CSSE CSSE CSSE Computer Systems Science & Engineering 0267-6192 Tech Science Press USA 17362 10.32604/csse.2021.017362 Article Inverse Length Biased Maxwell Distribution: Statistical Inference with an ApplicationInverse Length Biased Maxwell Distribution: Statistical Inference with an ApplicationInverse Length Biased Maxwell Distribution: Statistical Inference with an Application Al-Omari Amer Ibrahim 1 Alanzi Ayed R.A. 2 a.alanzi@mu.edu.sa Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan Department of Mathematics, College of Science and Human Studies at Hotat Sudair, Majmaah University, Majmaah 11952, Saudi Arabia *Corresponding Author: Ayed R. A. Alanzi. Email: a.alanzi@mu.edu.sa 02 6 2021 39 1 147 164 28 1 2021 14 3 2021 © 2021 Al-Omari and Alanzi 2021 Al-Omari and Alanzi This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we suggested and studied the inverse length biased Maxell distribution (ILBMD) as a new continuous distribution of one parameter. The ILBMD is obtained by considering the inverse transformation technique of the Maxwell length biased distribution. Statistical characteristics of the ILBMD such as the moments, moment generating function, mode, quantile function, the coefficient of variation, coefficient of skewness, Moors and Bowley measures of kurtosis and skewness , stochastic ordering, stress-strength reliability, and mean deviations are obtained. In addition, the Bonferroni and Lorenz curves, Gini index, the reliability function, the hazard rate function, the reverse hazard rate function, the odds function, and the distributions of order statistics for the ILBMD, are presented. The ILBMD parameter is estimated using the maximum likelihood method, the method of moments, the maximum product of spacing technique, the ordinary and weight least square procedures, and the Cramer-Von-Mises methods. The Fishers information, as well as the Rényi and q-entropies, are derived. To investigate the usefulness of the proposed lifetime distribution and to illustrate the purpose of the study, a real dataset of the relief times of 20 patients receiving an analgesic is used.

Maxell distribution inverse length biased Maxwell distribution Fisher’s information methods of estimation goodness of fit tests
Introduction

A random variable W follows a Maxwell distribution with scale parameter α , if its probability density function (pdf) and cumulative distribution function (cdf), respectively, are given by

fMD(w;α)=2πw2α3ew22α2,0<w<,α>0,

FMD(w;α)=Erf(wα2)wα2πew22α2,

where Erf(z)=2π0zet2dt . In the literature of the probability distributions, uni-modal and skewed to the right characteristics are very important to the distribution of interest. One of these distributions is the Maxwell distribution, which is a well-known lifetime distribution in physics and statistical mechanics. Reference Iriarte et al.  suggested a gamma-Maxwell distribution. The tail behavior of the generalized Maxwell distribution is considered by Huang et al. . Recently, Saghir et al.  suggested a length-biased Maxwell distribution (LBMD) as a modification of the base Maxwell distribution based on the weighted distribution suggested by Rao et al. , to obtain the probability density function given by:

fLBMD(w;α)=w32α4ew22α2,0<w<,α>0,

and a cumulative distribution function defined as

FLBMD(w;α)=1(w22α2+1)ew22α2,0<w<,α>0.

The mode and median of the LBMD are wM=α3 and E(W)=3α42π , respectively. For more information about the LBMD see . Due to the large number of data in these times, a large number of distributions are suggested based several philosophies, assuming that the suggested distributions are more flexible in modeling data. For example,  introduced Marshall–Olkin length-biased Maxwell distribution. Reference Singh et al.  suggested length-biased weighted Maxwell distribution and  considered estimation of the inverse Maxwell distribution parameter. Reference Garaibah et al.  suggested size-biased Ishita distribution and  introduced transmuted Ishita distribution. The Marshall-Olkin length-biased exponential distribution is proposed by Shraa et al. . A new mixture continuous Darna distribution is suggested by Al-Omari et al. [11,12] proposed length-biased Suja distribution. Reference Sharma et al.  studied the power size biased two-parameter Akash distribution with some statistical properties and real data applications. Reference Al-Omar et al.  proposed length and area biased Maxwell distributions. Reference Gharaibeh  suggested Top-Leone Mukherjee-Islam distribution and  proposed transmuted Aradhana distribution.

