Majority research studies in the literature determine the weighted coefficients of balanced loss function by suggesting some arbitrary values and then conducting comparison study to choose the best. However, this methodology is not efficient because there is no guarantee ensures that one of the chosen values is the best. This encouraged us to look for mathematical method that gives and guarantees the best values of the weighted coefficients. The proposed methodology in this research is to employ the nonlinear programming in determining the weighted coefficients of balanced loss function instead of the unguaranteed old methods. In this research, we consider two balanced loss functions including balanced square error (BSE) loss function and balanced linear exponential (BLINEX) loss function to estimate the parameter and reliability function of inverse Rayleigh distribution (IRD) based on lower record values. Comparisons are made between Bayesian estimators (SE, BSE, LINEX, and BLINEX) and maximum likelihood estimator via Monte Carlo simulation. The evaluation was done based on absolute bias and mean square errors. The outputs of the simulation showed that the balanced linear exponential (BLINEX) loss function has the best performance. Moreover, the simulation verified that the balanced loss functions are always better than corresponding loss function.

1. Introduction

Scholars always concern about how to find the best estimates of parameters and reliability function of the probability distributions. For this purpose, many methods were proposed. Some of these methods are classical, where they depend only on the sample information under study, assuming that the distribution parameter is fixed but unknown. There are other approaches (which are commonly known as Bayesian methods) that depend on merging prior information with sample information, assuming that the prior parameters behave as random variables, which are commonly known as prior probability distributions.

From a Bayesian perspective, the choice of the loss function is a crucial part of the estimation and prediction problems. To simplify the calculations, many authors prefer using a squared error loss function to produce Bayesian estimates. However, this loss function has mainly criticized where both of overestimation and underestimation are given equal importance, which does not agree with real practices. To deal with this situation, several asymmetric loss functions were proposed in the literature. For example, general entropy loss function (Abdel-Hamid ) and linear exponential (LINEX) loss function (Al-Duais and Alhagyan ; Khatun and Matin )

After that, the balanced loss function idea appeared in the literature which tried to reflect the desired criteria of two methods (see equation (13)), for example, balanced square error (BSE) loss function , balanced general entropy (BGE) loss function ), and balanced linear exponential (BLINEX) loss function  EL-Sagheer  EL-Sagheer .

However, the majority of proposed balanced loss functions in the literature determine the value of weighted coefficients ω1 and ω2 randomly without convinced mathematical justification. This motivated us to treat this issue by determining the weighted coefficients by using nonlinear programming. In this paper, we are going to use two balanced loss functions (i.e., BSE and BLINEX) to estimate the parameter and reliability function of inverse Rayleigh distribution (IRD) based on lower record values utilizing nonlinear programming in determining the best-weighted coefficients.

The IRD is considered as one of the important distributions. It has wide applications in the area of reliability theory, survival analysis, and life testing study. IRD under record value was studied by Muhammad ; Shawky and Badr ; Soliman et al. ; Manzoor et al. ; Rasheed and Aref ; and Abdullah and Aref .

The probability density function (pdf) and cumulative distribution function (cdf) of the IRD with scale parameter α are given, respectively, as follows:(1)fx;α=2αx3expαx2;x>0,α>0,(2)Fx;α=expαx2;x>0,α>0.

Moreover, the reliability function Rt at mission time t for the IRD is given by(3)Rt;α=1expαt2;t>0.

2. Record Values and Maximum Likelihood (ML) Estimation

Let X1,X2,X3, be a sequence of independent and identically distributed (iid) random variables with (cdf) F(x) and (pdf) f (x). Set Ym=minX1,X2,X3,,Xm,m1. We say that Xj is a lower record and denoted by XLi if Yj<Yj1,j>1.

Assuming that XL1,XL2,XL3,XLm are the first m lower record values arising from a sequence Xi of iid inverse Rayleigh distribution whose pdf and cdf are, respectively, given by (1) and (2). The joint density function of the ﬁrst m lower record values x¯xL1,xL2,xL3,xLm is given by(4)f1,2,3,,nxL1,xL2,xL3,xLn=fxLmi=1m1fxLi1FxLi,0xL1<xL2<xL3,<xLm<,where f. and F. are given, respectively, by (1) and (2) after replacing x by xLi. The likelihood function based on the m lower record values x is given by(5)α|x¯=2αmuexpαTD,u=i=1mxLi3,TD=xLm3.

