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

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.


Introduction
In reliability literature, the quantity R � P(X > 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. en, 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 of R has established, is generalized logistic (GL) distribution. is 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: which is denoted by GL(α, λ). Furthermore, its cumulative distribution function is 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 on R.
is method leads to asymptotic Confidence Interval (C.I) for R and analogously asymptotic Critical Region (C.R) for testing hypotheses on R. ese 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. erefore, 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). e mentioned methodology of this study has been applied for the first time in stressstrength literature. For this goal, we find UMPUT and its corresponding C.I for R in the case where components have exponential distribution.
en, 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. e rest of the paper is organized as follows. We obtain UMPUT and unbiased C.I for R 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.

) as n and m tend to infinity and
From this proposition, (1 − α) × 100 percentage C.I of R is given by We define three testing hypotheses about R as In continuation of this section, we find UMPUT for above tests. First, we consider problem of comparing parameters of two independent exponential populations in eorem 1.

Theorem 1. Consider problem of testing hypotheses
where and pbeta(b, n, m) is cumulative distribution function of distribution beta(n, m) in b.
Proof. First, notice that if random variable B has a beta distribution B ∼ beta(n, m), then P(B < qbeta(α, n, m)) � α and P (B < b) � pbeta(b, n, m). e joint distribution of X � (X 1 , . . . , X n ) and Y � (Y 1 , . . . , Y m ) is given by erefore, regarding Lehmann and Romano [13], in Section 3, there exists UMPUT of three hypotheses concerning the parameter θ � λ 2 − λ 1 . e tests are performed based on S � n i�1 X i conditionally on T � n i�1 X i + m j�1 Y j . Our computations show that the conditional distribution of S given T is as follows: In fact, S|(T � t) ∼ tbeta(n, m). Now, C.R of K E,0 is given by C.R UMPU,0 � S > c α,t and c α,t 0 f S|T (s, t) ds � 1 − α. Substitution of equation (7) in the above integral and using a change of variable z � (S/t) leads to which shows c α,t � tqbeta (1 − α, n, m). erefore, C.R UMPU,0 � S > Tqbeta(1 − α, n, m) completes the first part of proof. e CR of K E,1 named C.R UMPU,1 is earned by similar computations.
For proof of part (ii), we have C.R UMPU,2 � S < d α,t or S > c α,t , where d α,t and c α,t are determined by and E S I C.R UMPU,2 |t � α E( S|t).

by thw similar method with (i) leads to equation (4).
Solving equation (10) is the last pace of proof, first, we compute E(S|t) and E(SI C.R UMPU,3 |t) as follows: From these equations, we have which clearly leads to equation (5). Now, we can apply eorem 1 in order to find UMPUTs of hypotheses K 0 , K 1 , and K 2 .
Also, the acceptance region of UMPUT for K 2 is In which d α and c α are determined by the following equations: Computational Intelligence and Neuroscience pbeta c α , n, m − pbeta d α , n, m � 1 − α, where Proof. Notice that R � (λ 2 /λ 1 + λ 2 ) � r is equivalent with , by considering proof of eorem 1 is completed. For achieving an unbiased C.I for R, we check which one of values of r satisfies equations (15)- (17). ese values construct mentioned C.I.

Note 1. Besides three basic tests
and H 0 : R � r K 2 : R ≠ r , one may wish to test hypotheses H 3 : R ≤ r 1 or R ≥ r 2 K 3 : r 1 < R < r 2 and H 4 : r 1 ≤ R ≤ r 2 K 4 : R < r 1 or R > r 2 . However, these two hypotheses cannot be converted to tests based on X * and Y. In other words, there is no linear function between λ 1 and λ 2 which is equivalent with hypotheses H 3 , K 3 , H 4 , and K 4 . erefore, UMPUTdoes not exist for H 3 against K 3 and H 4 versus K 4 .

Testing R 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. is is assumed that two-scale parameters are equal and known. An asymptotic test has been computed for comparison with UMPUT.

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 for R was achieved. is 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 Section 1, 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. ese methods have an appropriate performance usually for large data sample size. e 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 (R G ), for GL distribution. Finding UMPUT for testing this quantity is an interesting work which may be done in future.
Recently, the study of R 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. e study on UMPUT of R for censoring data can be a challenging and interesting work for the future.

Data Availability
No data were used to support the findings of the study.

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