+1[E(Yn+1(k))j+1−E(Yn(k))j+1],(2.12)
after arranging (2.12) we get the yields given in (2.9).

Remark 2.2.

If we put

  1. β=λ=0 in (2.11) we get the recurrence relations for single moment of generalized upper records from Weibull distribution

    E(Yn+1(k))j+2=(1+j+22nα)E(Yn(k))j+2,
    at j′=j+2, we get similar result as obtained by Khan and Khan [18].

  2. β=λ=0 and α=12 in (2.11) we get the recurrence relations for single moment of generalized upper records from the exponential distribution

    E(Yn+1(k))j+2=(1+j+2n)E(Yn(k))j+2,
    at j′=j+2, we get similar result as obtained by Khan and Khan [18].

  3. β=λ=0 and α=1 in (2.11) we get the recurrence relations for single moment of generalized upper records from Rayleigh distribution

    E(Yn+1(k))j+2=(1+j+22n)E(Yn(k))j+2,
    at j′=j+2, we get similar result as obtained by Khan and Khan [18].

  4. λ=0 and α=1 in (2.11) we get the recurrence relations for single moment of generalized upper records from the LED

    EYn+1(k)j+2=(1+j+22n)E(Yn(k))j+2+(j+2)β(j+1)θ[(1+j+1n)E(Yn(k))j+1−E(Yn+1(k))j+1].

  5. β=0 in (2.8) we get the recurrence relations for single moment of generalized upper records from the generalized Rayleigh distribution

    E(Yn(k))j=nαθ(j+2)λ[(1+j+22nα)E(Yn(k))j+2−E(Yn+1(k))j+2].

Corollary 2.2.

The recurrence relation for single moments of the upper record values from GLED has the form

λE(XUn)j=nαθj+2[(1+j+22nα)E(XUn)j+2−E(XUn+1)j+2]+nαβj+1[(1+j+1nα)E(XUn)j+1−E(XUn+1)j+1].(2.13)

Remark 2.3.

If we put

  1. β=λ=0 in (2.12), we get the recurrence relations for single moment of upper records from Weibull distribution, which is similar to the result as obtained by Balakrishnan and Chan [25] for j′=j+2.

  2. β=λ=0 and α=12 and (2.12) in, the results for the single moments of upper records obtained by Balakrishnan and Chan [25] at j′=j+2 for the exponential distribution is deduced.

  3. β=λ=0 and α=1 in (2.12), the results for the single moments of upper records obtained by Balakrishnan and Chan [25] at j′=j+2 for Rayleigh distribution is deduced.

  4. λ=0 and α=1 in (2.12), we get the recurrence relations for single moments of upper records from the LED

    EXUn+1j+2=(1+j+22n)E(XUn) j+2+(j+2)β(j+∣1)θ[(1+j+1n)E(XUn) j+1−E(XUn+1)j+1].

  5. Setting β=0 in (2.12), we get the recurrence relations for single moments of upper records from the generalized Rayleigh distribution

    E(XUn)j=nαθ(j+2)λ[(1+j+22nα)E(XUn)j+2−E(XUn+1)j+2].

    Numerical computations for moments of k-th upper record values from the GLED for arbitrary chosen values of α, β, θ. Setting λ=0 without loss of generality and various sample size n=1,2,…10 are given in Tables 1 and 2.