The rest of this paper is organized as follows: In Section 2, we present the derivation of the suggested distribution. Section 3 deals with the main statistical properties of the ILBMD. Different methods of estimation for the distribution parameter are given in Section 4. In Section 5, a simulation study is conducted to investigate the distribution. An application of real data is presented in Section 6 and the paper is concluded in Section 7.

Derivation of the Suggested Model

If a random variable W has a LBMD with pdf given in (3), then the random variable X=1W is said to follow the inverse LBMD. The pdf and cdf of the inverse length-biased Maxwell distribution (ILBMD), respectively are given by

fILBMD(x;α)=12α4x5e12α2x2,0<x<,α>0,

and

FILBMD(x;α)=(1+12α2x2)e12α2x2,0<x<,α>0.

Plots of the pdf and cdf of the ILBMD are presented in Fig. 1 for various distribution parameter. Fig. 1, revealed that the pdf of the suggested distribution is skewed to the right and be more flatting as α values are increasing. Also, the pdf of the ILBMD can exhibit various behavior depending on the values of the parameter.

The ILBMD pdf and cdf plots for <inline-formula id="ieqn-7"> <!--<alternatives><inline-graphic xlink:href="ieqn-7.tif"/><tex-math id="tex-ieqn-7"><![CDATA[\alpha = 0.1,\,0.2,0.3,0.4,0.5]]></tex-math>--><mml:math id="mml-ieqn-7"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="thinmathspace"></mml:mspace><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.3</mml:mn><mml:mo>,</mml:mo><mml:mn>0.4</mml:mn><mml:mo>,</mml:mo><mml:mn>0.5</mml:mn></mml:math> <!--</alternatives>--></inline-formula>
Statistical Properties

In this section, the main properties if the proposed model are presented.

Reliability Analysis

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

HILBMD(x;α)=fILBMD(x;α)1FILBMD(x;α)=12α4x5e12α2x21(1+12α2x2)e12α2x2.

The hazard (A) and reliability (B) functions of the ILBMD plots for <inline-formula id="ieqn-24"> <!--<alternatives><inline-graphic xlink:href="ieqn-24.tif"/><tex-math id="tex-ieqn-24"><![CDATA[\alpha = 0.1,\,0.2,0.3,0.4,0.5]]></tex-math>--><mml:math id="mml-ieqn-24"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="thinmathspace"></mml:mspace><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.3</mml:mn><mml:mo>,</mml:mo><mml:mn>0.4</mml:mn><mml:mo>,</mml:mo><mml:mn>0.5</mml:mn></mml:math> <!--</alternatives>--></inline-formula> The reverse hazard (A) and odds (B) functions of the ILBMD plots for <inline-formula id="ieqn-25"> <!--<alternatives><inline-graphic xlink:href="ieqn-25.tif"/><tex-math id="tex-ieqn-25"><![CDATA[\alpha = 0.1,\,0.2,0.3,0.4,0.5]]></tex-math>--><mml:math id="mml-ieqn-25"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="thinmathspace"></mml:mspace><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.3</mml:mn><mml:mo>,</mml:mo><mml:mn>0.4</mml:mn><mml:mo>,</mml:mo><mml:mn>0.5</mml:mn></mml:math> <!--</alternatives>--></inline-formula>

To determine the shape of the HR function we followed the technique of  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)=5x1x3α2 and Φ/(x;α)=3α2x45x2 . 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

RILBMD(x;α)=1FILBMD(x;α)=1(1+12α2x2)e12α2x2

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

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

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

OILBMD(x;α)=FILBMD(x;α)1FILBMD(x;α)=2α2x2+12α2x2e12α2x22α2x21.

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 .

The Mode

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=lnfILBMD(x;α)=ln(2α4x5)12α21x2.

The derivative of Υ with respect to x yields Υx=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.

Moments and Quantile Function

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

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

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

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

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

where MeijerG((a1anan+1ap),(b1bmbm+1bq),z) is the Meijer G function Gpqmn(z|a1apb1bq) .

