# Stress-Strength Parameter Estimation under Small Sample Size: A Testing Hypothesis Approach

Authors: Hassan Alsuhabi; Mohammad Mehdi Saber; M. M. Abd El-Raouf
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In this paper, uniformly most powerful unbiased test for testing the stress-strength model has been presented for the first time. The end of the paper is recommending a method which is appropriate for no large data where a normal asymptotic distribution is not applicable. The previous methods for inference on stress-strength models use almost all the asymptotic properties of maximum likelihood estimators. The distribution of components is considered exponential and generalized logistic. A corresponding unbiased confidence interval is constructed, too. We compare presented methodology with previous methods and show the method of this paper is logically better than other methods. Interesting result is that our recommended method not only uses from small sample size but also has better result than other ones.


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## Body

## 1. Introduction

In reliability literature, the quantityR=PX>Y is often referred to the stress-strength model. In addition to reliability, this parameter has application in some scientific fields such as biostatistics, quality control, engineering, stochastic precedence, and probabilistic mechanical design. Kotz et al. [1] and Ventura and Racugno [2] have presented a comprehensive review on this matter, especially from its applications. Regarding [1] an instance of real practice of the stress-strength model is in a clinical study, where Y and X are assumed as the outcomes of a treatment and a control group, respectively. Then, the ineffectiveness of the treatment is measured by R. In terms of reliability, Y is considered the strength of a component, which is under X stress. Henceforth, two quantities R and (1−R) indicate the probabilities of system performance and system failure, respectively. Many distributions have been applied by authors for estimation of R. For instance, see Rezaei et al. [3] and Nadar et al. [4] for a nearly complete list of distributions used in this matter.A distribution, whose application in reliability and especially in estimation ofR has established, is generalized logistic (GL) distribution. This distribution, as one of three generalized forms of the standard logistic distribution, has been defined by Balakrishnan and Leung [5]. A random variable X is said to have a GL distribution if it has the following probability density function:(1)fx=αλe−λx1+e−λx−α−1,α>0,λ>0,x∈R,which is denoted by GLα,λ. Furthermore, its cumulative distribution function is(2)Fx=1+e−λx−α.Here,α and λ are the shape and scale parameters, respectively.It has also been called the skew-logistic distribution and is defined on−∞,∞. Estimation of GL distribution parameters has been received attention for practical usage by some works such as Balakrishnan [6], Asgharzadeh [7], and Gupta and Kundu [8]. Among the recent articles, Alkasasbeh and Raqab [9] estimated the unknown parameters of the GL lifetime model using different approaches and Vasudeva Rao et al. [10] obtained maximum likelihood estimation by a linear approximation. In stress-strength literature, when distribution of components is GL, Asgharzadeh et al. [11] and Rasekhi et al. [12] have investigated in statistical inference of R and its multicomponent version, respectively.In almost all studies, authors have applied maximum likelihood estimator (MLE) and its asymptotic distribution for estimating and inference onR. This method leads to asymptotic Confidence Interval (C.I) for R and analogously asymptotic Critical Region (C.R) for testing hypotheses on R. These asymptotic C.I and C.R have suitable efficiency for data with large sample size. In the other words, performance of any asymptotic method for small data is not appropriate. For small sample size, nonasymptotic and exact statistical methods are needed in order to reach reliable results. However, an exact C.I and C.R of R has not been presented in previous works. Therefore, we are motivated for estimation of R with small sample size based on a testing hypothesis approach and finding its Uniformly Most Powerful Unbiased Test (UMPUT). The mentioned methodology of this study has been applied for the first time in stress-strength literature. For this goal, we find UMPUT and its corresponding C.I for R in the case where components have exponential distribution. Then, by using of relationship between exponential distribution and GL distribution, our findings are applied and modified to the case of GL distributed components. We have chosen GL distribution since, by its relation with exponential distribution paper, it covers this distribution, too. On the contrary, GL distribution has been used in many other previous papers on stress-strength models, which is mentioned above. Also, GL distribution is flexible and can be fitted to numerous datasets.The rest of the paper is organized as follows. We obtain UMPUT and unbiased C.I forR when components have exponential distributions in Section 2. Section 3 is devoted to similar work as Section 2, however, for GL distributed components. A comparison between the introduced method and previous methods which are based on asymptotic distribution of maximum likelihood estimators has been provided in Section 4.


