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Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application

  • Hayrinisa Demirci Biçer EMAIL logo
Published/Copyright: July 24, 2019
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Open Mathematics
From the journal Open Mathematics Volume 17 Issue 1

Abstract

In the present paper, we introduce a new form of generalized Rayleigh distribution called the Alpha Power generalized Rayleigh (APGR) distribution by following the idea of extension of the distribution families with the Alpha Power transformation. The introduced distribution has the more general form than both the Rayleigh and generalized Rayleigh distributions and provides a better fit than the Rayleigh and generalized Rayleigh distributions for more various forms of the data sets. In the paper, we also obtain explicit forms of some important statistical characteristics of the APGR distribution such as hazard function, survival function, mode, moments, characteristic function, Shannon and Rényi entropies, stress-strength probability, Lorenz and Bonferroni curves and order statistics. The statistical inference problem for the APGR distribution is investigated by using the maximum likelihood and least-square methods. The estimation performances of the obtained estimators are compared based on the bias and mean square error criteria by a conducted Monte-Carlo simulation on small, moderate and large sample sizes. Finally, a real data analysis is given to show how the proposed model works in practice.

Keywords: Alpha power transformation; Maximum likelihood estimate; Least-square estimate; Shannon entropy; Generalized Rayleigh distribution
MSC 2010: 62E20; 62F10

1 Introduction

The famous distribution families have been successfully used in modeling real-world data sets, until recently. However, it is well known that the performances of these distributions in the modeling of complex real-world data sets are not always at the desired level. In recent years, a number of researchers who take into account this situation have focused on introducing the flexible distribution families in order to the modeling of data sets in a wide variety of complex structures and have made several breakthroughs by giving various continuous distribution generating methods, especially in lifetime distributions. These distribution produce methods lay out a new distribution taking a baseline distribution. The baseline distributions are always a special case of the newly obtained distribution. Hence, the produced distribution has the characteristics of the baseline distribution and provides better data fit than the baseline distribution. There are numerous papers in the literature that create a new distribution using a baseline distribution and draw attention to its advantages. We refer readers to [1, 2, 3, 4, 5] for further information on generating a new distribution family by using a baseline distribution.

The Rayleigh distribution, which has only a shape parameter, was originally introduced by a study of Rayleigh on a problem of acoustic. The distribution has a strong modeling ability of positive valued and skewed data obtained from many fields such as engineering, biology, life sciences, reliability and etc. The Rayleigh distribution is a distribution related to Gamma, Weibull, Exponential and Rice distributions. However, it has a disadvantage since the distribution has only a single shape parameter in which plays a crucial role in describing the various behaviors of the distribution. Fortunately, to overcome this disadvantage of Rayleigh distribution, there are many generalizations of the distribution such as generalized Rayleigh distribution [6], transmuted Rayleigh distribution [7], Weibull Rayleigh distribution [8], inverted exponentiated Rayleigh distribution [9] and the slashed exponentiated Rayleigh distribution [10]. The generalized Rayleigh distribution is the most widely used among these generalizations. The generalized Rayleigh distribution has also some important generalizations recently introduced to achieve optimal data fit, such as the Kumaraswamy generalized Rayleigh [11], the beta generalized Rayleigh [12], the slashed generalized Rayleigh [13] and the Marshall-Olkin extended generalized Rayleigh [14]. In the literature, there are also many published papers on the estimation of the parameters of Rayleigh and generalized Rayleigh distributions for the various data types, see [15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

The main motivation of this paper is to introduce a more flexible lifetime distribution than the Rayleigh and generalized Rayleigh distribution to be used for the modeling of data sets in wide variety structures. In the aim of this context, in the study, a new three-parameter family of Rayleigh distribution which is named alpha power generalized Rayleigh distribution (APGR) is derived using the alpha power transform (APT) method recently introduced by Mahdavi and Kundu [5]. Both Rayleigh and generalized Rayleigh distributions are the special cases of APGR distribution. Therefore, APGR distribution has more data modeling capability than the Rayleigh and generalized Rayleigh. Further, the APGR distribution is an important alternative to famous distributions like Gamma, Weibull, and exponential for modeling the data observed from industrial and physical phenomena.

The rest of the paper is organized as follows. In section 2, we introduce the APGR distribution. We discuss some important statistical characteristics of the APGR distribution in section 3. In section 4, statistical inference problem for the APGR distribution is investigated according to maximum likelihood (ML) and least-square (LSq) methods. Section 5 includes a comprehensive Monte-Carlo simulation study display the estimation performance of the estimators derived in section 4. A real-world data set is analyzed in section 6 for illustrative purposes. Finally, section 7 concludes the paper.

2 Definition and properties of the APGR Distribution

In this section, we derive the probability density function (pdf) and cumulative distribution function (cdf) of the APGR distribution by using the APT method given in [5] and study some distributional properties of the APGR distribution. Before progressing for further, we recall the generalized Rayleigh distribution. The pdf of the generalized Rayleigh distribution is

g(x;β,λ)=2βλ2xe−λx21−e−λx2β−1,x>0, (1)

and its cdf is

Gx,β,λ=1−e−λx2β,x>0, (2)

where β and λ is the positive and real valued scale parameter and shape parameters of the distribution, respectively. Generalized Rayleigh distribution was originally studied by Surles and Padgett [6] as the two-parameter Burr Type X distribution. Then, the distribution was called the generalized Rayleigh distribution by Raqab and Kundu [25].

Now, we introduce the APGR distribution by using generalized Rayleigh distribution as a baseline distribution in the APT method.

Definition 1

A random variable X is said to have a APGR distribution with parameters α, β and λ, if it has the following pdf and cdf

fAPGR(x;α,β,λ)=ln⁡αα−12βλ2xe−λx21−e−λx2β−1α1−e−λx2β, x>0, α>0∧α≠1 2βλ2xe−λx21−e−λx2β−1 ,x>0, α=1  0 otherwise (3)

and

FAPGR(x;α,β,λ)=α1−e−λx2β−1α−1, x>0, α>0∧α≠11−e−λx2β,x>0, α=0, (4)

respectively.