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

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

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

SkILBMD=2(π2)π(1π4)3/2=5.08834.

The coefficient of variation of the ILBMD is

CvILBMD=22π12α2π8α2α4(α2)3/2=4π1=0.522723,

which is a very small value.

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

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

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

lnk+12α2Q2(k)=ln(1+12α2Q2(k)).

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

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

Fishers Information

Theorem: Let XfILBMD(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

lnfILBMD(x;α)=5log[x]+log[12α4]12α2x2.

The first derivative of this function with respect to α yields

lnfILBMD(x;α)α=2e12x2α2x5(e12x2α22x7α72e12x2α2x5α5)α4.

Again differentiate the last equation with respect to α to get

2lnfILBMD(x;α)α2=8α3x5e12α2x2(e12α2x22α7x72e12α2x2α5x5)2αx3e12α2x2(e12α2x22α7x72e12α2x2α5x5)+2e12x2α2x5(e12x2α22x9α1011e12x2α22x7α8+10e12x2α2x5α6)α4.

Now, take the expectation of 2lnfILBMD(x;α)α2 as

E(2lnfILBMD(x;α)α2)=02lnfILBMD(x;α)α212α4x5e12α2x2dx=8α2.

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

Order Statistics

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

f(i:n)(x)=n!(i1)!(ni)![F(x)]i1[1F(x)]nif(x)=n!(i1)!(ni)!k=0ni(1)k(nik)12α4x5(1+12α2x2)k+i1(e12α2x2)k+i,

and the corresponding cdf is defined as

F(i:n)(x)=j=tn(nj)F(x)]j[1F(x)]nj=j=tnk=0ni(1)k(nj)(nik)[(1+12α2x2)e12α2x2]j+k.

Stochastic Ordering

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 (XmrlY) if mX(x)mY(x) for all x;

2) Likelihood ratio order (XlrY) if fX(x)fY(x) decreases in x;

3) Hazard rate order (XhrY) if hX(x)hY(x) for all x;

4) Stochastic order (XstY) if FX(x)FY(x) for all x.

Based on these relations, we have (XlrY)(XhrY)(XstY)(XmrlY) .

Theorem 2: Let XILBMDfX(x;α) and YILBMDfY(x;θ) . If θ<α , then (XlrY) , and hence (XhrY) , (XmrlY) and (XstY) .

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

fX(x;α)fY(x;θ)=12α4x5e12α2x212θ4x5e12θ2x2=2θ4x52α4x5e(12α2x212θ2x2)=θ4α4e12x2(1α21θ2) ,

where its logarithm is

lnfX(x;α)fY(x;θ)=ln[θ4α4e12x2(1α21θ2)]=ln(θ4α4)12x2(1α21θ2) .

The first derivative of this equation with respect to x is

xlnfX(x;α)fY(x;θ)=1x3(1α21θ2)=1x3(θ2α2α2θ2)=1x3((θα)(θ+α)α2θ2) .

Now, if θ<α , then xln(fX(x;α)fY(x;θ))<0 , and hence (XlrY) , and the other relations are holds, i.e., (XhrY) , (XmrlY) and (XstY) .

Mean and Median Deviations

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)20μxf(x)dx and ΦM=μ20Mxf(x)dx ,

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

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

Φμ=e4/π2πα(πe4/ππErfc[2π])=0.212229α,

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

ΦM=2πα2M2(2αe12M2α2+M2πα2Erfc[12Mα])4α3M.

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

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

q=(1+12α2xq2)e12α2xq2 and the quantile is the solution of the ln(q)=ln(1+12α2xq2)12α2xq2 .

Gini Index and Some Curves

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

G(x)=11μ0(1FILBMD(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

B(p)=1pμ0qxfILBMD(x;α)dx=e12q2α22π+qαErfc[12qα]pqα,p>0,q>0,α>0,

q=F1(p) and p(0,1] .

The Lorenz curve for the ILBMD is defined as

L(x)=1μ0qxfILBMD(x;α)dx=e12q2α22π+qαErfc[12qα]qα.

