The reliability is a well-known in engineering where it gives the probability for surviving at least time t of a product operate, while the hazard function shows the nature of failure rate related to the product. Generally, the reliability and hazard functions are fundamental to study the characteristics of the time to event data. Figs. 2 and 3 are the plots of the hazard, reliability reversed hazard, and the odds functions of the ILBMD for α=0.1,0.2,0.3,0.4,0.5 .

Hazard rate function: The hazard rate (HR) function is a very important property in characterizing any lifetime distribution. The HR of the ILBMD is given by

(7) HILBMD(x;α)=fILBMD(x;α)1−FILBMD(x;α)=12α4x5e−12α2x21−(1+12α2x2)e−12α2x2.

To determine the shape of the HR function we followed the technique of [17] which is defined as Φ(x;α)=−f/(x;α)f(x;α) which only depends on the pdf of the distribution. He proved that if Φ/(x)>0 for all x∈(0,x0) , while Φ/(x0)=0 , and Φ/(x)<0 for all x∈(x0,∞) , the distribution has upside down bathtub hazard rate (UBT). For the ILBMD we have Φ(x;α)=5x−1x3α2 and Φ/(x;α)=3α2x4−5x2 . Now, it is found that Φ(3/5α;α)=0 , that is x0=3/5α , and hence for the ILBMD we have that Φ(x;α) is increasing on the interval (0,3/5α) and it is decreasing on the interval (3/5α,∞) as illustrated in Fig. 2A. Therefore, the proposed ILBMD is useful in reliability data and medical fields due its skewness to the right with UBT shape of hazard rate function.

Reliability function: The reliability function of the ILBMD distribution is

(8) RILBMD(x;α)=1−FILBMD(x;α)=1−(1+12α2x2)e−12α2x2

Fig. 2B shows that the reliability plots of the ILBMD intersect at the point f(x)=1 for x=0 , while as x goes to infinity, the reliability function decreases and goes to zero.

Reversed hazard function: The reversed hazard function of the ILBMD is defined as

(9) RHILBMD(x;α)=fILBMD(x;α)FILBMD(x;α)=12α2x2+1

Odds function: The odds function of the ILBMD is given by

(10) OILBMD(x;α)=FILBMD(x;α)1−FILBMD(x;α)=2α2x2+12α2x2e12α2x2−2α2x2−1.

Based on Fig. 3 it can be noted that the reversed hazard decreases with negative J-shaped distribution, while the odds function increases taking the J-shaped with more flatting for small amounts of the parameter α .

In this section, the mode of the ILBMD is derived. Since the pdf fILBMD(x;α) of the model and its logarithm are maximized at the same point, then for simple calculation, take the derivative of the logarithm of the function fILBMD(x;α) as

Υ=ln⁡fILBMD(x;α)=−ln⁡(2α4x5)−12α21x2.

The derivative of Υ with respect to x yields ∂Υ∂x=−5x+12α22x3. Equating the preceding derivative to 0 leads to −5x+12α22x3=0 and xMode=±15α. But since α>0 , then the mode of the ILBMD is xMode=15α. It is clear that the distribution is a unimodal and the mode decreases with increases values of α.

In this section, we derived the various moments of the suggested ILBMD as

Let X∼fILBMD(x;α) , then the rth moment of X is

(11) E(XILBMDr)=Γ(2−r2)2r2αr,α>0,r=1,2,3,...

If X∼fILBMD(x;α) , then the moment generating function of X is

(12) MILBMD(t)=t4log⁡[e]4MeijerG[{{},{}},{{−2,−32,0},{}},t2log⁡[e]28α2]64πα4,α>0,

where MeijerG((a1…anan+1…ap),(b1…bmbm+1…bq),z) is the Meijer G function Gpqmn(z|a1…apb1…bq) .

From Eq. (11), the first and second moments of the ILBMD, respectively, are given as

E(XILBMD1)=12απ2 and E(XILBMD2)=12α2 .

The variance of the ILBMD is

(13) Var(XILBMD)=E(XILBMD2)−(E(XILBMD1))2=12α2−(12απ2)2=4−π8α2.

The degree of long-tail is measured by skewness (Sk) and for the ILBMD it is given by

(14) SkILBMD=2(π−2)π(−1π−4)3/2=5.08834.