## 2. UMPUT forR in Exponential Distribution

Suppose random variablesX and Y are independent and X∼Eλ1 and Y∼Eλ2; then, R=PX>Y=λ2/λ1+λ2. Throughout this section, this supposes that X1,…,Xn∼X and Y1,…,Ym∼Y. An asymptotic C.I for R has been obtained in the following proposition.Proposition 1.
The ML estimators for parameters areλ^1=1/X¯ and λ^2=1/Y¯, so R^=λ^2/λ^1+λ^2. Also, R^−R/σ^1⟶N0,1 as n and m tend to infinity and n/m⟶d, for 0<d<+∞, where σ^12=n+m/nmλ^12λ^22/λ^1+λ^24.From this proposition,1−α×100 percentage C.I of R is given by(3)R∈R^−z1−α/2σ^1,R^+z1−α/2σ^1.We define three testing hypotheses aboutR as H0:R=rK0:R>r, H0:R=rK1:R<r, and H0:R=rK2:R≠r. The critical region (C.R) for these tests based on a normal approximation are C.Ras,0=Z>z1−α, C.Ras,1=Z<zα, and C.Ras,2=Z>z1−α/2, respectively where Z=R^−r/σ^1.In continuation of this section, we find UMPUT for above tests. First, we consider problem of comparing parameters of two independent exponential populations in Theorem1.Theorem 1.
Consider problem of testing hypothesesHE,0:λ2=λ1KE,0:λ2>λ1, HE,0:λ2=λ1KE,1:λ2<λ1, and HE,0:λ2=λ1KE,2:λ2≠λ1. Then,(i)
C.Rs of UMPUTs forKE,0 and KE,1 are given by C.RUMPU,0=W>qbeta1−α,n,m and C.RUMPU,1=W<qbetaα,n,m, respectively. Here, qbetaα,n,m is αth quantile of distribution betan,m.(ii)
Also, the acceptance region of UMPUT forKE,2 is A.RUMPU,2=dα<W<cα, which dα and cα are determined by the following equations:(4)pbetacα,n,m−pbetadα,n,m=1−α,(5)pbetacαT,n+1,m−pbetadαT,n+1,m=1−αpbeta1T,n+1,m,whereW=S/T, S=∑i=1nXi, T=∑i=1nXi+∑j=1mYj, and pbetab,n,m is cumulative distribution function of distribution betan,m in b.Proof.
First, notice that if random variableB has a beta distribution B∼betan,m, then PB<qbetaα,n,m=α and PB<b=pbetab,n,m.
The joint distribution ofX=X1,…,Xn and Y=Y1,…,Ym is given by(6)fX,Yx,y=λ1nλ2me−λ1∑i=1nxi−λ2∑j=1myj⋉eλ2−λ1∑i=1nxi−λ2∑j=1myj+∑i=1nxi.
Therefore, regarding Lehmann and Romano [13], in Section 3, there exists UMPUT of three hypotheses concerning the parameter θ=λ2−λ1. The tests are performed based on S=∑i=1nXi conditionally on T=∑i=1nXi+∑j=1mYj. Our computations show that the conditional distribution of S given T is as follows:(7)fS|Ts,t=Γn+mΓnΓm1tstn−11−stm−1.
In fact,S|T=t∼tbetan,m.
Now, C.R ofKE,0 is given by C.RUMPU,0=S>cα,t and ∫0cα,tfS|Ts,tds=1−α. Substitution of equation (7) in the above integral and using a change of variable z=S/t leads to(8)∫0cα,t/tΓn+mΓnΓmzn−11−zm−1dz=1−α,which shows cα,t=tqbeta1−α,n,m. Therefore, C.RUMPU,0=S>Tqbeta1−α,n,m completes the first part of proof. The CR of KE,1 named C.RUMPU,1 is earned by similar computations.
For proof of part (ii), we haveC.RUMPU,2=S<dα,torS>cα,t, where dα,t and cα,t are determined by(9)EIC.RUMPU,2|t=αand(10)ESIC.RUMPU,2|t=αES|t.
From equation (9), we have ∫dα,tcα,tfS|Ts,tds=1−α which by thw similar method with (i) leads to equation (4).
Solving equation (10) is the last pace of proof, first, we compute ES|t and ESIC.RUMPU,3|t as follows:(11)ES|t=∫01sfS|Ts,tds=t∫01/tΓn+mΓnΓmzn1−zm−1dz=nn+mtpbeta1t,n+1,m,ESIC.RUMPU,3|t=t∫0dα,t/tΓn+mΓnΓmzn1−zm−1dz+∫cα,t/t1/tΓn+mΓnΓmzn1−zm−1dz=t∫01/tΓn+mΓnΓmzn1−zm−1dz−∫dα,t/tcα,t/tΓn+mΓnΓmzn1−zm−1dz.
From these equations, we have(12)αpbeta1t,n+1,m=pbeta1t,n+1,m−∫dα,t/tcα,t/tΓn+mΓnΓmzn1−zm−1dz,which clearly leads to equation (5).
Now, we can apply Theorem1 in order to find UMPUTs of hypotheses K0, K1, and K2.Theorem 2.
The CRs of UMPUTs forK0 and K1 are(13)C.RUMPU,0∗=W∗>qbeta1−α,n,mand(14)C.RUMPU,1∗=W∗<qbetaα,n,m,respectively.Also, the acceptance region of UMPUT forK2 is(15)A.RUMPU,2∗=dα<W∗<cα,In which dα and cα are determined by the following equations:(16)pbetacα,n,m−pbetadα,n,m=1−α,(17)pbetacαT∗,n+1,m−pbetadαT∗,n+1,m=1−αpbeta1T∗,n+1,m,where S∗=1−r/r∑i=1nXi, T∗=1−r/r∑i=1nXi+∑j=1mYj, and W∗=S∗/S∗+T∗.Proof.
Notice thatR=λ2/λ1+λ2=r is equivalent with λ2=λ1∗, where λ1∗=r/1−rλ1. First, we convert Xi∗=1−r/rXi, for i=1,…,n. Since Xi∗∼Eλ1∗ we can perform H0:R=rK0:R>r, H0:R=rK1:R<r, and H0:R=rK2:R≠r based on independent samples X1∗,…,Xn∗ and Y1,…,Ym. The corresponding hypothesis are H0:λ2=λ1∗K0:λ2>λ1∗, H0:λ2=λ1∗K1:λ2<λ1∗, and H0:λ2=λ1∗K2:λ2≠λ1∗, by considering proof of Theorem 1 is completed.
For achieving an unbiased C.I forR, we check which one of values of r satisfies equations (15)–(17). These values construct mentioned C.I.
Note 1. Besides three basic tests H0:R=rK0:R>r, H0:R=rK1:R<r, and H0:R=rK2:R≠r, one may wish to test hypotheses H3:R≤r1orR≥r2K3:r1<R<r2 and H4:r1≤R≤r2K4:R<r1orR>r2. However, these two hypotheses cannot be converted to tests based on X∗ and Y. In other words, there is no linear function between λ1 and λ2 which is equivalent with hypotheses H3, K3, H4, and K4. Therefore, UMPUT does not exist for H3 against K3 and H4 versus K4.