E(X) E(X)2
α = 0.5, θ = 1, β = 2 α = 0.5, θ = 1, β = 2
n k = 1 k = 2 k = 3 k = 1 k = 2 k = 3
1 0.736241 0.2210499 0.1040861 2.110072 0.2316011 0.0561998
2 1.947826 0.6230462 0.3010611 8.417389 1.015631 0.2581783
3 3.449258 1.16617 0.5780411 20.40594 2.670637 0.7090046
4 5.12462 1.816237 0.9222053 39.00304 5.470104 1.511246
5 6.905298 2.546131 1.321817 64.75762 9.630955 2.758801
6 8.751285 3.335303 1.766684 97.98972 15.31757 4.533193
7 10.63908 4.168577 2.248265 138.8874 22.65138 6.90277
8 12.55456 5.03488 2.759559 187.5635 31.72096 9.923529
9 14.48904 5.926162 3.294916 244.0877 42.59071 13.64067
10 16.43696 6.836557 3.849815 308.5043 55.30755 18.09037
Table 1
Moments of the generalized linear exponential distribution(GLED).
E(X)3 E(X)4
α = 0.5, θ = 1, β = 2 α = 0.5, θ = 1, β = 2
n k = 1 k = 2 k = 3 k = 1 k = 2 k = 3
1 10.71348 0.479532 0.064216 77.53965 1.505019 0.116498
2 55.65176 2.772377 0.393788 490.8589 10.64161 0.879686
3 168.2068 9.18847 1.377652 1762.711 42.06369 3.692394
4 388.0785 22.94524 3.613889 4734.55 122.7896 11.38394
5 758.6281 48.00968 7.906454 10597.46 295.4639 28.7853
6 1325.249 88.92437 15.24622 20908.68 620.8854 63.26942
7 2134.446 150.654 26.78464 37600.07 1179.848 125.224
8 3233.338 238.4644 43.80586 62982.53 2074.428 228.4448
9 4669.385 357.8329 67.7008 99748.23 3428.868 390.451
10 6490.247 514.3848 99.94468 150971.8 5390.161 632.7313
Table 2
Moments of the generalized linear exponential distribution(GLED).

3. RELATIONS FOR PRODUCT MOMENTS

In this section, we derived the recurrence relations for product moments of generalized upper record values. Before proving the main result, we shall prove the following lemma:

Lemma 3.1.

Fix a positive integer k≥1, for 1≤m≤n−2 and i,j=0,1,…,

E[(Ym+1(k))i(Yn(k))j]−E[(Ym(k))i(Yn(k))j]=iknm!(n−m−1)!∫​​−∞∞∫​​−∞yxi−1yj×[−lnF¯(x)]m[lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy.(3.1)

Proof.

From (1.5), we have

E[(Ym+1(k))i(Yn(k))j]−E[(Ym(k))i(Yn(k))j]=kn(m−1)!(n−m−1)!×∫​​−∞∞∫​​−∞yxiyj[−lnF¯(x)]m−1f(x)F¯(x)[lnF¯(x)−lnF¯(y)]n−m−2× [F¯(y)]k−1f(y){[−lnF¯(x)](n−m−1)m−[lnF¯(x)−lnF¯(y)]}dx dy.(3.2)

Let

h(x,y)=1m[lnF¯(x)−lnF¯(y)]n−m−1[−lnF¯(x)]m(3.3)
∂∂xh(x,y)=−[−lnF¯(x)]m−1f(x)F¯(x)[lnF¯(x)−lnF¯(y)]n−m−2×{[−lnF¯(x)](n−m−1)m−[lnF¯(x)−lnF¯(y)]}.(3.4)

On using the value of (3.4) in (3.2), we find that

E[(Ym+1(k))i(Yn(k))j]−E[(Ym(k))i(Yn(k))j]=−kn(m−1)!(n−m−1)!×∫​​−∞∞yj[F¯(y)]k−1f(y){∫​​−∞yxi∂∂xh(x,y)dx}dy.(3.5)

Now, in view of (3.3)

∫−∞yxi∂∂xh(x,y)dx=−im∫−∞yxi−1−lnF¯(x)mlnF¯(x)−lnF¯(y)n−m−1dx.(3.6)

On substituting (3.6) in (3.5) and simplifying, we get the required result.

Theorem 3.1.

For the distribution given in (1.1) and k≥1, m≥1 and i,j=0,1,…

E[(Ym(k))i(Ym+1(k))j]=mαθ(i+2)λ[(1+i + 22mα)E[Ym(k)i+2Ym+1(k)j]−E[(Ym+1(k))i+j+2]]+mαβ(i+1)λ[(1+i + 1mα)E[(Ym(k))i+1(Ym+1(k))j]−E[(Ym+1(k))i+j+1]](3.7)
and for 1≤m≤n−2, and i,j = 0,1,…
E[(Ym(k))i(Yn(k))j]=mαθ(i+2)λ[(1+i + 22mα)E[(Ym(k))i+2(Yn(k))j]−E[(Ym+1(k))i+2(Yn(k))j]]+mαβ(i+1)λ[(1+i + 1mα)E[(Ym(k))i+1(Yn(k))j]−E[(Ym+1(k))i+1(Yn(k))j]].(3.8)

Proof.