Stress-Strength Reliability

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

R=P(Y<X)=0P(Y<X|X=x)f(x)dx=0f(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

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

Entropies

The Rényi entropy is defined as

RE(η)=11ηlog(0f(x)ηdx), where η>0 and η1 .

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

RE(η)=11ηlog[(18α)(1η)η12(15η)Γ[5η212]],ηα2>0,η>15.

The q-entropy, say QE(q) is given by QE(q)=1q1log(1f(x)qdx),q>0,q1 . For the ILBMD we have

QE(q)=1q1log{1(18α)(1q)q12(15q)Γ[5q212]}.

Different Methods of Estimation

In this section, we discuss different estimation procedures for estimating the unknown suggested model parameter.

Maximum Likelihood Method

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=1n12α4xi5e12α2xi2=(12α4)ni=1n1xi5i=1ne12α2xi2 .

The log likelihood function is given by

Ψ=lnL(θ)=ln((12α4)ni=1n1xi5i=1ne12α2i=1nxi2)=nln(2α4)+ln(i=1n1xi5)+12α2i=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 .

Method of Moments

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 .

Cramèr–von-Mises Estimation

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. . 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)2i12n)2.

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

Cv(α)=112n+i=1n[(1+12α2x(i:n)2)e12α2x(i:n)22i12n]2,

with respect to α .

Ordinary and Weighted Least Squares Methods

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)  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

LS(α)=i=1n[(1+12α2x(i:n)2)e12α2x(i:n)2in+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(ni+1)(F(x(i:n),α)in+1)2,

with respect to α . For the ILBMD we have

WLS(α)=i=1n(n+2)(n+1)2i(ni+1)[(1+12α2x(i:n)2)e12α2x(i:n)2in+1]2,

with respect to α .

Method of Maximum Product of Spacing

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.  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(i1: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+1i=1n+1logDi(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

GM(α)=i=1n+1[(1+12α2x(i:n)2)e12α2x(i:n)2(1+12α2x(i1:n)2)e12α2x(i1:n)2]n+1.

Simulation Study

In this section, we compared the various suggested estimators of the model parameters. We selected the values of the parameters α=1, 2,3 with samples sizes n=10, 20,40,60, 80, 100 and 200. The results are presented in Tabs. 1 and 2 foe the parameter estimates (Es) and the corresponding mean squared errors (MSE).