The coefficient of variation of the ILBMD is

(15) CvILBMD=22π12α2−π8α2α4(α2)3/2=4π−1=0.522723,

which is a very small value.

Let X∼fILBMD(x;α) , then the rth order inverse moment about the origin of X is

(16) E(1XILBMDr)=2r2αrΓ(2+r2),r=1,2,3,...

Based on Eq. (6), the harmonic mean of the ILBMD distribution can be obtained for r=1 as given by

(17) E(1XILBMD)=32π2α.

If Q(k) is the quantile function of order k of the ILBM random variable, then it can be the solution of the equation

(18) ln⁡k+12α2Q2(k)=ln⁡(1+12α2Q2(k)).

The Moors and Bowley measures of kurtosis and skewness, respectively, are given by

(19) Mk=Q(78)−Q(58)+Q(38)−Q(18)Q(68)−Q(28)andBsk=Q(34)−2Q(12)+Q(14)Q(34)−Q(14).

Theorem: Let X∼fILBMD(x;α) , then the Fisher’s information of α is FIILBMD(α)=8α2.

Proof: To find the Fisher’s information of the ILBMD, we have

ln⁡fILBMD(x;α)=−5log⁡[x]+log⁡[12α4]−12α2x2.

The first derivative of this function with respect to α yields

∂ln⁡fILBMD(x;α)∂α=2e12x2α2x5(e−12x2α22x7α7−2e−12x2α2x5α5)α4.

Again differentiate the last equation with respect to α to get

∂2ln⁡fILBMD(x;α)∂α2=8α3x5e12α2x2(e−12α2x22α7x7−2e−12α2x2α5x5)−2αx3e12α2x2(e−12α2x22α7x7−2e−12α2x2α5x5)+2e12x2α2x5(e−12x2α22x9α10−11e−12x2α22x7α8+10e−12x2α2x5α6)α4.

Now, take the expectation of ∂2ln⁡fILBMD(x;α)∂α2 as

−E(∂2ln⁡fILBMD(x;α)∂α2)=−∫0∞∂2ln⁡fILBMD(x;α)∂α212α4x5e−12α2x2dx=8α2.

This information is very helpful in determining the variance of estimator or lower bound of an estimator.

Let X(1:n),X(2:n),...,X(n:n) be the order statistics of the random sample X1,X2,...,Xn selected from a pdf and cdf fILBMD(x;α) and FILBMD(x;α) , respectively. The pdf of the ith order statistics say X(i:n) , is

(20) f(i:n)(x)=n!(i−1)!(n−i)![F(x)]i−1[1−F(x)]n−if(x)=n!(i−1)!(n−i)!∑k=0n−i(−1)k(n−ik)12α4x5(1+12α2x2)k+i−1(e−12α2x2)k+i,

and the corresponding cdf is defined as

(21) F(i:n)(x)=∑j=tn(nj)F(x)]j[1−F(x)]n−j=∑j=tn∑k=0n−i(−1)k(nj)(n−ik)[(1+12α2x2)e−12α2x2]j+k.

The stochastic ordering can be considered to compare the behavior of two random variables. Let X and Y be two random variables, then X is said to be smaller than in

1) Mean residual life order (X≤mrl⁡Y) if mX(x)≤mY(x) for all x;

2) Likelihood ratio order (X≤lr⁡Y) if fX(x)fY(x) decreases in x;

3) Hazard rate order (X≤hr⁡Y) if hX(x)≥hY(x) for all x;

4) Stochastic order (X≤st⁡Y) if FX(x)≥FY(x) for all x.

Based on these relations, we have (X≤lr⁡Y)⇒(X≤hr⁡Y)⇓(X≤st⁡Y)⇒(X≤mrl⁡Y) .

Theorem 2: Let XILBMD∼fX(x;α) and YILBMD∼fY(x;θ) . If θ<α , then (X≤lr⁡Y) , and hence (X≤hr⁡Y) , (X≤mrl⁡Y) and (X≤st⁡Y) .