The log-likelihood function is written as(6)lnα|x¯=mln2ααTD3i=1mlnxLi.

By differentiating (6) with respect to the parameterα and equating to zero, the maximum likelihood estimate (MLE), under lower record value, say α^ML, was obtained as(7)α^ML=mTD.

By using the invariance property of the maximum likelihood estimator, the maximum likelihood estimator of reliability function R^tML of Rt given by (3) after replacing α by α^ML is(8)R^tML=1expα^MLt2,t0.

3. Loss Functions

From a Bayesian perspective, the choice of loss function is an essential part in the estimation and prediction problems. In this work, we use three main types of loss function including squared error loss function, LINEX loss function, and balanced loss functions.

3.1. Squared Error (SE) Loss Function

SE loss function is a symmetric loss function. The SE loss function is expressed as follows:(9)Lϕ^,ϕ=ϕ^ϕ2,where ϕ^ is the estimation of parameter ϕ. The Bayes estimator of ϕbased on SE loss function denoted byϕ^SE is obtained as follows:(10)ϕ^SE=Eπϕ^|x¯.

3.2. Linear Exponential (LINEX) Loss Function

LINEX loss function is an asymmetric loss function. The LINEX loss function is expressed as follows (see Varian ): (11)LΔexpcΔcΔ1,where Δ=ϕ^ϕ. The sign and magnitude of c reflect the direction and degree of asymmetry, respectively. The Bayes estimator related to LINEX loss function, denoted by ϕ^L, is given by(12)ϕ^L=1clnEϕexpcϕ,c0,provided that Eϕexpcϕ exists and finite, where Eϕdenotes the expected value.

3.3. Balanced Loss Function (BLF)

BLF is a mix of two estimators. In general, BLF is expressed as follows (see Jozani et al. ):(13)Lρ,ω,γoϕ^,γ=ω1ρϕ,γ0+ω2ρϕ,γω1+ω2=1,Lρ,ω,γoϕ^,γ=ω1ρϕ,γ0+ω2ρϕ,γω1+ω2=1,where ρ is an arbitrary loss function, while γ0 is a chosen a prior target estimator of ϕ that can be obtained by several methods like maximum likelihood, least squares, or unbiasedness, and ω1 and ω2 represent weighted coefficient ω1 and ω2ϵ0,1. In this work, we focus on two types of BLF, including balanced squared error (BSE) loss function and balanced LINEX (BLINEX) loss function.

3.3.1. Balanced Squared Error (BSE) Loss Function

BSE loss function is obtained by choosingρϕ,γ=γϕ2, so equation (13) will be on the form (see Ahmadi et al. ):(14)Lω,γ0ϕ,γ=ω1γγ02+ω2γϕ2,and the corresponding Bayes estimate of the unknown parameter ϕ is given by(15)γω,γ0x¯=ω1γ0+ω2Eϕ|x¯.

Note that when ω1=0, then BSE loss function is just an SE loss function.

3.3.2. Balanced Linear Exponential (BLINEX) Loss Function

The BLINEX loss function is obtained by choosing ρϕ,γ=expcϕγcϕγ1 in equation (13) as follows (see Zellner ):(16)Lω,γ0ϕ,γ=ω1expcγγ0cγγ01+ω2expcγϕcγϕ1.

And the corresponding Bayes estimate of the unknown parameter ϕ is given by(17)γω,γ0x¯=1clnω1expcγ0x¯+ω2Eexpcϕ|x¯.

It is worth noting, when ω1=0 then BLINEX loss function is just a LINEX loss function.

4. Bayes Estimation

In this section, we derive the Bayes estimates of the scale parameter α and the reliabilityRt function of the IRD by using balanced loss functions (BLF). Furthermore, we assume gamma ϰ,b as a conjugate prior distribution for α as follows:(18)gα=bϰΓϰαϰ1expbα;b>0,α>0.