Estimates and MSEs with MLE, CV, and MOM methods for the ILBMD with <inline-formula id="ieqn-142"> <!--<alternatives><inline-graphic xlink:href="ieqn-142.tif"/><tex-math id="tex-ieqn-142"><![CDATA[\alpha = 1,]]></tex-math>--><mml:math id="mml-ieqn-142"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math> <!--</alternatives>--></inline-formula>2,3 and <inline-formula id="ieqn-143"> <!--<alternatives><inline-graphic xlink:href="ieqn-143.tif"/><tex-math id="tex-ieqn-143"><![CDATA[n = 10,]]></tex-math>--><mml:math id="mml-ieqn-143"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>,</mml:mo></mml:math> <!--</alternatives>--></inline-formula>20,40, 60, 80, 100, 200, 400
MLECV
n α Es MSE Es MSE Es MSE
10 1 0.995924 0.011847 0.991184 0.012989 0.980571 0.013476
2 1.981047 0.046646 1.971264 0.051318 1.973994 0.053759
3 2.977570 0.103597 2.954931 0.134482 2.954117 0.124594
20 1 0.997368 0.005989 0.989289 0.007110 0.991652 0.005833
2 1.998237 0.025302 1.982299 0.025994 1.978287 0.027209
3 3.001566 0.054498 2.963013 0.064315 2.970309 0.059139
40 1 0.999047 0.003172 0.991716 0.003276 0.991394 0.003234
2 1.998907 0.012212 1.993159 0.013333 1.983139 0.012917
3 2.992754 0.026943 2.971542 0.032232 2.982549 0.030891
60 1 0.997824 0.002377 0.994213 0.002429 0.996004 0.002219
2 2.002695 0.008112 1.991705 0.009028 1.985409 0.008619
3 2.990985 0.018646 2.982247 0.020553 2.980146 0.019963
80 1 0.999361 0.001616 0.997906 0.001802 0.996385 0.001648
2 1.997114 0.006034 1.994376 0.006188 1.993271 0.006873
3 2.997377 0.013123 2.994897 0.014442 2.986911 0.014781
100 1 1.000008 0.001280 0.997574 0.001243 0.997230 0.001419
2 1.995360 0.004856 1.998231 0.005106 1.993223 0.005391
3 2.998770 0.010524 2.985528 0.012388 2.988051 0.010962
200 1 0.998602 0.000652 0.996814 0.000705 0.998669 0.000628
2 2.002684 0.002847 1.995579 0.002652 1.996316 0.002650
3 2.996923 0.005726 2.993280 0.006111 2.995057 0.006072
400 1 0.999881 0.000312 0.999214 0.000362 0.999735 0.000308
2 1.999506 0.001259 1.997887 0.001297 1.999012 0.001315
3 3.001122 0.002856 2.997404 0.003018 2.997452 0.003068
Estimates and MSEs with LS, WLS, and MPS methods for the ILBMD with <inline-formula id="ieqn-145"> <!--<alternatives><inline-graphic xlink:href="ieqn-145.tif"/><tex-math id="tex-ieqn-145"><![CDATA[\alpha = 1,]]></tex-math>--><mml:math id="mml-ieqn-145"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math> <!--</alternatives>--></inline-formula>2,3 and <inline-formula id="ieqn-146"> <!--<alternatives><inline-graphic xlink:href="ieqn-146.tif"/><tex-math id="tex-ieqn-146"><![CDATA[n = 10,]]></tex-math>--><mml:math id="mml-ieqn-146"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>,</mml:mo></mml:math> <!--</alternatives>--></inline-formula>20,40, 60, 80, 100, 200, 400
LSWLEMPS
n α Es MSE Es MSE Es MSE
10 1 1.007925 0.015105 1.006955 0.014096 1.130497 0.033357
2 2.012742 0.060097 2.005761 0.056658 2.247161 0.124619
3 3.016037 0.143091 3.038500 0.133337 3.411764 0.314966
20 1 0.999859 0.007603 1.005104 0.006716 1.066338 0.011475
2 2.006540 0.030309 2.006146 0.027890 2.137371 0.046024
3 3.011123 0.062426 2.997492 0.059193 3.195782 0.104628
40 1 1.005808 0.003683 1.001200 0.003192 1.038072 0.004547
2 2.001974 0.014647 1.995240 0.012789 2.075144 0.019910
3 3.001535 0.033451 2.987103 0.030327 3.118552 0.046022
60 1 1.002426 0.002372 0.997947 0.002258 1.024475 0.002672
2 2.000718 0.009925 1.998646 0.008446 2.047890 0.010844
3 2.997560 0.022334 3.002908 0.020097 3.080202 0.024972
80 1 0.999352 0.001885 1.001006 0.001669 1.019037 0.001871
2 1.998633 0.007305 1.995840 0.007117 2.037926 0.007875
3 3.005454 0.017804 2.998899 0.015568 3.056238 0.018755
100 1 1.000232 0.001459 0.998625 0.001338 1.015747 0.001519
2 2.001972 0.005673 2.002987 0.004788 2.030212 0.006199
3 3.003503 0.014207 2.999699 0.011628 3.044901 0.013372
200 1 0.998634 0.000707 1.000297 0.000622 1.009054 0.000715
2 2.001705 0.002883 1.996734 0.002457 2.014366 0.002870
3 2.994524 0.006424 3.001192 0.005964 3.028472 0.006385
400 1 1.007925 0.015105 0.999890 0.000336 1.130497 0.033357
2 2.012742 0.060097 2.000354 0.001277 2.247161 0.124619
3 3.016037 0.143091 3.000917 0.002992 3.411764 0.314966

The bias of the suggested estimators is very small and goes to zero for all cases considered in this study. Also, as the samples sizes increase the MSE of all proposed estimators decreases.