Proof: Based on the concept of the likelihood ratio order, we have

fX(x;α)fY(x;θ)=12α4x5e−12α2x212θ4x5e−12θ2x2=2θ4x52α4x5e−(12α2x2−12θ2x2)=θ4α4e−12x2(1α2−1θ2) ,

where its logarithm is

lnfX(x;α)fY(x;θ)=ln⁡[θ4α4e−12x2(1α2−1θ2)]=ln⁡(θ4α4)−12x2(1α2−1θ2) .

The first derivative of this equation with respect to x is

∂∂xlnfX(x;α)fY(x;θ)=1x3(1α2−1θ2)=1x3(θ2−α2α2θ2)=1x3((θ−α)(θ+α)α2θ2) .

Now, if θ<α , then ∂∂xln⁡(fX(x;α)fY(x;θ))<0 , and hence (X≤lr⁡Y) , and the other relations are holds, i.e., (X≤hr⁡Y) , (X≤mrl⁡Y) and (X≤st⁡Y) .

This section, introduced the mean and median deviations of the ILBMD, Φμ and ΦM , respectively. It is a measure of the scatter in the population the mean deviation about the mean and the mean deviation about the median, where

Φμ=2μF(μ)−2∫0μxf(x)dx and ΦM=μ−2∫0Mxf(x)dx ,

where μ and M are the population mean and median, respectively.

Theorem: Let X∼fILBMD(x;α), the mean and median deviations about the mean and median, respectively, are

(22) Φμ=e−4/π2πα(π−e4/ππErfc[2π])=0.212229α,

where Erfc is the complementary function of the error function erfc, and

(23) ΦM=2πα2M−2(2αe−12M2α2+M2πα2Erfc[12Mα])4α3M.

Since the median of the ILBMD is M=0.545813α , then ΦM=0.200744α .

The qth quantile x_q of the ILBMD can be found by solving the equation q=FILBMD(xq;α) , that is

q=(1+12α2xq2)e−12α2xq2 and the quantile is the solution of the ln⁡(q)=ln⁡(1+12α2xq2)−12α2xq2 .

Let X be a non-negative random variable with a continuous twice differentiable cumulative distribution function F(x) . In this section, we want to obtain the Gini index, Bonferroni and Lorenz curves for the ILBMD. The Gini coefficient is developed by the Italian statistician Gin in (1912). The Gini index measures the inequality among values of a frequency distribution, for example levels of total income. The Gini index value running from zero to one. A Gini index of zero value indicates perfect equality (that is all values are the same, a group has the same monthly income), while most unequal group where a single person receives Gini index of 1 of the total.

The Gini index for the ILBMD is given by

(24) G(x)=1−1μ∫0∞⁡(1−FILBMD(x;α))2dx=0.237437.

It is clear that the Gini index value is small and it is about 0.24. The Bonferroni curve for the ILBMD is defined as

(25) B(p)=1pμ∫0q⁡xfILBMD(x;α)dx=e−12q2α22π+qαErfc[12qα]pqα,p>0,q>0,α>0,

q=F−1(p) and p∈(0,1] .

The Lorenz curve for the ILBMD is defined as

(26) L(x)=1μ∫0q⁡xfILBMD(x;α)dx=e−12q2α22π+qαErfc[12qα]qα.

Let X and Y be independent random variables observed from the pdf f(x) . The stress-strength reliability clarify the life of a component that has a random strength Y which is subjected to a random stress X, where

(27) R=P(Y<X)=∫0∞P(Y<X|X=x)f(x)dx=∫0∞f(x;α)F(x;δ)dx.

Theorem: Let the random variables X and Y be independent selected from the ILBMD. The stress-strength reliability is given by

(28) RILBMD(α,ω)=ω4(3α2+ω2)(α2+ω2)3,1α2+1ω2>0.

The Rényi entropy is defined as

RE(η)=11−ηlog⁡(∫0∞f(x)ηdx), where η>0 and η≠1 .

Let X∼fILBMD(x;α) , then the and Rényi entropy of X is defined as

(29) RE(η)=11−ηlog⁡[(18α)(1−η)η12(1−5η)Γ[5η2−12]],ηα2>0,η>15.

The q-entropy, say QE(q) is given by QE(q)=1q−1log⁡(1−∫−∞∞f(x)qdx),q>0,q≠1 . For the ILBMD we have

(30) QE(q)=1q−1log⁡{1−(18α)(1−q)q12(1−5q)Γ[5q2−12]}.