Considering the cdf given by equation (4), the survival and hazard functions of the APGR distribution can be easily written as in the following forms:

SAPGR(x;α,β,λ)=α−α1−e−x2λ2βα−1,   x>0, α>0∧α≠11−1−e−λx2β, x>0, α=0 (5)

and

hAPGR(x;α,β,λ)=2βλ2xe−λx21−e−λx2β−1α1−e−λx2β−11−α1−e−λx2β−1ln⁡α  x>0, α>0∧α≠1 2βλ2xe−λx21−e−λx2β−11−1−e−λx2βx>0, α=0. (6)

From now on, a random variable X distributed the APGR with parameters α, β and λ will be indicated as X ∼ APGR(α, β, λ). By considering the equation (11) in [5], the p-th quantile of the APGR distribution, say Qp, is immediately obtained as below

Qp=−ln1−ln−α1−α+11−α+αp−pln⁡(α)1/β1/2λ. (7)

Thus, when α ≠ 1, the median of the APGR distribution is obtained as

M=Q0.5=1λ−ln1−lnα+12ln⁡α1β1/2 (8)

and when α = 1, the median of the APGR distribution is equal to median of the generalized Rayleigh distribution.

Now, we discuss the shape behavior of the pdf fAPGR(x; α, β, λ). When X tends to 0 and X tends to ∞, the pdf fAPGR(x; α, β, λ) comply with the following behaviors

limx→0+fAPGR(x;α,β,λ)=0

and

limx→∞fAPGR(x;α,β,λ)=0,

respectively.

Theorem 1

The APGR distribution is unimodal.

Proof

First derivative of the pdf fAPGR(x; α, β, λ) given by equation (3) is

fAPGR′(x;α,β)=β2ln⁡(α)e−2βxα1−e−βx(βx+β+1)β+1β2(x+1)2ln⁡(α)−(β+1)eβx(βx+β−1)(α−1)(β+1)2,α≠1β2e−βx1−β−βxβ+1,α=1. (9)

When α = 1, that is the distribution is a generalized Rayleigh, mode of the distribution can be easily obtained from solution of the equation

β2e−βx1−β−βxβ+1=0. (10)

When α ≠ 1, the derivative fAPGR′ (x; α, β, λ) is a strictly decreasing and continuous function of x and limx→0+fAPGR′ (x; α, β, λ) is positive and fAPGR′ (x; α, β, λ) takes negative values as x → ∞. Thus, we can say the fAPGR′ (x; α, β, λ) has only one zero according to intermediate value theorem and the pdf fAPGR(x; α, β, λ) is unimodal. □

We present a figure to show the shape behavior of the APGR distribution for illustrative purposes. Fig.1 a, b, c display the some of the possible shapes of the pdf of the APGR distribution for different values of the parameters α, β and λ.

Figure 1 The pdf of the APGR distribution, when (a): α = 0.25, 0.5, 1.5, 2., β = 2 and λ = 2; (b): α = 0.25, β = 0.5, 1, 2, 4 and λ = 2, (c): α = 0.25, β = 2 and λ = 0.5, 1, 2, 4
Figure 1

The pdf of the APGR distribution, when (a): α = 0.25, 0.5, 1.5, 2., β = 2 and λ = 2; (b): α = 0.25, β = 0.5, 1, 2, 4 and λ = 2, (c): α = 0.25, β = 2 and λ = 0.5, 1, 2, 4

3 Some Important Characteristics of the APGR Distribution

In this section, the moments, moment generating function and related measures such as mean, variance, skewness and kurtosis are obtained for the APGR distribution. In addition, the distribution of order statistics, stress-strength probability and Shannon and Rényi entropies and the Lorenz and Bonferroni curves of the APGR distribution are also obtained in this section.

Let we first introduce the Lemma 1 to obtain the moments of APGR distribution.

Lemma 1

Let X be a random variable with pdf given by equation (3). For any real numbers a > 0, b > 0, L > 0, r ≥ 0 and δ ≥ 0, the integral

ξa,b,L,r,δ=∫0∞xr+1e−Lx21−e−λx2b−1a1−e−Lx2beδxdx (11)

is calculated as

ξa,b,L,r,δ=∑i=0∞∑j=0ib∑k=0b−1log⁡aii!−1j+kibjb−1k12L2(j+k+1)12(−r−3)×δΓr+321F1r+32;32;δ24(j+k+1)L2+Γr2+1L2(j+k+1)1F1r+22;12;δ24(j+k+1)L2 (12)

where 1F1(.; .; .) is indicate the hypergeometric function, see [27]

Proof

See Appendix A for proof of Lemma 1. □

Obviously, by using the Lemma 1, the r-th moment, moment generating function, characteristic function, mean and variance of the APGR distribution are easily obtained as

μr′=EXr=ln⁡αα−12βλ2ξα,β,λ,r,0, (13)
MXt=Eetx=ln⁡αα−12βλ2ξα,β,λ,0,t, (14)
ΦXt=Eeitx=ln⁡αα−12βλ2ξα,β,λ,0,it (15)
μ=μ1′=EX=ln⁡αα−12βλ2ξα,β,λ,1,0, (16)

and σ2 = Var(X) = μ2′−μ1′2 , respectively.

Now, we derive the central moments and cumulants of the APGR distribution. By using the raw moments given in equation (13), r-th central moment of the APGR distribution is obtained as follow

μr=∑j=0rrj−1r−jμj′μ1′r−j=∑j=0rrj−1r−jln⁡αα−12βλ2ξα,β,λ,j,0ln⁡αα−12βλ2ξα,β,λ,1,0r−j. (17)

Therefore, using the central moments given by equation (17), the second, third and fourth cumulants κ2, κ3 and κ4 can be expressed as κ2 = μ2, κ3 = μ3 and κ4 = μ4 − 3μ22 , respectively. The skewness and the kurtosis coefficients of the APGR distribution are calculated by γ1=κ3/κ23/2  and  γ2=κ4/κ22 .