Real Data Application

In this section, we use lifetime data set to compare the fit of the suggested ILBMD distribution with four competitors distributions: Rani, length-biased Maxwell distribution, Rama, and exponential defined as

1) Rani distribution (Rn) suggested by Shanker et al.  with pdf given by

f(x;α)=α5α5+24(α+x4)eαx;α>0,x>0 .

2) Length-biased Maxwell distribution (LBM), f(x;α)=12α4x3ex22α2;α>0,x>0.

3) Rama distribution (Rm) suggested by Gross et al.  with pdf given by

fRD(x;α)=α4α3+6(x3+1)eαx x>0,α>0.

4) Exponential distribution (Exp), f(x;α)=αeαx;x>0,α>0.

The data set given in this section represents the relief times of 20 patients receiving an analgesic. This data set was taken from  and it is: 1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8 ,1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0.

In order to compare the two models, we consider the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Hannan-Quinn Information Criterion (HQIC), and Bayesian Information Criterion (BIC). The generic formulas for finding AIC, CAIC, HQIC, and BIC are respectively, given as AIC=2logL+2m, CAIC=2logL+2mnnm1, BIC=2logL+κlog(n), HQIC=2log{log(n)[m2logL]} , where -2logL is the negative maximized log-likelihood values. The Kolmogorov-Smirnov (K-S) test statistic where these measures are defined as KS=Supn|Fn(x)F(x)| , where Fn(x)=1ni=1nIxix and m is the number of parameters and n is the sample size. Also, the Kolmogorov-Smirnov (KS) test is empirical distribution function and F(x) is cumulative distribution function. The best distribution corresponds to lower values of -2lnL, AIC, AICC, BIC, HIQC and KS statistic. The results are displayed in Tab. 3.

Model comparison using AIC, CAIC, BIC, HQIC, -2logL, and the KS test criterion for the 20 patients data
ILBM Rm Exp Rn LBM
MLE 0.30091 1.52130 0.52617 1.71928 1.01012
Error 0.02379 0.15231 0.11766 0.12778 0.07986
AIC 35.43390 61.70660 67.67416 67.30852 40.61917
CAIC 35.65611 61.92882 67.89638 67.53074 40.84139
BIC 36.42962 62.70233 68.66989 68.30425 41.61490
HQIC 35.62827 61.90097 67.86853 67.50290 40.81354
-2LogL 16.71695 29.85330 32.83708 32.65426 19.30958
KS 0.15809 0.35667 0.43942 0.35355 0.17891
P-value 0.69950 0.01233 0.00088 0.01348 0.54400

Hence, we can deduce that the inverse length biased Maxwell distribution leads to a better fit than the Rama, Rani, length biased Maxwell and exponential distribution. The Kolmogorov Smirnov p-value suggests that inverse length biased Maxwell distribution fits statistically better than other distributions considered in this example to the 20 patients data set. Plots of the fitted densities and the histogram are given in Fig. 4.

The fitted pdfs of the Rama, Rani, LBM, Exp and ILBM models
Conclusions

In this article, we introduced and studied the ILBMD. Some statistical properties of the ILBMD are derived and discussed. The reliability and hazard functions of the distribution are analyzed. Also, the distribution of order statistics, mode, harmonic mean, Fisher's information, the stochastic ordering and the mean deviations about the mean and median are presented. The distribution parameter is estimated using different estimation methods includes the maximum likelihood estimation, method of moments, maximum product of spacing, ordinary and weight least square procedures, and the Cramer-Von-Mises methods. The q and Rényi entropies are derived as well as the stress strength reliability is obtained. A real data sets is considered to support the paper objectives. It is revealed that the ILBMD is more power than its competitors used in this study. As a future works the distribution parameter can be estimated based on ranked set sampling method, see .

The authors are grateful to the Editor and anonymous reviewers for their valuable comments and suggestions.

Funding Statement: A.R.A. Alanzi would like to thank the Deanship of Scientific Research at Majmaah University for financial support and encouragement.

Conflicts of Interest: The authors declare that they have no conflicts of interest.

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