From (1.5), for m ≤ n − 1, we have

θ2E[(Ym(k))i+2(Yn(k))j]+βE[(Ym(k))i+1(Yn(k))j]−λE[(Ym(k))i(Yn(k))j]
=kn(m−1)!(n−m−1)!∫​​​∞Ψ∫​​​yΨ(θ2x2+βx−λ)xiyj[−lnF¯(x)]m−1f(x)F¯(x)× [lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy.

In view of (1.3),

θ2E[(Ym(k))i+2(Yn(k))j]+βE[(Ym(k))i+1(Yn(k))j]−λE[(Ym(k))i(Yn(k))j]
=αkn(m−1)!(n−m−1)!∫Ψ∞∫Ψy(β+θx)xiyj[−lnF¯(x)]m× [lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy.=αβkn(m−1)!(n−m−1)!∫Ψ∞∫Ψyxiyj[−lnF¯(x)]m[lnF¯(x)−lnF¯(y)]n−m−1× [F¯(y)]k−1f(y)dx dy+αθkn(m−1)!(n−m−1)!∫Ψ∞∫Ψyxi+1yj[−lnF¯(x)]m× [lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy.

Using Lemma 3.1, we get

θ2E[(Ym(k))i+2(Yn(k))j]+βE[(Ym(k))i+1(Yn(k))j]−λE[(Ym(k))i(Yn(k))j]
=mαθi + 2E[(Ym+1(k))i+2(Yn(k))j]−E[(Ym(k))i+2(Yn(k))j]+mαβi + 1E[(Ym+1(k))i+1(Yn(k))j]−E[(Ym(k))i+1(Yn(k))j],(3.9)
arranging (3.9) we get the result as given in (3.8).

Now putting n = m + 1 and noting that E[(Ym(k))i(Ym(k))j]=E[(Ym(k))i+j], the recurrence relation given in (3.7) can be easily obtained on the same line of proof (3.8).

Remark 3.1.

If we put

  1. β = λ = 0 in (3.9), we get the recurrence relations for product moment of generalized upper records from Weibull distribution

    E[(Ym+1(k))i+2(Yn(k))j]=(1+i+22αm)E[(Ym(k))i+2(Yn(k))j],
    at i′ = i + 2, we get similar result as obtained by Khan and Khan [18].

  2. β = λ = 0 and α=12 in (3.9), we get the recurrence relations for product moment of generalized upper records from the exponential distribution

    E[(Ym+1(k))i+2(Yn(k))j]=(1+i+2m)E[(Ym(k))i+2(Yn(k))j],
    at i′ = i + 2, we get similar result as obtained by Khan and Khan [18].

  3. β = λ = 0 and α = 1 in (3.9), we get the recurrence relations for product moment of generalized upper records from Rayleigh distribution

    E[(Ym+1(k))i+2(Yn(k))j]=(1+i+22m)E[(Ym(k))i+2(Yn(k))j],
    at i′ = i + 2, we get similar result as obtained by Khan and Khan [18].

  4. λ = 0 and α = 1 in (3.9), we get the recurrence relations for product moment of generalized upper records from the LED

    E[(Ym+1(k))i+2(Yn(k))j]=(1+i+22m)E[(Ym(k))i+2(Yn(k))j]+(i+2)β(i+1)θ[(1+i+1m)E[(Ym(k))i+1(Yn(k))j]−E[(Ym+1(k))i+1(Yn(k))j]].

  5. Setting β = 0 in (3.8) we get the recurrence relations for product moment of generalized upper records from the generalized Rayleigh distribution

    E[(Ym(k))i(Yn(k))j]=mαθ(i+2)λ[(1+i+22mα)E[(Ym(k))i+2(Yn(k))j]−E[(Ym+1(k))i+2(Yn(k))j]].