3.1 Order Statistics

Let X1, X2, …, Xn be a random sample from APGR( α, β, λ) distribution and X(1), X(2), … X(n) denote their order statistics. The pdf of the random variable X(r), (r = 1, 2, …, n) is obtained as

fX(r)x=n!(r−1)!(n−r)!FAPGR(x,α,β,λ)r−1fAPGR(x,α,β,λ)(1−FAPGR(x,α,β,λ))n−r=n!(r−1)!(n−r)!ln⁡αα−12βλ2xe−λx2α1−e−λx2β−1α−1r−1  1−α1−e−λx2β−1α−1n−r1−e−λx2β−1α1−e−λx2β=n!(r−1)!(n−r)!2βλ2xe−λx2ln⁡αα−1αζ−1α−1r−11−αζ−1α−1n−rζ1−1βαζ (18)

where, ζ = (1 − e−(λx)2)β. In particular, pdf of the first and n-th order statistics can be easily derived from equation (18) as

fX(1)x=n2βλ2xe−λx2ln⁡αα−11−αζ−1α−1n−1ζ1−1βαζ (19)

and

fX(n)x=n2βλ2xe−λx2ln⁡αα−1αζ−1α−1n−1ζ1−1βαζ, (20)

respectively.

3.2 Stress-strength probability

We suppose that X and Y be random variables from APGR( α1, β1, λ1) and Y ∼ APGR(α2, β2, λ2) distributions, respectively. In this situation, the stress-strength probability is calculated by R = P(Y < X), where Y represents the ‘stress’ and X represents the ‘strength’ to sustain the stress. For APGR distribution, stress-strength probability P(Y < X) is obtained as below

R=PY<X=∫0∞PY<X | X=xfXxdx=∫0∞fXxFYxdx=∫0∞ln⁡α1α1−12β1λ12xe−λ1x21−e−λ1x2β1−1α11−e−λ1x2β1FYxdx=∫0∞ln⁡α1α1−12β1λ12xe−λ1x21−e−λ1x2β1−1α1−e−λ1x2β1α21−e−λ2x2β2−1α2−1dx=1α2−12β1λ12ln⁡α1α1−1∫0∞xe−λ1x21−e−λ1x2β1−1α11−e−λx2βα21−e−λ2x2β2dx−EX. (21)

Further, using the Lemma 1, we have

R=1α2−1Λα2,β2,λ2−2β1λ12ln⁡α1α−1α2−1ξα1,β1,λ1,1,0, (22)

where Λα2,β2,λ2=Eα21−e−λ2x2β2 .

3.3 Shannon and Rényi Entropies

The entropy is quite important as a measure of variation or uncertainty of a random variable. In this section, we investigate the Shannon and Rényi entropies of the APGR distribution. The Shannon entropy of a random variable X with pdf f(x) is defined as, see [26],

HX=E−ln⁡fx. (23)

Hence, the Shannon entropy of APGR(α, β, λ) distribution is obtained as

HX=−∫0∞fAPGRx,α,β,λlnfAPGRx,α,β,λdx=−∫0∞fAPGRx,α,β,λlnln⁡α−lnα−1+ln⁡2+ln⁡β+2ln⁡λ+ln⁡x−λ2x2+β−1ln1−e−λx2+ln⁡α1−e−λx2βdx=−lnln⁡α−lnα−1+ln⁡2+ln⁡β+2ln⁡λ+Elnx−λ2Ex2+β−1Eln1−e−λx2+ln⁡αE1−e−λx2β. (24)

By applying the Lemma 1 to equation (24), the Shannon entropy 𝓗(X) is written as

HX=−lnln⁡α−lnα−1+ln⁡2+ln⁡β+2ln⁡λ+YX−ln⁡αα−12βλ4ξα,β,λ,2,0+β−1ϑX+ln⁡αςX, (25)

where YX = E(ln(X)), ϑX = E[ln(1 − e−(λx)2)] and ςX = E[(1 − e−(λx)2)β] and these expectations can be easily calculated numerically.

Now, we calculate the Rényi entropy of the APGR distribution. We first recall the definition of the Rényi entropy. The Rényi entropy of a random variable X with pdf f is given by

REXξ=11−ξln∫−∞∞fxξdx. (26)

By using the pdf (3) in the equation (26), Rényi entropy of the APGR distribution is obtained as

REXξ=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∑j=0iβ+β−1iβ+β−1j−1j12Γξ+12(j+1)λ2ξ12(−ξ−1), (27)

see Appendix B for calculation of the Rényi entropy of the APGR distribution.

3.4 Lorenz and Bonferroni Curves

Lorenz and Bonferroni curves are two graphical representations to the measure inequality of distribution of a random variable. The Lorenz and Bonferroni curves for a random variable X are defined as the plot of

L(p)=1μ∫0qxfxdx, (28)

and

B(p)=1pμ∫0qxfxdx, (29)

respectively, against F(x), where μ is indicate the expectation of the random variable X and q = F−1(p) also L(p) and B(p) are called the Lorenz index and Bonferroni index, respectively. If the expectation (16) and pdf (3) are used in the equation (28), the Lorenz index of APGR distribution is obtained as

Lp=1ln⁡αα−12βλ2ξα,β,λ,1,0∫0qxfAPGRxdx=1ln⁡αα−12βλ2ξα,β,λ,1,0∫0qxln⁡αα−12βλ2xe−λx21−e−λx2β−1α1−e−λx2βdx=1ξα,β,λ,1,0∫0qxln⁡αα−12βλ2xe−λx21−e−λx2β−1α1−e−λx2βdx (30)

Following steps of the proof of Lemma 1, the Lorenz index (30) is immediately written as

Lp=1ξα,β,λ,1,0∑i=0∞∑j=0iβ+β−1iβ+β−1j−1k×ln⁡αiα−1πerfj+1λq−2j+1λqe−(j+1)λ2q24(j+1)3/2λ3, (31)

where erf(.) is indicate the error function, see [27]. Similarly, the Bonferroni index of APGR distribution is also obtained as

Bp=1pξα,β,λ,1,0∑i=0∞∑j=0iβ+β−1iβ+β−1j−1k×ln⁡αiα−1πerfj+1λq−2j+1λqe−(j+1)λ2q24(j+1)3/2λ3 (32)

4 Inference

In this section, we consider the statistical inference problem for APGR(α, β, λ) distribution. We employ the ML and LSq methods to obtaining the estimators of the unknown parameters α, β, and λ.