Let X1,X2,...,Xn be a random sample of size n selected from the ILBMD with parameter α>0 . The maximum likelihood estimator for the ILBMD parameter can be derived based on the likelihood function as:

LILBMD(α)=∏i=1nf(xi,α)=∏i=1n12α4xi5e−12α2xi2=(12α4)n∏i=1n1xi5∏i=1ne−12α2xi2 .

The log likelihood function is given by

Ψ=lnL(θ)=ln((12α4)n∏i=1n1xi5∏i=1ne−12α2∑i=1nxi2)=−nln(2α4)+ln(∏i=1n1xi5)+12α2∑i=1nxi2.

The derivative of Ψ with respect to α is Ψ′=−4nα+∑i=1n1xi2α3. Setting the last equation to zero to get the MLE of α is α^MLE=±∑i=1n1xi22n , and since x>0 , then α^MLE=∑i=1n1xi22n .

The mean of the ILBMD random variable is E(XILBMD1)=12απ2 and X¯=∑i=1nXin. Let E(XILBMD1)=X¯ to get the MOM estimator of α is α^MOM=12απ2 .

Let x1,x2,...,xn be the observed values of a random sample of size n selected from a the fILBMD(x;α) , and let x(1:n),x(2:n),...,x(n:n) be the order statistics of the sample. The Cramèr-von Mises estimation method (Cv) is suggested by Swain et al. [18]. The Cv method is based on the minimum difference between the cumulative and empirical distribution functions. The Cv estimator of the parameter can be calculated by minimizing

Cv(α)=112n+∑i=1n(F(x(i:n),α)−2i−12n)2.

For the proposed ILBMD, the Cv estimator, α^Cv of α , can be obtained by minimizing the equation

(31) Cv(α)=112n+∑i=1n[(1+12α2x(i:n)2)e−12α2x(i:n)2−2i−12n]2,

with respect to α .

Let X(1:n),X(2:n),...,X(n:n) be the order statistics of the random sample X1,X2,...,Xn selected from a the fILBMD(x;α) . The least square estimator (LSE) [19] can be obtained by minimizing the residual sum of the square, which is defined as the differences of theoretical cdf and empirical cdf as

LS(α)=∑i=1n(F(x(i:n),α)−in+1)2.

For the ILBMD, the LS estimator, α^LS of α can be obtained by minimizing the equation

(32) LS(α)=∑i=1n[(1+12α2x(i:n)2)e−12α2x(i:n)2−in+1]2,

with respect to α . Similarly, the weighted least squares (WLS) estimate of α denoted by α^WLS can be obtained by minimizing the function

WLS(α)=∑i=1n(n+2)(n+1)2i(n−i+1)(F(x(i:n),α)−in+1)2,

with respect to α . For the ILBMD we have

(33) WLS(α)=∑i=1n(n+2)(n+1)2i(n−i+1)[(1+12α2x(i:n)2)e−12α2x(i:n)2−in+1]2,

with respect to α .

The maximum product of spacing (MPS) method as suggested by Cheng et al. [20,21] is a powerful alternative to the MLE method for estimating the parameters of continuous distributions. Reference Shanker et al. [22] showed that the MPS method possess similar properties as the MLE method.

Let uniform spacing’s of a random sample of size n uniform spacing’s is given as be

Di(α)=F(x(i:n)|α)−F(x(i−1:n)|α),i=1,2,...,n,

where F(x(0:n)|α)=0 , F(x(n+1:n)|α)=1 and ∑i=1n+1Di(α)=1. The MPS can be calculated by maximizing the geometric mean (GM) of the spacing’s defined as

GM(α)=(∏i=1n+1Di(α))1n+1 ,

Or, equivalently, by maximizing the function S(α)=1n+1∑i=1n+1log⁡Di(α) , with respect to α .

Now, the MPS estimate of the ILBMD parameter α denoted by α^MPS can be obtained by maximizing the equation

(34) GM(α)=∏i=1n+1[(1+12α2x(i:n)2)e−12α2x(i:n)2−(1+12α2x(i−1:n)2)e−12α2x(i−1:n)2]n+1.