  6. Putting j = 0 in (3.8) we get the recurrence relation for single moment of generalized record value from the GLED.

Corollary 3.1.

The recurrence relation for product moments of the upper record values from GLED has the form

λE[(XUm)i(XUn)j]=mαθi+2[(1+i+22mα)E[(XUm)i+2(XUn)j]−E[(XUm+1)i+2(XUn)j]]+mαβi+1[(1+i+1mα)E[(XUm)i+1(XUn)j]−E[(XUm+1)i+1(XUn)j]].(3.10)

Remark 3.2.

If we put

  1. β = λ = 0 in (3.10), we get the recurrence relation of upper records from Weibull distribution, which is similar to the result as given by Balakrishnan and Chan [25] for i′ = i + 2.

  2. β = λ = 0 and α=12 in (3.10), we get the recurrence relation of upper records from the exponential distribution, which is similar to the result as given by Balakrishnan and Chan [25] for i′ = i + 2.

  3. β = λ = 0 and α = 1 in (3.10), we get the recurrence relation of upper records from Rayleigh distribution, which is similar to the result as given by Balakrishnan and Chan [25] for i′ = i + 2.

  4. λ = 0 and α = 1 in (3.10), we get the recurrence relations for upper records from the LED

    E[(XUm+1)i+2(XUn)j]=(1+i + 22m)E[(XUm)i+2(XUn)j]+(i + 2)β(i+1)θ[(1+i + 1m)E[(XUm)i+i(XUn)j]−E[(XUm+1)i+1(XUn)j]].

  5. Setting β = 0 in (3.10) we get the recurrence relations of upper records from the generalized Rayleigh distribution

    E[(XUm)i(XUn)j]=mαθ(i + 2)λ[(1+i + 22mα)E[(XUm)i+2(XUn)j]−E[(XUm+1)i+2(XUn)j]].

  6. Putting j = 0 in (3.10) we get (2.12), the recurrence relation for single moment of upper record value from the GLED.

4. CHARACTERIZATION

This section contains the characterizations of GLED, we start with the following result of Lin [26].

PROPOSITION. Let n0 be any fixed non-negative integer, −∞≤a<b≤∞ and g(x)≥0 an absolutely continuous function with g′(x)≠0 a.e. on (a,b). Then the sequence of functions {(g(x))n e−g(x),n≥n0} is complete in L(a,b) if g(x)is strictly monotone on(a,b).

Theorem 4.1.

Fix a positive integer k ≥ 1 and let j be any non-negative integer. A necessary and sufficient condition for a random variable X to be distributed with pdf given by (1.1) is that

E(Yn(k))j=nαθ(j + 2)λ[(1+j + 22nα)E(Yn(k))j+2−E(Yn+1(k))j+2]+nαβ(j + 1)λ[(1+j + 1nα)E(Yn(k))j+1−E(Yn+1(k))j+1](4.1)
for n = 1, 2, ….

Proof.

The necessary part follows from (2.9). On the other hand if the recurrence relation (4.1) is satisfied, then on rearranging (4.1)

θ2[E(Yn(k))j+2]+β[E(Yn(k))j+1]−λ[E(Yn(k))j]=nαθj + 2[E(Yn+1(k))j+2−E(Yn(k))j+2]+nαβj + 1[E(Yn+1(k))j+1−E(Yn(k))j+1]
in view of Khan et al. [19], we have
kn(n−1)!∫Ψ∞(θ2x2+βx−λ)xj[−lnF¯(x)]n−1[F¯(x)]k−1f(x)dx
=nαθj+2[(j+2)knn!∫Ψ∞xj+1[−lnF¯(x)]n[F¯(x)]kdx]+nαβj+1[(j+1)knn!∫Ψ∞xj[−lnF¯(x)]n[F¯(x)]kdx],
which implies
kn(n−1)!∫Ψ∞xj[−lnF¯(x)]n−1[F¯(x)]k−1×{(θ2x2+βx−λ)f(x)−α(θx+β)[−lnF¯(x)][F¯(x)]}dx=0.

