# Selection of Efficient Parameter Estimation Method for Two-Parameter Weibull Distribution

Authors: Mohammed M. A. Almazah; Muhammad Ismail
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Several studies have considered various scheduling methods and reliability functions to determine the optimum maintenance time. These methods and functions correspond to the lowest cost by using the maximum likelihood estimator to evaluate the model parameters. However, this paper aims to estimate the parameters of the two-parameter Weibull distribution (α, β). The maximum likelihood estimation method, modified linear exponential loss function, and Wyatt-based regression method are used for the estimation of the parameters. Minimum mean square error (MSE) criterion is used to evaluate the relative efficiency of the estimators. The comparison of the different parameter estimation methods is conducted, and the efficiency of these methods is observed, both mathematically and experimentally. The simulation study is conducted for comparison of samples sizes (10, 50, 100, 150) based on the mean square error (MSE). It is concluded that the maximum likelihood method was found to be the most efficient method for all sample sizes used in the research because it achieved the least MSE compared with other methods.


---

## Body

## 1. Introduction

The Weibull distribution is the continuous probability distribution. It has been frequently used as a failure model in reliability studies to describe all phases of machine failure [1, 2]. Waloddi Weibull used the probabilistic Weibull model to indicate the distribution of the breaking strength of materials and to illustrate the performance of systems or events that have some degree of variability [3]. It has frequently been utilized to evaluate the reliability of the product, examine life data, and model failure times [4].The Weibull distribution is commonly utilized in radar systems to model the variation of the signal level triggered by clutters. It is also used in wireless communications [4]. Further, the Weibull distribution has significant importance in reliability and life testing. It is the most frequently used distribution in the reliability theory. When employed, its greater effects generate more accurate results and have crucial applications for studying failure models [5]. Several studies have used this distribution for the analysis [4, 6–8]. Hallinan [9] gave a perceptive appraisal by introducing numerous chronological facts and various characteristics of this distribution. A detailed chapter on the systematic behavior of this distribution is contributed by Coleman et al. [10]. More recently, Lai et al. [2] wrote a monograph that comprises all aspects concerning the Weibull distribution and its expansions. Moreover, for the relevant information and details on applications, we recommend the readers to read several studies proposed in [4, 6, 8, 11–14]. The Bayesian estimation is commonly used in various scientific areas [15–17]. There exists much literature on the subject; due to the limitations of the paper, we are incapable of relating all related information.The previous paper [3] uses scheduling techniques and reliability function to establish the optimum maintenance time, which corresponds to the lowest cost, by employing the maximum likelihood estimator (MLE) to estimate model parameters. However, the current study is initiated to estimate the scale parameter (α) of the two-parameter Weibull distribution (α, β) in a mathematical method and obtain an estimator that comprises a minimum mean square error (MSE) and determines the relative efficiency of the estimators and comparability of estimators.


## 2. Materials and Methods

### 2.1. The Two-Parameter Weibull Distribution

The Weibull distribution contains one, two, or three parameters. These parameters are shape parameterβ and scale parameter α, and there may be a third parameter, the location γ parameter [5]. The distribution function of two-parameter Weibull distribution can be expressed as follows:(1)ft=βα−βtβ−1e−t/αβ,β>0,α>0,t>0,where(1)
αis considers as the scale parameter, which is recognized as the Weibull slope and takes a positive value.(2)
β is considered as the shape parameter. It defines the form of distribution and takes a positive value.(3)
γ is the location parameter γ=0.The cumulative distribution function (CDF) can be written as follows:(2)Ft=1−e−t/αβ,and the survival function can be written as follows:(3)Rt=1−Ft=e−t/αβ,and the failure function for the two-parameter Weibull distribution can be written as follows:(4)ht=ftRt=βα−βtβ−1,where(1)
β>1 means the failure rate is increasing.(2)
β<1 means the failure rate is decreasing.(3)
β=1 means fixed failure rate.Therth moment of the Weibull distribution is(5)Mr=αrΓrβ+1,and when r=1,2, we get the first and second moments M1=αΓ1/β+1 and M2=α2Γ2/β+1.Thus, the mean and variance of the two-parameter Weibull distribution are, respectively,(6)Mean=αΓ1+1β,Var=α2Γβ+2β−Γβ+1β2.The standard deviation isσ=α2Γβ+2/β−Γβ+1/β2.The reliability function can be written as(7)Rt=e−t/αβ.