## 3. TestingR in GL Distribution with Known and Common Scale Parameter

In this section, we compute asymptotic test and UMPUT of the stress-strength model for GL distribution. This is assumed that two-scale parameters are equal and known. An asymptotic test has been computed for comparison with UMPUT.Suppose random variablesV and U are independent and V∼GLα1,λ and U∼GLα2,λ; then,(18)R=PV>U=α1α1+α2.First, we find asymptotic test by using MLEs of parameters and their asymptotic distribution. LetV1,…,Vm∼GLα1,λ and U1,…,Un∼GLα2,λ; then, the likelihood function is(19)ι=mlnα1+nlnα2+m+nlnλ−λ∑i=1mvi+∑j=1nuj−α1+1S1v,λ−α2+1S1u,λ,where S1w,a=∑i=1qln1+e−awi and w=w1,…,wq. By derivation with respect to parameters, we have(20)α^1=mS1V,λ,α^2=nS1U,λ.By substitutionη^=α^1,α^2 in equation (3), we have(21)R^=mS1U,λmS1U,λ+nS1V,λ.Fisher’s information matrix and its inverse areJη=m/α1200n/α22 and J−1η=α12/m00α22/n.Sinceη^⟶N2η,J−1η, by the multivariate Delta method, one can see that R^⟶NR,σ2η, where σ2η=∂RG/∂ηJ−1η∂RG/∂ηT, ∂RG/∂η=∂RG/∂α1,∂RG/∂α2, ∂RG/∂α1=α2/α1+α22, and ∂RG/∂α2=−α1/α1+α22. Therefore, R^−R/σ^GL⟶N0,1 as n and m tend to infinity and n/m⟶d, for a 0<d<+∞, where(22)σ^GL2=n+mnmα^12α^22α^1+α^24.Now, the1−α×100 percentage C.I of R and its corresponding tests based on normal approximation for hypotheses K0, K1, and K2 are the same with Section 2 with σ^GL2 in equation (22) instead of σ^12.