4.1 ML estimation

Let X1, X2, …, Xn be a random sample from APGR(α, β, λ) distribution. The log-likelihood function of the random variables Xi, i = 1, 2, …, n can be easily written from equation (3)) as

Lα,β,λ; X1,X2,...,Xn=nlnln⁡α−lnα−1+ln⁡2+ln⁡β+2ln⁡λ+∑i=1nln⁡xi−λ2∑i=1nxi2+β−1∑i=1nln1−e−λxi2+ln⁡α∑i=1n1−e−λxi2β. (33)

Thus, by derivating the log-likelihood function given in equation (33) with respect to parameters α, β and λ, we can write the following likelihood equations

∂L∂α=−nαln⁡αα−1αln⁡α−α+1+1α∑i=1n1−e−λxi2β=0, (34)
∂L∂β=nβ+∑i=1nln1−e−λxi2+βln⁡α∑i=1n1−e−λxi2β−1=0 (35)

and

∂L∂λ=2nλ−2λ∑i=1nxi2−2λβ−1∑i=1nxi2e−λ2xi2e−λ2xi2−1−2βλln⁡α∑i=1nxi2e−xi2λ2e−xi2λ2−11−e−xi2λ2β=0 (36)

Unfortunately, the ML estimators of the parameters α, β and λ cannot be explicitly derived from equations (34), (35) and (36). However, we can obtain the ML estimates of the parameters α, β and λ, say α̂ML, β̂ML and λ̂ML, respectively, from the simultaneous numerical solution of equations (34), (35) and (36).

4.2 LSq Estimation

The LSq estimation method was firstly used by Swain et al. [28] as a nonlinear method in estimation of the parameters of the Beta distribution. Especially, when the maximum likelihood estimators cannot be obtained in an explicit form, the LSq estimates are quite important with regard to provide an initial estimation for numerical methods which use in obtaining the maximum likelihood estimations.

The LSq estimations of the parameters α, β and λ, say α̂LSq, β̂LSq and λ̂LSq, respectively, are obtained by minimizing the equation

∑i=1nFAPGR(x(i),α,β,λ)−F^xi2, (37)

with respect to parameters α, λ and β, where x(i), (i = 1, 2, …, n) is the ith element of the ordered observations x1, x2, …, xn and F̂(.) is indicate the observations’ empirical cumulative distribution function (ecdf) calculated as

F^.=in+1. (38)

By using the equations (4) and (38) in equation (37), we have

∑i=1nα1−e−λxi2β−1α−1−in+12. (39)

Note that both LSq estimates and ML estimates of the unknown parameters can be obtained using the numerical methods.

5 Monte-Carlo Simulation Study

In this section, some simulation studies are presented in order to compare the estimation efficiencies of the ML and LSq estimators obtained in the previous section. In the simulation studies, two different cases α < 1 and α > 1 are considered.

In the first case, the parameter α is chosen as 0.25 and also the values of the parameters β, λ are set as β = 0.5, 1, 2 and λ = 0.5, 1, 2, respectively. The ML and LSq estimates of the parameters (Est.) are obtained with the simulations performed by 1000 replications for the different sample of sizes n = 30, 50, 100 and 200. In addition, through the simulation study, the bias (Bias) and mean-squared error (MSE) values of the ML and LSq estimators are obtained. The simulated results are given in Table 1.