Now follow from the above proposition, we get

(θ2x2+βx−λ)f(x)=α(β+θx)[−lnF¯(x)][F¯(x)]
which proves that f(x) has the form as given in (1.3).

Theorem 4.2.

For a positive integer k, i and j be a non-negative integer, a necessary and sufficient condition for a random variable X to be distributed with pdf given by (1.1) and for 1 ≤ m ≤ n − 2, is that

E[(Ym(k))i(Yn(k))j]=mαθ(i + 2)λ[(1+i + 22mα)E[(Ym(k))i+2(Yn(k))j]−E[(Ym+1(k))i+2(Yn(k))j]]+mαβ(i + 1)λ[(1+i + 1mα)E[(Ym(k))i+1(Yn(k))j]−E[(Ym+1(k))i+1(Yn(k))j]].(4.2)

Proof.

The necessary part follows from (3.8). On the other hand if the relation in (4.2) is satisfied, then arranging the (4.2)

θ2E[(Ym(k))i+2(Yn(k))j]+βE[(Ym(k))i+1(Yn(k))j]−λE[(Ym(k))i(Yn(k))j]
=mαθi+2[E[(Ym+1(k))i+2(Yn(k))j]−E[(Ym(k))i+2(Yn(k))j]]+mαβi+1[E[(Ym+1(k))i+1(Yn(k))j]−E[(Ym(k))i+1(Yn(k))j]].

On using Lemma 3.1, we have

kn(m−1)!(n−m−1)!∫Ψ∞∫Ψy(θ2x2+βx−λ)xiyj−lnF¯(x)m−1f(x)F¯(x)
× [lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy=αθkn(m−1)!(n−m−1)!×∫Ψ∞∫Ψyxi+1yj[−lnF¯(x)]m[lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy+αβkn(m−1)!(n−m−1)!+∫Ψ∞∫Ψyxiyj[−lnF¯(x)]m[lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)dx dy.
which implies
∫Ψ∞∫Ψyxiyj[−lnF¯(x)]m−1[lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)]k−1f(y)×{(θ2x2+βx−λ)f(x)[F¯(x)]−α(θx+β)[−lnF¯(x)]}dx dy=0.

Now, follow from the above proposition with

(θ2x2+βx−λ)f(x)=α(β+θx)[−lnF¯(x)][F¯(x)]
which proves that f(x) has the form as given in (1.3).

Theorem 4.3.

Let X be a non-negative random variable having an absolutely continuous df F (x) and F (0) = 0 and 0 ≤ F (x) ≤ 1 for all x > 0, then

E[ξ(Yn(k))∣Ym(k)=x]=ξ(x)(kk+1)n−m,(4.3)
if and only if
F¯(x)=exp{−(θ2x2+βx−λ)α}, x>Ψ.
where
ξ(t)=exp{−(θ2t2+βt−λ)α}, t>Ψ.

Proof.

From (1.6), we have

E[ξ(Yn(k))∣Ym(k)=x]=kn−m(n−m−1)!×∫​​x∞ξ(y)[lnF¯(x)−lnF¯(y)]n−m−1[F¯(y)F¯(x)]k−1f(y)F¯(x)dy.(4.4)

By setting u=F¯(y)F¯(x)=exp{−(θ2y2+βy−λ)α}exp{−(θ2x2+βx−λ)α} from (1.2) in (4.4), we have

E[ξ(Yn(k))∣(Ym(k))=x]=kn−m(n−m−1)!ξ(x)∫01uk[−lnu]n−m−1du.(4.5)

Therefore we have (see Gradshteyn and Ryzhik [27], p. 551)

∫01[−lnx]μ−1xν−1dx=Γμνμ, μ>0, ν>0.(4.6)

On using (4.6) in (4.5), we have the result given in (4.3).

To prove sufficient part, we have

kn−m(n−m−1)!∫x∞ξ(y)[−lnF¯(y)+lnF¯(x)]n−m−1[F¯(y)]k−1f(y)dy=[F¯(x)]kgn∣m(x),(4.7)
where
gn∣m(x)=ξ(x)(kk+1)n−m.