### 3.1. UMPUT forR in GL Distribution

For UMPUT ofK0, K1, and K2, we only need to convert original data to Xi=ln1+e−λUi, for i=1,…,n, and Yj=ln1+e−λVj, for j=1,…,m. This can be simply seen as X1,…,Xn∼Eα2 and Y1,…,Ym∼Eα1. From this point, all results of Theorems 1 and 2 can be used here with substitution α2 and α1 instead λ1 and λ2. To put it more clearly, first of all, we define three statistics UGL=1−rS1U,λ/1−rS1U,λ+rS1V,λ, SGL∗=1−r/rS1U,λ, and TGL∗=1−r/rS1U,λ+S1V,λ. Now, CRs of UMPUTs for K0 and K1 are, respectively,(23)C.RUMPU,0=UGL>qbeta1−α,n,mand(24)C.RUMPU,1=UGL<qbetaα,n,m.Also, the acceptance region of UMPUT for hypothesisK2 is achieved from the following equation:(25)A.RUMPU,2=c1α,TGL∗<SGL∗<c2α,TGL∗,where two values c1α,TGL∗ and c2α,TGL∗ are determined by the following equations:(26)pbetac2α,TGL∗TGL∗,n,m−pbetac1α,TGL∗TGL∗,n,m=1−α,(27)pbetac2α,TGL∗TGL∗,n+1,m−pbetac2α,TGL∗TGL∗,n+1,m=1−αpbeta1TGL∗,n+1,m.Again similar with the previous section, for getting an unbiased C.I. forR in case of GL model, we check which one of the values of r satisfies equations (25)–(27). Then, one can have an unbiased C.I. Also, Note 1 remains right here in case of GL distribution.


## 3.1. UMPUT forR in GL Distribution

For UMPUT ofK0, K1, and K2, we only need to convert original data to Xi=ln1+e−λUi, for i=1,…,n, and Yj=ln1+e−λVj, for j=1,…,m. This can be simply seen as X1,…,Xn∼Eα2 and Y1,…,Ym∼Eα1. From this point, all results of Theorems 1 and 2 can be used here with substitution α2 and α1 instead λ1 and λ2. To put it more clearly, first of all, we define three statistics UGL=1−rS1U,λ/1−rS1U,λ+rS1V,λ, SGL∗=1−r/rS1U,λ, and TGL∗=1−r/rS1U,λ+S1V,λ. Now, CRs of UMPUTs for K0 and K1 are, respectively,(23)C.RUMPU,0=UGL>qbeta1−α,n,mand(24)C.RUMPU,1=UGL<qbetaα,n,m.Also, the acceptance region of UMPUT for hypothesisK2 is achieved from the following equation:(25)A.RUMPU,2=c1α,TGL∗<SGL∗<c2α,TGL∗,where two values c1α,TGL∗ and c2α,TGL∗ are determined by the following equations:(26)pbetac2α,TGL∗TGL∗,n,m−pbetac1α,TGL∗TGL∗,n,m=1−α,(27)pbetac2α,TGL∗TGL∗,n+1,m−pbetac2α,TGL∗TGL∗,n+1,m=1−αpbeta1TGL∗,n+1,m.Again similar with the previous section, for getting an unbiased C.I. forR in case of GL model, we check which one of the values of r satisfies equations (25)–(27). Then, one can have an unbiased C.I. Also, Note 1 remains right here in case of GL distribution.