Table 1

Parameter estimates, Bias and MSE values, when α = 0.25

α β λ
α β λ n Method Est. Bias MSE Est. Bias MSE Est. Bias MSE
0.25 0.50 0.50 30 ML 0.3310 0.0810 0.0829 0.5339 0.0339 0.0128 0.5331 0.0331 0.0122
LSq 0.4547 0.2047 0.1884 0.4843 -0.0157 0.0146 0.4960 -0.0040 0.0227
50 ML 0.2985 0.0485 0.0511 0.5237 0.0237 0.0069 0.5234 0.0234 0.0071
LSq 0.4221 0.1721 0.1561 0.4893 -0.0107 0.0098 0.4928 -0.0072 0.0143
100 ML 0.2640 0.0140 0.0104 0.5094 0.0094 0.0031 0.5103 0.0103 0.0028
LSq 0.4055 0.1555 0.1303 0.4845 -0.0155 0.0061 0.5023 0.0023 0.0104
200 ML 0.2635 0.0135 0.0056 0.5038 0.0038 0.0012 0.5059 0.0059 0.0013
LSq 0.3845 0.1345 0.0827 0.4827 -0.0173 0.0033 0.5115 0.0115 0.0057
1.00 30 ML 0.3224 0.0724 0.0971 0.5268 0.0268 0.0118 1.0522 0.0522 0.0487
LSq 0.4221 0.1721 0.1840 0.4879 -0.0121 0.0159 0.9644 -0.0356 0.0931
50 ML 0.2905 0.0405 0.0494 0.5160 0.0160 0.0067 1.0317 0.0317 0.0264
LSq 0.4396 0.1896 0.1731 0.4760 -0.0240 0.0085 0.9901 -0.0099 0.0762
100 ML 0.2687 0.0187 0.0190 0.5093 0.0093 0.0037 1.0062 0.0062 0.0100
LSq 0.4172 0.1672 0.1409 0.4834 -0.0166 0.0059 0.9966 -0.0034 0.0457
200 ML 0.2560 0.0060 0.0054 0.5059 0.0059 0.0015 1.0102 0.0102 0.0040
LSq 0.3654 0.1154 0.0842 0.4888 -0.0112 0.0033 1.0143 0.0143 0.0280
2.00 30 ML 0.3508 0.1008 0.1126 0.5311 0.0311 0.0123 2.1191 0.1191 0.2319
LSq 0.4029 0.1529 0.1659 0.4881 -0.0119 0.0133 1.8829 -0.1171 0.4014
50 ML 0.3233 0.0733 0.0590 0.5125 0.0125 0.0064 2.0609 0.0609 0.1113
LSq 0.4304 0.1804 0.1723 0.4836 -0.0164 0.0098 1.9363 -0.0637 0.2866
100 ML 0.2771 0.0271 0.0219 0.5096 0.0096 0.0033 2.0237 0.0237 0.0397
LSq 0.3729 0.1229 0.1164 0.4872 -0.0128 0.0054 1.9441 -0.0559 0.1902
200 ML 0.2688 0.0188 0.0204 0.5019 0.0019 0.0016 2.0029 0.0029 0.0154
LSq 0.3591 0.1091 0.0865 0.4881 -0.0119 0.0033 1.9935 -0.0065 0.1155
1.00 0.50 30 ML 0.4193 0.1693 0.2044 1.0525 0.0525 0.0565 0.5234 0.0234 0.0108
LSq 0.4877 0.2377 0.2023 0.9628 -0.0372 0.0845 0.5050 0.0050 0.0116
50 ML 0.3653 0.1153 0.0830 1.0500 0.0500 0.0358 0.5268 0.0268 0.0064
LSq 0.4403 0.1903 0.1651 0.9802 -0.0198 0.0439 0.5050 0.0050 0.0077
100 ML 0.2990 0.0490 0.0339 1.0125 0.0125 0.0151 0.5074 0.0074 0.0032
LSq 0.4238 0.1738 0.1423 0.9645 -0.0355 0.0226 0.5039 0.0039 0.0054
200 ML 0.2732 0.0232 0.0093 1.0076 0.0076 0.0074 0.5056 0.0056 0.0013
LSq 0.3890 0.1390 0.0929 0.9676 -0.0324 0.0127 0.5085 0.0085 0.0032
1.00 30 ML 0.5775 0.3275 0.5573 1.0270 0.0270 0.0666 1.0490 0.0490 0.0547
LSq 0.4907 0.2407 0.2161 0.9589 -0.0411 0.0817 0.9866 -0.0134 0.0540
50 ML 0.4350 0.1850 0.2384 1.0246 0.0246 0.0356 1.0375 0.0375 0.0344
LSq 0.4388 0.1888 0.1860 0.9794 -0.0206 0.0453 0.9958 -0.0042 0.0366
100 ML 0.3179 0.0679 0.0481 1.0107 0.0107 0.0148 1.0133 0.0133 0.0168
LSq 0.3941 0.1441 0.1358 0.9699 -0.0301 0.0218 0.9920 -0.0080 0.0250
200 ML 0.2729 0.0229 0.0137 1.0123 0.0123 0.0077 1.0081 0.0081 0.0055
LSq 0.3690 0.1190 0.1078 0.9780 -0.0220 0.0121 0.9955 -0.0045 0.0183
2.00 30 ML 0.5230 0.2730 0.4866 1.0636 0.0636 0.0830 2.1203 0.1203 0.2022
LSq 0.4464 0.1964 0.2029 0.9863 -0.0137 0.1071 1.9768 -0.0232 0.2354
50 ML 0.4479 0.1979 0.2545 1.0163 0.0163 0.0358 2.0363 0.0363 0.1255
LSq 0.4397 0.1897 0.1860 0.9671 -0.0329 0.0403 1.9618 -0.0382 0.1595
100 ML 0.3651 0.1151 0.1078 1.0098 0.0098 0.0183 2.0244 0.0244 0.0798
LSq 0.4319 0.1819 0.1677 0.9706 -0.0294 0.0226 1.9988 -0.0012 0.1115
200 ML 0.2980 0.0480 0.0330 0.9982 -0.0018 0.0084 2.0104 0.0104 0.0308
LSq 0.3954 0.1454 0.1261 0.9634 -0.0366 0.0138 1.9947 -0.0053 0.0728
2.00 0.50 30 ML 0.6306 0.3806 1.1199 2.1752 0.1752 0.4756 0.5261 0.0261 0.0090
LSq 0.4746 0.2246 0.1810 2.0750 0.0750 0.9362 0.5126 0.0126 0.0075
50 ML 0.3896 0.1396 0.1791 2.0603 0.0603 0.1600 0.5053 0.0053 0.0055
LSq 0.4653 0.2153 0.1629 1.9650 -0.0350 0.2443 0.5084 0.0084 0.0051
100 ML 0.3223 0.0723 0.0492 2.0399 0.0399 0.0773 0.5042 0.0042 0.0030
LSq 0.4474 0.1974 0.1374 1.9857 -0.0143 0.1249 0.5142 0.0142 0.0036
200 ML 0.2839 0.0339 0.0170 2.0171 0.0171 0.0352 0.5033 0.0033 0.0011
LSq 0.4171 0.1671 0.1012 1.9704 -0.0296 0.0536 0.5139 0.0139 0.0023
1.00 30 ML 0.7320 0.4820 1.5364 2.1230 0.1230 0.3462 1.0528 0.0528 0.0346
LSq 0.4745 0.2245 0.1941 1.9797 -0.0203 0.4356 1.0050 0.0050 0.0277
50 ML 0.5323 0.2823 0.5206 2.0073 0.0073 0.1550 1.0196 0.0196 0.0258
LSq 0.4783 0.2283 0.1855 1.9078 -0.0922 0.2113 1.0075 0.0075 0.0182
100 ML 0.3521 0.1021 0.0891 2.0137 0.0137 0.0709 1.0044 0.0044 0.0154
LSq 0.4389 0.1889 0.1481 1.9450 -0.0550 0.1009 1.0155 0.0155 0.0134
200 ML 0.2776 0.0276 0.0239 2.0078 0.0078 0.0342 0.9966 -0.0034 0.0073
LSq 0.4016 0.1516 0.1105 1.9548 -0.0452 0.0511 1.0155 0.0155 0.0105
2.00 30 ML 0.7357 0.4857 1.4241 2.1062 0.1062 0.2895 2.1069 0.1069 0.1367
LSq 0.4806 0.2306 0.2067 2.0005 0.0005 0.4792 2.0186 0.0186 0.1419
50 ML 0.4790 0.2290 0.3525 2.0337 0.0337 0.1646 2.0350 0.0350 0.0762
LSq 0.4346 0.1846 0.1688 1.9244 -0.0756 0.2082 1.9907 -0.0093 0.0667
100 ML 0.3738 0.1238 0.1281 2.0218 0.0218 0.0851 2.0085 0.0085 0.0663
LSq 0.4098 0.1598 0.1439 1.9477 -0.0523 0.1132 1.9959 -0.0041 0.0600
200 ML 0.3109 0.0609 0.0568 2.0054 0.0054 0.0404 1.9894 -0.0106 0.0388
LSq 0.4122 0.1622 0.1380 1.9563 -0.0437 0.0543 2.0094 0.0094 0.0498

For the second case of the simulation study, the α parameter is set as 4. Also, the values of the parameters β and λ are chosen β = 0.5, 1, 2 and λ = 0.5, 1, 2, respectively, as in the firs case. The simulated results are given by Table 2.