### 2.2. The Parameter Estimation of the Weibull Distribution

In the following subsections, we describe different parameter estimation methods for Weibull distribution.


#### 2.2.1. Maximum Likelihood Estimation (MLE)

MLE is the most commonly used method for the estimation of parameters. It makes the possible function based on the maximum. It demonstrates the function of the maximum likelihood, so the most plausible results can be obtained from relationship (1) by the following equation:(8)Lti,β,α=∏i=1nfxi,β,α=βnαn∏i=1ntαβ−1e∑i=1nt/αβ.Equation (8) can be linearized by taking the logarithm on both sides:(9)lnLti,β,α=nlnβ−nlnα+β−1∑i=1nlntiα−∑i=1ntiαβ.MLE of the parameter is the value of the parameter that maximizesL (likelihood). MLE for the Weibull distribution can be obtained by solving the equations resulting from setting the two partial derivatives of L (α, β) to zero:(10)nβ^+∑i=1nlnti−1α∑i=1ntiβlnti=0,(11)−nα+1α2∑i=1ntiβ=0.Then,β^ is the solution of(12)1β^+1n∑i=1nlnti−∑i=1ntiβlnti∑i=1ntiβ=0.Traditional methods cannot be used to solve equation (11). We can use the Newton–Raphson method [18, 19] to find a numerical solution of this nonlinear equation when the shape parameter is known, i.e.,(13)α^MLE=∑i=1ntiβn1/β^,and(14)β^=∑i=1ntiβ^lnti∑i=1ntiβ^−1−1n∑i=1nlnti−1.The obtained MLE estimates ofα then β are often biased when sample sizes are small, i.e., n<30. The MLE estimates become unbiased and achieve constancy property when the sample size is large [20]. Since the property of constancy characterizes the MLE, by using this property, we can get the MLE estimates of the reliability function, i.e.,(15)R^t=exp−tβ^α^.


#### 2.2.2. Method of Minimax (MOM)

This estimation method primarily depends on Lemann's theorem; letτ=Fα,α∈Θ be a family of distribution functions and D be a class of estimators of α. Suppose that δ∗∈D is a Bayes estimator against a prior distribution ξ∗α on Θ, the risk function Rδ∗,α equals constant on Θ, and δ∗ is a minimax estimator [21]. Assuming a quadratic loss function L=Loss=α−δ1/α2, we can find the estimator δ1∗ as the initial distribution of the parameter α distributed as a gamma distribution [21, 22].(16)πα=1Γθβθαθ−1e−α/β,,α>0,θ,β>0.A formula is derived for the estimatorδ1∗ of the parameter α; after, it is found that the subsequent distribution of the parameter α with a random sample X=x1,x2,x3,…,xn is(17)πα|t=fα,tft=1/αθ+n+1e−1/α∑i=1ntiβ+1/λ∫0∞1/αθ+n+1e−∑i=1ntiβ/α+1/λαdα,=∑i=1ntiβ+1/λn+θΓn+θ1αθ+n+1e−−1/α∑i=1ntiβ+1/λπα|t∼Gamman+θ,∑i=1ntiβ+1λ.Depending on the quadratic loss functionψϱβ1=α−δ1∗/α2,(18)Loss=α−δ1α2,Rδ,α=ELoss=∫0∞α−δ1α2πα|tdα,Rδ,α=παΙt−2δ1E1α+δ12E1α12,∂Rδ,α∂d=−2E1α+2δ1E1α12=0,⇒δ1∗=E1/α|tE1/α12|t,where(19)E1α|t=∫0∞1απα|tdα=n+θ∑i=0ntiβ+1/λ,1α12|t=∫0∞1α2πα|tdα=n+θn+θ+1∑i=0ntiβ+1/λ2,δ1∗=E1/αE1/α12=∫0∞1/απαΙtdα∫0∞1/α2παΙtdα=n+θ/∑i=0ntiβ+1/λn+θn+θ+1/∑i=0ntiβ+1/λ2,δ1∗=∑i=0ntiβ+1/λn+θ+1=Μ1η+1λ,where(20)Μ1=1n+θ+1,η=∑i=1ntiβ.