By combining the likelihood function in equation (5) with the prior pdf of α in equation (18), we get the posterior distribution of α as(19)πα|x¯=Lα;x¯gα0Lα;x¯gαdα=νm+ϰΓm+ϰαm+ϰ1expαν,where(20)x¯xL1,xL2,xL3,,xLm,ν=b+TD.

4.1. Estimates Based on Balanced Squared Error (BSE) Loss Function

Based on BSE loss function and by using equation (15), the Bayes estimate of a function ϕ where ϕ can be α or Rt is given by(21)ϕBSE=ω1ϕ^ML+ω2Eϕ|x¯,where ϕ^ML is the ML estimate ofϕ and Eϕ|x¯ can be obtained by(22)Eϕ|x¯=0ϕπϕ|x¯ dϕ.

Based on the BSE loss function and by using equation (21), the Bayes estimator α^BSE forα is(23)α^BSE=ω1α^ML+ω2Eα|x¯,where α^ML is the ML estimate ofα, which can be obtained using equation (7) and Eα|x¯ can be obtained using the following equation:(24)Eα|x¯=0ανm+ϰΓm+ϰαm+ϰ1expανdα=ϰ+mν.

Similarly, the Bayes estimate R^tBSE of the reliability Rt at a mission time t related to BSE loss function is(25)R^tBSE=ω1R^tML+ω2ERt|x¯,where R^tML is the ML estimate of Rt which can be obtained using equation (8) and ERt|x¯ can be obtained using the following equation:(26)ERt|x¯=01expαt2νm+ϰΓm+ϰαm+ϰ1expανdα=1νν+t2m+ϰ;t.

In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient ω1 and ω2in equation (21):(27)minimize:MSEϕ^BSE=Eϕ^BSEϕ2=Eω1ϕ^ML+ω2Eϕ|x¯ϕ2subject toω1+ω2=1,0ω1<1,0ω2<1.

4.2. Estimates Based on Balanced Linear Exponential (BLINEX) Loss Function

Based on BLINEX loss function and by using equation (16), the Bayes estimate of a function ϕ where ϕ can be α or Rt is given by(28)ϕ^BL=1clnω1expcϕ^ML+ω2Eexpcϕ|x¯,where ϕ^MLis the ML estimate of ϕ and Eexpcϕ|x¯ can be obtained by(29)Eexpcϕ|x¯=0expcϕ|x¯πϕ|x¯dϕ.

Based on BLINEX loss function and by using equation (28), the Bayes estimator α^BL for α is given as(30)α^BL=1clnω1expcα^ML+ω2Eexpcα|x¯,where α^ML is the ML estimate of α which can be obtained using equation (7) and Eexpcα|x¯ can be obtained using the following integral:(31)Eexpcα|x¯=0expcα|x¯πα|x¯ dα=0expcανm+ϰΓm+ϰαm+ϰ1expανdα=1+cνm+ϰ.

Similarly, the Bayes estimate R^tBL of the reliability Rt at a mission time t related to BLINEX loss function is(32)R^tBL=1clnω1expcR^tML+ω2EexpcRt|x¯,where R^tML is the ML estimate of Rt which can be obtained using equation (8) and EexpcRt|x¯can be obtained using the following integral:(33)EexpcRt|x¯=0expcRt|x¯πα|x¯ dα=0expc1expαt2νm+ϰΓm+ϰαm+ϰ1expαν dα=expc+expcνm+ϰi=1ncii!ν+it2m+ϰ.

In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient ω1 and ω2 in equation (28):(34)minimize:MSEϕ^BL=Eϕ^Blϕ2=1clnω1expcϕ^ML+ω2Eexpcϕ|x¯ϕ2subject toω1+ω2=1,0ω1<1,0ω2<1,ϕ^BL=1clnω1expcϕ^ML+ω2Eexpcϕ|x¯.

5. Simulation Study and Comparisons

All estimation methods, mentioned in Section 4, are used to estimate the parameter and reliability function of IRD. To examine the performance of these estimation methods, the Monte Carlo simulation study is conducted. The simulation consists of four steps as follows:

For the given values of prior parameters b=2,ϰ=1, generate a random value α=1.383 from the Gamma prior pdf in equation (18) .