Differentiating (4.7) both sides with respect to x, we get

−kn−m(n−m−2)!f(x)F¯(x)∫x∞ξ(y)[−lnF¯(y)+lnF¯(x)]n−m−2[F¯(y)]k−1f(y)dy=gn∣m′(x)[F¯(x)]k−kgn∣m(x)[F¯(x)]k−1f(x)
or
−k gn|m+1(x)[F¯(x)]k−1f(x)=gn|m′(x)[F¯(x)]k−k gn|m(x) [F¯(x)]k−1f(x).

Therefore,

f(x)F¯(x)=−gn|m′(x)k[gn|m+1(x)−gn|m(x)]=−ξ′(x)ξ(x),(4.8)
where
gn|m′(x)=ξ′(x)(kk+1)n−m,gn|m+1(x)−gn|m(x)=1kξ(x)(kk+1)n−m,

Integrating both the sides (4.8) with respect to x between (Ψ, y), which gives F¯(x) = ξ(x), the sufficiency part is proved.

5. CONCLUSION

In this study, we demonstrate the explicit expression as well as recurrence relation for the moments of k-th upper record values from the GLED. These relations can be used to reduce the amount of direct computation and moments of any order can be calculated easily. To verify the designed models which is required in probability distribution, we used the results of the characterization. At the different values of parameters, we reduced some well-known results. We can explore our study for generalized order statistics which contains several models of order random variates.

CONFLICTS OF INTEREST

The authors are declare no competing interests.

AUTHORS' CONTRIBUTIONS

Mahfooz Alam proposed the concept the paper and calculate the existing results. M.A. Khan contributed in writing—original draft preparation. R.U. Khan performed revision and improve the quality of the draft. All authors have read and agreed to the published version of the manuscript.

Funding Statement

This research received no external funding.

ACKNOWLEDGMENTS

The authors acknowledge with thanks the concrete suggestions and comments of the learned Referees, which led to an overall improvement of the paper. The authors are also grateful to Dr. Z. Akhter, Department of Statistics, University of Delhi for his help and suggestions throughout the preparation of this paper.

REFERENCES

4.M. Ahsanullah, Record Statistics, Nova Science Publishers, New York, NY, USA, 1995.
11.Z. Grudzień, Characterization of Distribution of Time Limits in Record Statistics as Well as Distributions and Moments of Linear Record Statistics from the Samples of Random Numbers, Praca Doktorska, UMCS, Lublin, Poland, 1982.
12.Z. Grudzień and D. Szynal, Prob. Stat. Decis. Theory A., 1983, pp. 119-127.
13.Z. Grudzień and D. Szynal, J. Appl. Stat. Sci., Vol. 5, 1997, pp. 259-266.
14.P. Pawlas and D. Szynal, J. Appl. Stat. Sci., Vol. 7, 1998, pp. 53-62.
16.P. Pawlas and D. Szynal, J. Appl. Stat. Sci., Vol. 10, 2000, pp. 17-26.
17.S. Minimol and P. Thomos, J. Appl. Stat. Sci., Vol. 10, 2014, pp. 1726.
21.B. Singh, R.U. Khan, and M.A.R. Khan, Thail. Stat., Vol. 17, 2019, pp. 93-103.
23.M. Ahsanullah, Pak. J. Stat., Vol. 3A, 1987, pp. 17-40.
24.M. Ahsanullah, Int. J. Stat. Sci., Vol. 11, 2011, pp. 111-123.
25.N. Balakrishnan and P.S. Chan, Nat. Inst. Standard Technol. J. Res. Special Publ., Vol. 899, 1993, pp. 41-51.
27.I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series of Products, Academic Press, New York, NY, USA, 2007.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
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Pages
289 - 303
Publication Date
2021/03/01
ISSN (Online)
2214-1766
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risenwbib
TY  - JOUR
AU  - M. Alam
AU  - M. A. Khan
AU  - R. U. Khan
PY  - 2021
DA  - 2021/03/01
TI  - On Upper k-Record Values from the Generalized Linear Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 289
EP  - 303
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210216.001
DO  - 10.2991/jsta.d.210216.001
ID  - Alam2021
ER  -