#### 2.2.3. Modified Linear Exponential (MLINEX) Loss Function

It is a modified linear exponential loss function. It is frequently used in statistical estimation and prediction problems. It can be expressed as follows:(21)Lα,δ2=kδ2αc−clnδ2α−1,where δ2 is the estimator of the parameter α and k, c are constants in the loss function.The estimator in minimax of the parameterα under the loss function (21) is δ2 which is symbolized as ψMLβ1:(22)δ2=ψMLβ1=Eα−c−1/c,Eα−c=∫0∞α−cπα|tdα,=∫0∞∑i=1ntiβ+1/λn+θΓn+θα−θ+n+c+1e−1/α∑i=1ntiβ+1/λdα=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c.The second estimator minimax under a modified quadratic loss function is(23)δ2∗=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c=Μ2η+1λ,where(24)Μ2=Γn+θΓθ+n+c1/c,η=∑i=1ntiβ.


#### 2.2.4. White's Method

This method is mainly based on the cumulative distribution function presented in equation (2) to formulate a simple linear slope model as follows:(25)Fti=1−e−ti/αβ,1−Fti=e−ti/αβ,11−Fti=eti/αβ.Equation (25) can be linearized by taking the logarithm twice on both sides:(26)ln−ln1−Fti=βlnti−βlnα.Now, we can write equation (26) as a linear regression model:(27)Yi=a+bTi,where(28)Yi=ln−ln1−Fti,Ti=lnti,a=−βlnα,andb=β.Using the ordinary least squares method (OLSM), we can estimate the parameters of the regression model as follows:(29)b^LS=∑i=1nTiYi−nT¯Y¯∑i=1nTi2−∑i=1nTi2,a^OLS=Y¯−b^LST¯,i.e.,(30)α^=e−a^LS,β^=b^LS.It can also be obtained by estimating the first order of the Taylor series for ascending order data as follows, assuming that(31)Zi=ln−ln1−pi,tΡi=β+αZi,∂t∂β=1,∂t∂α=Zi,fa+b=fa+h1f′a+h2f=a,∴fa=β0+α0Zi,Eti=β0+α0Zi+β−β0.1+θ−θ0Zi,=β0+α0Zi+β−β0+θZi−θ0Zi=β+θZi.By using the method of least squares, dependent on the matrix approach, we can estimate the parameters of the Weibull distribution by extracting the estimated values ofα,β from the following formula:(32)β^α^=HTH−1HTtZi,where(33)H=1z11z2⋮⋮1zn.


## 2.1. The Two-Parameter Weibull Distribution

The Weibull distribution contains one, two, or three parameters. These parameters are shape parameterβ and scale parameter α, and there may be a third parameter, the location γ parameter [5]. The distribution function of two-parameter Weibull distribution can be expressed as follows:(1)ft=βα−βtβ−1e−t/αβ,β>0,α>0,t>0,where(1)
αis considers as the scale parameter, which is recognized as the Weibull slope and takes a positive value.(2)
β is considered as the shape parameter. It defines the form of distribution and takes a positive value.(3)
γ is the location parameter γ=0.The cumulative distribution function (CDF) can be written as follows:(2)Ft=1−e−t/αβ,and the survival function can be written as follows:(3)Rt=1−Ft=e−t/αβ,and the failure function for the two-parameter Weibull distribution can be written as follows:(4)ht=ftRt=βα−βtβ−1,where(1)
β>1 means the failure rate is increasing.(2)
β<1 means the failure rate is decreasing.(3)
β=1 means fixed failure rate.Therth moment of the Weibull distribution is(5)Mr=αrΓrβ+1,and when r=1,2, we get the first and second moments M1=αΓ1/β+1 and M2=α2Γ2/β+1.Thus, the mean and variance of the two-parameter Weibull distribution are, respectively,(6)Mean=αΓ1+1β,Var=α2Γβ+2β−Γβ+1β2.The standard deviation isσ=α2Γβ+2/β−Γβ+1/β2.The reliability function can be written as(7)Rt=e−t/αβ.