## 4. Comparison between UMPUT and Asymptotic Test

In this section, we compare two tests for hypothesesK0, K1, and K2 based on UMPUT and asymptotic test. First, we show tests done by asymptotic distribution of R^ are unbiased for three mentioned hypotheses.Lemma 1.
LetX1,…,Xn∼Eλ1 and Y1,…,Ym∼Eλ2 and C.Rs for testing hypotheses K0, K1, and K2 are C.Ras,0, C.Ras,1, and C.Ras,2. Then, these tests are unbiased.Proof.
We prove theorem for three hypotheses separately by computation of its power function for a fixed pointr1 in alternative hypothesis βr1=PR=r1C.R. For hypothesis K0:R>r, C.R is C.Ras,0=Z>z1−α for Z=R^−r/σ^1. So, the power function is(28)βr1=PR=r1R^−rσ^1>z1−α=PZ>z1−α−r1−rσ1=1−Φz1−α−r1−rσ1.
Sincer1>r, we have βr1>α. For hypothesis K1:R<r, similar to the previous case, we have βr1=Φzα−r1−r/σ1 which for r1<r is clearly bigger than α. For the last case, the proof is complicated. For K2:R≠r, one can show βr1=PZ>d1+PZ<d2, where d1=z1−α/2−r1−r/σ1 and d2=−z1−α/2−r1−r/σ1. Now, we consider two cases r1>r and r1<r separately. If r1>r, we have(29)βr1=PZ>z1−α/2+PZ<−z1−α/2+Pd1<Z<z1−α/2−Pd2<Z<−z1−α/2.
By these facts that two intervalsd1,z1−α/2 and d2,−z1−α/2 have equal lengths and second interval is placed in tail of Standard Normal distribution, it has less probability than the first interval. This demonstrates that βr1>α. For case of r1<r, the power function is as follows:(30)βr1=PZ>z1−α/2+PZ<−z1−α/2+P−z1−α/2<Z<d2−Pz1−α/2<Z<d1.
By similar logic as case ofr1>r, one can show that βr1>α. These results complete the proof.
Now, we can compare two tests, asymptotic and UMPUT, in the following theorem.Theorem 3.
LetX1,…,Xn∼Eλ1 and Y1,…,Ym∼Eλ2 and asymptotic C.Rs for testing hypotheses K0, K1, and K2 be C.Ras,0, C.Ras,1, and C.Ras,2. Then, UMPUTs based on equations (13)–(17) are more powerful than these asymptotic tests.Proof.
The proof comes from unbiasedness of asymptotic tests in Lemma1 and definition of UMPUT.
Theorem3 guaranties that the unbiased C.I earned by using of acceptance region of K2 is more accurate than asymptotic C.I.
All results achieved about comparison between asymptotic test and UMPUT for exponential distribution are satisfied for GL distribution, too. In the other words, Lemma1 and Theorem 3 are held for case of GL distributed components. The most important matter which has value for repeat is this point that UMPUTs based on equations (23)–(27) are more powerful than the asymptotic tests by using of statistic Z=R^−r/σ^GL.


## 5. Conclusion and Future Works

In this paper, we found UMPUT for stress-strength quantity in case of exponential and GL distributed components, respectively. By using this test in two sides’ case, C.I forR was achieved. This has been proved that UMPUT is more powerful than the asymptotic test. Our methodology has been used on stress-strength models for the first time.As we mentioned in Section1, a numerous distributions have been applied to estimation of stress-strength quantity. In almost all of these papers, estimation is performed by MLE and its consistency property. Also, in some cases the Bayesian estimation is performed. These methods have an appropriate performance usually for large data sample size. The methodology introduced in this article can be applied to other distributions such as generalized exponential, generalized Pareto, Kumaraswamy, etc. (see Alshanbari et al. [14]).Saber and Yousof [15] surveyed a generalization of stress-strength models named generalized stress-strength models (RG),(31)RG=PY<X<Z,for GL distribution. Finding UMPUT for testing this quantity is an interesting work which may be done in future.Recently, the study ofR by censoring data has been expanded by many authors. For instance, Abu-Moussa et al. [16] and Almongy et al. [17] studied R under progressive censoring data for Rayleigh and Weibull extended distributed components, respectively. The study on UMPUT of R for censoring data can be a challenging and interesting work for the future.



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