Table 2

Parameter estimates, Bias and MSE values, when α = 4

α β λ
α β λ n Method Est. Bias MSE Est. Bias MSE Est. Bias MSE
4.00 0.50 0.50 30 ML 4.0241 0.0241 0.2554 0.5273 0.0273 0.0131 0.5177 0.0177 0.0037
LSq 4.0346 0.0346 0.2556 0.5320 0.0320 0.0187 0.5174 0.0174 0.0048
50 ML 4.0299 0.0299 0.2391 0.5143 0.0143 0.0074 0.5063 0.0063 0.0020
LSq 4.0325 0.0325 0.2399 0.5149 0.0149 0.0089 0.5068 0.0068 0.0025
100 ML 4.0130 0.0130 0.0475 0.5047 0.0047 0.0021 0.5020 0.0020 0.0007
LSq 4.0133 0.0133 0.0475 0.5045 0.0045 0.0024 0.5014 0.0014 0.0009
200 ML 4.0009 0.0009 0.0075 0.5029 0.0029 0.0010 0.5034 0.0034 0.0004
LSq 4.0012 0.0012 0.0075 0.5028 0.0028 0.0012 0.5032 0.0032 0.0005
1.00 30 ML 4.0532 0.0532 0.4908 0.5340 0.0340 0.0151 1.0276 0.0276 0.0128
LSq 4.0941 0.0941 0.4905 0.5595 0.0595 0.0304 1.0528 0.0528 0.0219
50 ML 4.0322 0.0322 0.2618 0.5182 0.0182 0.0091 1.0166 0.0166 0.0062
LSq 4.0597 0.0597 0.2865 0.5225 0.0225 0.0113 1.0234 0.0234 0.0092
100 ML 4.0120 0.0120 0.0412 0.5065 0.0065 0.0026 1.0092 0.0092 0.0026
LSq 4.0184 0.0184 0.0416 0.5089 0.0089 0.0041 1.0117 0.0117 0.0038
200 ML 4.0071 0.0071 0.0219 0.5044 0.0044 0.0018 1.0041 0.0041 0.0014
LSq 4.0083 0.0083 0.0219 0.5057 0.0057 0.0024 1.0030 0.0030 0.0021
2.00 30 ML 4.0844 0.0844 0.4919 0.5378 0.0378 0.0172 2.0478 0.0478 0.0348
LSq 4.1138 0.1138 0.4949 0.5550 0.0550 0.0272 2.0749 0.0749 0.0485
50 ML 4.0352 0.0352 0.1014 0.5319 0.0319 0.0099 2.0316 0.0316 0.0215
LSq 4.0524 0.0524 0.0986 0.5343 0.0343 0.0115 2.0414 0.0414 0.0281
100 ML 4.0074 0.0074 0.0515 0.5117 0.0117 0.0038 2.0078 0.0078 0.0066
LSq 4.0127 0.0127 0.0488 0.5132 0.0132 0.0044 2.0138 0.0138 0.0096
200 ML 3.9978 -0.0022 0.0147 0.5061 0.0061 0.0014 2.0008 0.0008 0.0036
LSq 3.9991 -0.0009 0.0148 0.5070 0.0070 0.0018 2.0014 0.0014 0.0049
1.00 0.50 30 ML 4.0057 0.0057 0.0471 1.0075 0.0075 0.0254 0.5048 0.0048 0.0020
LSq 4.0122 0.0122 0.0502 1.0173 0.0173 0.0379 0.5073 0.0073 0.0024
50 ML 4.0038 0.0038 0.0579 1.0181 0.0181 0.0167 0.5042 0.0042 0.0011
LSq 4.0060 0.0060 0.0585 1.0201 0.0201 0.0185 0.5052 0.0052 0.0012
100 ML 3.9978 -0.0022 0.0080 1.0080 0.0080 0.0054 0.5013 0.0013 0.0005
LSq 3.9983 -0.0017 0.0080 1.0083 0.0083 0.0057 0.5011 0.0011 0.0006
200 ML 4.0006 0.0006 0.0039 1.0016 0.0016 0.0022 0.5007 0.0007 0.0003
LSq 4.0007 0.0007 0.0039 1.0018 0.0018 0.0023 0.5007 0.0007 0.0003
1.00 30 ML 3.9814 -0.0186 0.0329 1.0426 0.0426 0.0724 1.0036 0.0036 0.0069
LSq 4.0030 0.0030 0.0849 1.0489 0.0489 0.1037 1.0116 0.0116 0.0088
50 ML 4.0021 0.0021 0.0313 1.0150 0.0150 0.0242 1.0089 0.0089 0.0042
LSq 4.0105 0.0105 0.0325 1.0157 0.0157 0.0294 1.0074 0.0074 0.0054
100 ML 3.9991 -0.0009 0.0323 1.0040 0.0040 0.0117 1.0027 0.0027 0.0020
LSq 4.0001 0.0001 0.0323 1.0044 0.0044 0.0124 1.0020 0.0020 0.0024
200 ML 3.9991 -0.0009 0.0036 1.0068 0.0068 0.0042 1.0014 0.0014 0.0011
LSq 3.9998 -0.0002 0.0037 1.0061 0.0061 0.0045 1.0012 0.0012 0.0012
2.00 30 ML 3.9995 -0.0005 0.0347 1.0524 0.0524 0.0644 2.0090 0.0090 0.0177
LSq 4.0183 0.0183 0.0349 1.0597 0.0597 0.0691 2.0163 0.0163 0.0255
50 ML 3.9938 -0.0062 0.0120 1.0406 0.0406 0.0519 2.0138 0.0138 0.0124
LSq 4.0082 0.0082 0.0144 1.0562 0.0562 0.0569 2.0202 0.0202 0.0212
100 ML 4.0014 0.0014 0.0039 1.0130 0.0130 0.0172 2.0056 0.0056 0.0045
LSq 4.0049 0.0049 0.0047 1.0161 0.0161 0.0196 2.0091 0.0091 0.0071
200 ML 3.9990 -0.0010 0.0020 1.0144 0.0144 0.0076 2.0044 0.0044 0.0020
LSq 4.0001 0.0001 0.0020 1.0155 0.0155 0.0088 2.0046 0.0046 0.0024
2.00 0.50 30 ML 4.0007 0.0007 0.0692 2.0598 0.0598 0.1088 0.5010 0.0010 0.0011
LSq 4.0007 0.0007 0.0692 2.0598 0.0598 0.1089 0.5012 0.0012 0.0011
50 ML 4.0149 0.0149 0.0601 2.0287 0.0287 0.0718 0.5033 0.0033 0.0007
LSq 4.0151 0.0151 0.0602 2.0293 0.0293 0.0719 0.5032 0.0032 0.0007
100 ML 4.0008 0.0008 0.0145 2.0182 0.0182 0.0212 0.5028 0.0028 0.0004
LSq 4.0008 0.0008 0.0145 2.0182 0.0182 0.0212 0.5028 0.0028 0.0004
200 ML 4.0008 0.0008 0.0179 2.0033 0.0033 0.0086 0.5003 0.0003 0.0002
LSq 4.0008 0.0008 0.0179 2.0033 0.0033 0.0086 0.5003 0.0003 0.0002
1.00 30 ML 4.0163 0.0163 0.2120 2.1333 0.1333 0.3086 1.0133 0.0133 0.0056
LSq 4.0181 0.0181 0.2113 2.1332 0.1332 0.3090 1.0149 0.0149 0.0062
50 ML 4.0004 0.0004 0.0454 2.0808 0.0808 0.1425 1.0084 0.0084 0.0025
LSq 4.0012 0.0012 0.0448 2.0807 0.0807 0.1424 1.0095 0.0095 0.0027
100 ML 3.9903 -0.0097 0.0398 2.0413 0.0413 0.0636 1.0036 0.0036 0.0014
LSq 3.9902 -0.0098 0.0398 2.0404 0.0404 0.0637 1.0043 0.0043 0.0014
200 ML 3.9978 -0.0022 0.0125 2.0100 0.0100 0.0137 1.0017 0.0017 0.0006
LSq 3.9979 -0.0021 0.0125 2.0103 0.0103 0.0137 1.0014 0.0014 0.0006
2.00 30 ML 4.0296 0.0296 0.1536 2.1908 0.1908 0.6677 2.0249 0.0249 0.0206
LSq 4.0440 0.0440 0.1859 2.1871 0.1871 0.6694 2.0276 0.0276 0.0219
50 ML 3.9930 -0.0070 0.0438 2.0927 0.0927 0.2209 2.0126 0.0126 0.0095
LSq 3.9937 -0.0063 0.0436 2.0908 0.0908 0.2225 2.0138 0.0138 0.0099
100 ML 3.9876 -0.0124 0.0250 2.0371 0.0371 0.0760 2.0054 0.0054 0.0046
LSq 3.9892 -0.0108 0.0221 2.0368 0.0368 0.0760 2.0061 0.0061 0.0046
200 ML 3.9912 -0.0088 0.0140 2.0192 0.0192 0.0337 2.0013 0.0013 0.0022
LSq 3.9915 -0.0085 0.0138 2.0194 0.0194 0.0337 2.0013 0.0013 0.0022