## 2.2. The Parameter Estimation of the Weibull Distribution

In the following subsections, we describe different parameter estimation methods for Weibull distribution.


### 2.2.1. Maximum Likelihood Estimation (MLE)

MLE is the most commonly used method for the estimation of parameters. It makes the possible function based on the maximum. It demonstrates the function of the maximum likelihood, so the most plausible results can be obtained from relationship (1) by the following equation:(8)Lti,β,α=∏i=1nfxi,β,α=βnαn∏i=1ntαβ−1e∑i=1nt/αβ.Equation (8) can be linearized by taking the logarithm on both sides:(9)lnLti,β,α=nlnβ−nlnα+β−1∑i=1nlntiα−∑i=1ntiαβ.MLE of the parameter is the value of the parameter that maximizesL (likelihood). MLE for the Weibull distribution can be obtained by solving the equations resulting from setting the two partial derivatives of L (α, β) to zero:(10)nβ^+∑i=1nlnti−1α∑i=1ntiβlnti=0,(11)−nα+1α2∑i=1ntiβ=0.Then,β^ is the solution of(12)1β^+1n∑i=1nlnti−∑i=1ntiβlnti∑i=1ntiβ=0.Traditional methods cannot be used to solve equation (11). We can use the Newton–Raphson method [18, 19] to find a numerical solution of this nonlinear equation when the shape parameter is known, i.e.,(13)α^MLE=∑i=1ntiβn1/β^,and(14)β^=∑i=1ntiβ^lnti∑i=1ntiβ^−1−1n∑i=1nlnti−1.The obtained MLE estimates ofα then β are often biased when sample sizes are small, i.e., n<30. The MLE estimates become unbiased and achieve constancy property when the sample size is large [20]. Since the property of constancy characterizes the MLE, by using this property, we can get the MLE estimates of the reliability function, i.e.,(15)R^t=exp−tβ^α^.


### 2.2.2. Method of Minimax (MOM)

This estimation method primarily depends on Lemann's theorem; letτ=Fα,α∈Θ be a family of distribution functions and D be a class of estimators of α. Suppose that δ∗∈D is a Bayes estimator against a prior distribution ξ∗α on Θ, the risk function Rδ∗,α equals constant on Θ, and δ∗ is a minimax estimator [21]. Assuming a quadratic loss function L=Loss=α−δ1/α2, we can find the estimator δ1∗ as the initial distribution of the parameter α distributed as a gamma distribution [21, 22].(16)πα=1Γθβθαθ−1e−α/β,,α>0,θ,β>0.A formula is derived for the estimatorδ1∗ of the parameter α; after, it is found that the subsequent distribution of the parameter α with a random sample X=x1,x2,x3,…,xn is(17)πα|t=fα,tft=1/αθ+n+1e−1/α∑i=1ntiβ+1/λ∫0∞1/αθ+n+1e−∑i=1ntiβ/α+1/λαdα,=∑i=1ntiβ+1/λn+θΓn+θ1αθ+n+1e−−1/α∑i=1ntiβ+1/λπα|t∼Gamman+θ,∑i=1ntiβ+1λ.Depending on the quadratic loss functionψϱβ1=α−δ1∗/α2,(18)Loss=α−δ1α2,Rδ,α=ELoss=∫0∞α−δ1α2πα|tdα,Rδ,α=παΙt−2δ1E1α+δ12E1α12,∂Rδ,α∂d=−2E1α+2δ1E1α12=0,⇒δ1∗=E1/α|tE1/α12|t,where(19)E1α|t=∫0∞1απα|tdα=n+θ∑i=0ntiβ+1/λ,1α12|t=∫0∞1α2πα|tdα=n+θn+θ+1∑i=0ntiβ+1/λ2,δ1∗=E1/αE1/α12=∫0∞1/απαΙtdα∫0∞1/α2παΙtdα=n+θ/∑i=0ntiβ+1/λn+θn+θ+1/∑i=0ntiβ+1/λ2,δ1∗=∑i=0ntiβ+1/λn+θ+1=Μ1η+1λ,where(20)Μ1=1n+θ+1,η=∑i=1ntiβ.