Usingα obtained in Step 1, we generatemm=3,4,5,6,7 lower record values from inverse Rayleigh distribution whose pdf is given by equation (1).

The different estimates of α and Rt at time t (chosen to be 4) are computed.

Steps 1 to 3 are repeated 10,000 times.

The evaluation is done depending on the absolute bias in addition to the mean square error (MSE)(35)absolute bias ϕ^=110000i=110000ϕ^ϕ,MSE ϕ^=110000i=110000ϕ^ϕ2,

where ϕ^ is the estimate at the ithrun.

The results are listed in Tables 14.

Absolute bias of the estimates of α

mα^MLα^SEα^BSEα^Lα^BL
c=0.001c=1c=2c=0.001c=1c=2
31.037870.606930.568080.606620.398630.315140.567870.398490.28874
40.788280.553940.519680.553720.392720.321240.519540.392570.29764
50.653420.505960.475340.505780.378100.317040.475250.377930.29541
60.558810.460090.431950.459960.356610.304540.431880.356440.28420
70.480790.410940.385000.410830.328500.286770.384950.328320.26636

MSEs of the estimates of α

mα^MLα^SEα^BSEα^Lα^BL
c=0.001c=1c=2c=0.001c=1c=2
34.550020.667800.651870.667060.261090.148300.651200.261090.14235
41.828890.583230.582700.582700.265260.159280.568830.265260.15391
51.130610.496050.483810.495660.251130.158270.483480.251130.15330
60.773550.416910.405810.416620.227850.148910.405570.227850.14420
70.543740.344280.334370.344070.198970.133740.334200.198970.12880

Absolute bias of the estimates of Rt at t=4

mR^tMLR^tSER^tBSER^tLR^tBL
c=2c=0.001c=2c=2c=0.001c=2
30.053430.032760.030900.034440.032760.031200.031950.030900.02985
40.042510.030130.028480.031370.030130.028960.029180.028480.02777
50.035820.027690.026220.028660.027690.026770.026700.026210.02571
60.030930.025290.023940.034440.032760.031200.031950.030900.02985
70.026780.022670.021420.031370.030130.028960.029180.028480.02777

MSEs of the estimates of Rtat t=4

mR^tMLR^tSER^tBSER^tLR^tBL
c=2c=0.001c=2c=2c=0.001c=2
30.008150.001880.001840.002090.001880.002090.002030.001840.00167
40.004570.001660.001630.001810.001660.001530.001760.001630.00151
50.003040.001430.001400.001540.001430.001330.001490.001400.00131
60.002160.001210.001190.001300.001210.001140.001260.001190.00112
70.001580.001010.000990.001070.001010.000950.001040.000990.00094
6. Concluding Remarks

In this paper, nonlinear programming was employed to get the best values of weighted coefficients (ω1 and ω2) of the balanced loss function. The Bayesian and non-Bayesian estimates of the parameter α and reliability function R(t) of the lifetimes follow the inverse Rayleigh distribution. The estimations were conducted depending on lower record values.

The results are listed in Tables 14. The main observations are stated in the following points:

All tables showed that the Bayes estimates under BLINEX loss function are the best according to the smallest values of absolute bias and MSE comparing with the estimates under LINEX loss function, BSE loss function, SE loss function, or MLEs. Bayes estimates under the BSE loss function came in the second level of accuracy. The third, fourth, and fifth levels of accuracy were for Bayes estimates under the LINEX loss function, the estimates under SE loss function, and ML estimations, respectively.

The results showed that the values of all MSEs and all absolute bias decrease as m increases. This means here is an inverse relationship between the evaluation functions and the number of recorded values.

In order to show the effect of the shape parameter of the asymmetric loss function, we examined different values of c. One can observe that when the value of c is closed to zero, then the values of MSE of both Bayes estimates under LINEX and BLINEX loss function are almost the same. This means that BLINEX loss function is generated to LINEX loss function.

Data Availability

The data were generated by simulation done by using mathematical software.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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