When the results given by Tables 1 and 2 are examined, it is seen that as the sample size n increases, both the estimations are close to actual values of the parameters and the ML and LSq estimators have smaller bias and MSE values for all cases. Furthermore, for both cases, it is concluded that the ML estimators outperform the LSq estimators with smaller MSE values according to the results given in Tables 1 and 2.

6 Application to Real Data

In this section, we present an analysis on a real-life data set called the coal mining disaster data set to illustrate the modeling behavior of the APGR distribution in comparison with Rayleigh and generalized Rayleigh distributions. The data set includes 191 observation dealing with the intervals in days between successive coal mining disasters in Great Britain [29].

Firstly, we investigate the underlying distribution of the data set. We apply the Kolmogorov-Smirnov (KS) test statistic to check whether this data set follows the APGR and most popular lifetime distributions such as Rayleigh, generalized Rayleigh, exponential, Gamma, Weibull, Log-Normal. The computed values of the KS statistic and corresponding p-values for each model are tabulated in Table 3.

Table 3

KS test results of the possible models for the coal mining disaster data set.

Model
APGR Rayleigh Generalized Rayleigh Exponential Weibull Gamma Log-Normal
KS 0.0370 0.4530 0.1531 0.1040 0.0464 0.0558 0.0742
p-value 0.9483 7.85E-35 2.36E-04 0.0340 0.7907 0.5753 0.2352

By Table 3, we can say that the underlying distribution of the coal mining disaster data set is compatible with the APGR, Gamma, Weibull and Log-Normal distributions.

Now, we apply the APGR, Gamma, Weibull and Log-Normal distributions as a model to coal mining disaster data set and obtain the negative log-likelihood (Neg. Log-Lik) and Akaike information criterion (AIC) values for deciding the optimal distribution model to this data set. The ML and LSq estimations of the parameters with the obtained AIC and Neg. Log-Lik values are summarized in Table 4.

Table 4

Model comparison and parameter estimates for the coal mining disaster data set.

Model
APGR Weibull Gamma Log-Normal
Neg. Log-Lik. 1197.6 1198.6 1201.4 1204.2
AIC 2401.1 2401.2 2406.8 2412.3
ML Estimations α 0.0045 αW 184.8301 αG 0.7211 μLN 4.5286
β 0.4134 θW 0.7928 βG 295.9621 σLN 1.4772
λ 0.0007

According to Table 4, it is concluded that the APGR distribution gives the better fit to the dataset than the Weibull, Gamma and Log-Normal distributions since it has smaller AIC and Neg. Log-Lik values. The data fitting performance of the APGR distribution can be clearly seen from Figure 2, which plots the ecdf and the cdf fitted by APGR distribution. As can be seen from Figure 2, the fitted cdf strongly follows the empirical cdf of the observations and this is the desired case in real-life applications.