### 2.2.3. Modified Linear Exponential (MLINEX) Loss Function

It is a modified linear exponential loss function. It is frequently used in statistical estimation and prediction problems. It can be expressed as follows:(21)Lα,δ2=kδ2αc−clnδ2α−1,where δ2 is the estimator of the parameter α and k, c are constants in the loss function.The estimator in minimax of the parameterα under the loss function (21) is δ2 which is symbolized as ψMLβ1:(22)δ2=ψMLβ1=Eα−c−1/c,Eα−c=∫0∞α−cπα|tdα,=∫0∞∑i=1ntiβ+1/λn+θΓn+θα−θ+n+c+1e−1/α∑i=1ntiβ+1/λdα=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c.The second estimator minimax under a modified quadratic loss function is(23)δ2∗=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c=Μ2η+1λ,where(24)Μ2=Γn+θΓθ+n+c1/c,η=∑i=1ntiβ.


### 2.2.4. White's Method

This method is mainly based on the cumulative distribution function presented in equation (2) to formulate a simple linear slope model as follows:(25)Fti=1−e−ti/αβ,1−Fti=e−ti/αβ,11−Fti=eti/αβ.Equation (25) can be linearized by taking the logarithm twice on both sides:(26)ln−ln1−Fti=βlnti−βlnα.Now, we can write equation (26) as a linear regression model:(27)Yi=a+bTi,where(28)Yi=ln−ln1−Fti,Ti=lnti,a=−βlnα,andb=β.Using the ordinary least squares method (OLSM), we can estimate the parameters of the regression model as follows:(29)b^LS=∑i=1nTiYi−nT¯Y¯∑i=1nTi2−∑i=1nTi2,a^OLS=Y¯−b^LST¯,i.e.,(30)α^=e−a^LS,β^=b^LS.It can also be obtained by estimating the first order of the Taylor series for ascending order data as follows, assuming that(31)Zi=ln−ln1−pi,tΡi=β+αZi,∂t∂β=1,∂t∂α=Zi,fa+b=fa+h1f′a+h2f=a,∴fa=β0+α0Zi,Eti=β0+α0Zi+β−β0.1+θ−θ0Zi,=β0+α0Zi+β−β0+θZi−θ0Zi=β+θZi.By using the method of least squares, dependent on the matrix approach, we can estimate the parameters of the Weibull distribution by extracting the estimated values ofα,β from the following formula:(32)β^α^=HTH−1HTtZi,where(33)H=1z11z2⋮⋮1zn.


## 2.2.1. Maximum Likelihood Estimation (MLE)

MLE is the most commonly used method for the estimation of parameters. It makes the possible function based on the maximum. It demonstrates the function of the maximum likelihood, so the most plausible results can be obtained from relationship (1) by the following equation:(8)Lti,β,α=∏i=1nfxi,β,α=βnαn∏i=1ntαβ−1e∑i=1nt/αβ.Equation (8) can be linearized by taking the logarithm on both sides:(9)lnLti,β,α=nlnβ−nlnα+β−1∑i=1nlntiα−∑i=1ntiαβ.MLE of the parameter is the value of the parameter that maximizesL (likelihood). MLE for the Weibull distribution can be obtained by solving the equations resulting from setting the two partial derivatives of L (α, β) to zero:(10)nβ^+∑i=1nlnti−1α∑i=1ntiβlnti=0,(11)−nα+1α2∑i=1ntiβ=0.Then,β^ is the solution of(12)1β^+1n∑i=1nlnti−∑i=1ntiβlnti∑i=1ntiβ=0.Traditional methods cannot be used to solve equation (11). We can use the Newton–Raphson method [18, 19] to find a numerical solution of this nonlinear equation when the shape parameter is known, i.e.,(13)α^MLE=∑i=1ntiβn1/β^,and(14)β^=∑i=1ntiβ^lnti∑i=1ntiβ^−1−1n∑i=1nlnti−1.The obtained MLE estimates ofα then β are often biased when sample sizes are small, i.e., n<30. The MLE estimates become unbiased and achieve constancy property when the sample size is large [20]. Since the property of constancy characterizes the MLE, by using this property, we can get the MLE estimates of the reliability function, i.e.,(15)R^t=exp−tβ^α^.