Figure 2 For the coal mining disaster data set, empirical and the fitted cdf with APGR distribution.
Figure 2

For the coal mining disaster data set, empirical and the fitted cdf with APGR distribution.

7 Conclusion

In this study, a new life-time distribution named the APGR distribution is introduced. The pdf and cdf of the introduced distribution are derived using the APT method. The behavior of the pdf of APGR distribution is displayed in Figure 1 for different values of the model parameters. The expressions for basic characteristics of the APGR distribution such as hazard function, survival function, moments, characteristic function, skewness, kurtosis, order statistics, Shannon entropy, and stress-strength probability and Lorenz and Bonferroni curves are derived in the paper. Also, the estimators of the model parameters α, β and λ are obtained using two different methods the ML and LSq. The efficiencies of the ML and LSq estimators are also compared by comprehensive simulation studies on the different sample of sizes small, moderate and large. The simulation results show that the efficiencies of both estimators are quite satisfactory according to bias and MSE criteria for all sample sizes. Further, the ML and LSq estimators are asymptotically unbiased and consistent since, when the sample size increases, both bias and MSE values converge to zero.

The APGR distribution presents better fit to the coal mining disaster data than Gamma, Weibull and Log-Normal distributions, with the smaller Neg. Log-Lik. and AIC values. Thus, we can say that the APGR distribution provides the quite preferable modeling performance for life-time data and is a powerful alternative to the famous life-time distributions such as Gamma, Weibull and Log-Normal. Further, by information from real data application carried out using the coal mining disaster data set, it can be said that the APGR distribution has displayed more flexible data modeling performance than the baseline distributions Generalized Rayleigh and Rayleigh. Because while the APGR distribution is a suitable model for the coal mining disaster data set according to the obtained results of the KS test statistic given in Table 3, the Generalized Rayleigh and Rayleigh distributions aren’t appropriate models. Therefore, it can be said that the APGR distribution has capable of modeling more data types than the baseline distributions generalized Rayleigh and Rayleigh.

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A Appendix: Proof of Lemma 1

By using the power expansion formula, equation (11) can be written as

ξa,b,r,L,δ=∫0∞xr+1e−Lx21−e−Lx2b−1a1−e−Lx2beδxdx=∑i=0∞log⁡aii!∫0∞xr+1e−Lx21−e−Lx2b−11−e−Lx2bieδxdx=∑i=0∞∑j=0ibibj−1jlog⁡aii!∫0∞xr+1e−Lx21−e−Lx2b−1e−Lx2jeδxdx=∑i=0∞∑j=0ib∑k=0b−1ibjb−1k−1j−1klog⁡aii!∫0∞xr+1e−Lx2e−Lx2ke−Lx2jeδxdx=∑i=0∞∑j=0ib∑k=0b−1ibjb−1k−1j−1klog⁡aii!∫0∞xr+1e−Lx2e−kLx2e−jLx2eδxdx=∑i=0∞∑j=0ib∑k=0b−1ibjb−1k−1k+jlog⁡aii!∫0∞xr+1e−Lx2k+j+1eδxdx (40)

and by applying the gamma function in the last equation, we have

ξa,b,L,r,δ=∑i=0∞∑j=0ib∑k=0b−1log⁡aii!−1j+kibjb−1k12L2(j+k+1)12(−r−3)×δΓr+321F1r+32;32;δ24(j+k+1)L2+Γr2+1L2(j+k+1)1F1r+22;12;δ24(j+k+1)L2 (41)

B Appendix: Calculation of the Rényi entropy of the APGR distribution

The Rényi entropy of the APGR distribution is

REXξ=11−ξln⁡∫0∞fxξdx=11−ξln⁡∫0∞ln⁡αα−12βλ2xe−λx21−e−λx2β−1α1−e−λx2βξdx=11−ξln2βλ2ln⁡αα−1ξ∫0∞xe−λx21−e−λx2β−1α1−e−λx2βξdx. (42)

Applying the power expansion formula, the equation (42) is written as

REXξ=11−ξln2βλ2ln⁡αα−1ξ∫0∞xe−λx21−e−λx2β−1α1−e−λx2βξdx=11−ξln2βλ2ln⁡αα−1ξ∫0∞xe−λx21−e−λx2β−1∑i=0∞ln⁡αii!1−e−λx2iβξdx=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∫0∞xe−λx21−e−λx2β−11−e−λx2iβξdx=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∫0∞xe−λx21−e−λx2β−1+iβξdx=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∫0∞xe−λx2∑j=0β−1+iββ−1+iβj−1je−λx2jξdx=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∑j=0β−1+iββ−1+iβj−1j∫0∞xe−λx2e−λx2jξdx=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∑j=0β−1+iββ−1+iβj−1j∫0∞xξe−ξj+1λx2dx (43)

By applying the gamma function to equation (43), we have

REXξ=11−ξln2βλ2ln⁡αα−1ξ∑i=0∞ln⁡αii!∑j=0iβ+β−1iβ+β−1j−1j12Γξ+12(j+1)λ2ξ12(−ξ−1) (44)
Received: 2018-11-25
Accepted: 2019-05-16
Published Online: 2019-07-24

© 2019 Biçer, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

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  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
  10. A new smoothing method for solving nonlinear complementarity problems
  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
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  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
  45. On Diophantine approximation by unlike powers of primes
  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
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  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
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  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
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  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
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  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
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  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
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https://doi.org/10.1515/math-2019-0057
Keywords for this article
Alpha power transformation; Maximum likelihood estimate; Least-square estimate; Shannon entropy; Generalized Rayleigh distribution
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
  10. A new smoothing method for solving nonlinear complementarity problems
  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
  22. On split Lie color triple systems
  23. Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
  45. On Diophantine approximation by unlike powers of primes
  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
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