## 2.2.2. Method of Minimax (MOM)

This estimation method primarily depends on Lemann's theorem; letτ=Fα,α∈Θ be a family of distribution functions and D be a class of estimators of α. Suppose that δ∗∈D is a Bayes estimator against a prior distribution ξ∗α on Θ, the risk function Rδ∗,α equals constant on Θ, and δ∗ is a minimax estimator [21]. Assuming a quadratic loss function L=Loss=α−δ1/α2, we can find the estimator δ1∗ as the initial distribution of the parameter α distributed as a gamma distribution [21, 22].(16)πα=1Γθβθαθ−1e−α/β,,α>0,θ,β>0.A formula is derived for the estimatorδ1∗ of the parameter α; after, it is found that the subsequent distribution of the parameter α with a random sample X=x1,x2,x3,…,xn is(17)πα|t=fα,tft=1/αθ+n+1e−1/α∑i=1ntiβ+1/λ∫0∞1/αθ+n+1e−∑i=1ntiβ/α+1/λαdα,=∑i=1ntiβ+1/λn+θΓn+θ1αθ+n+1e−−1/α∑i=1ntiβ+1/λπα|t∼Gamman+θ,∑i=1ntiβ+1λ.Depending on the quadratic loss functionψϱβ1=α−δ1∗/α2,(18)Loss=α−δ1α2,Rδ,α=ELoss=∫0∞α−δ1α2πα|tdα,Rδ,α=παΙt−2δ1E1α+δ12E1α12,∂Rδ,α∂d=−2E1α+2δ1E1α12=0,⇒δ1∗=E1/α|tE1/α12|t,where(19)E1α|t=∫0∞1απα|tdα=n+θ∑i=0ntiβ+1/λ,1α12|t=∫0∞1α2πα|tdα=n+θn+θ+1∑i=0ntiβ+1/λ2,δ1∗=E1/αE1/α12=∫0∞1/απαΙtdα∫0∞1/α2παΙtdα=n+θ/∑i=0ntiβ+1/λn+θn+θ+1/∑i=0ntiβ+1/λ2,δ1∗=∑i=0ntiβ+1/λn+θ+1=Μ1η+1λ,where(20)Μ1=1n+θ+1,η=∑i=1ntiβ.


## 2.2.3. Modified Linear Exponential (MLINEX) Loss Function

It is a modified linear exponential loss function. It is frequently used in statistical estimation and prediction problems. It can be expressed as follows:(21)Lα,δ2=kδ2αc−clnδ2α−1,where δ2 is the estimator of the parameter α and k, c are constants in the loss function.The estimator in minimax of the parameterα under the loss function (21) is δ2 which is symbolized as ψMLβ1:(22)δ2=ψMLβ1=Eα−c−1/c,Eα−c=∫0∞α−cπα|tdα,=∫0∞∑i=1ntiβ+1/λn+θΓn+θα−θ+n+c+1e−1/α∑i=1ntiβ+1/λdα=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c.The second estimator minimax under a modified quadratic loss function is(23)δ2∗=Γθ+n+cΓθ+n∑i=1ntiβ+1λ−c=Μ2η+1λ,where(24)Μ2=Γn+θΓθ+n+c1/c,η=∑i=1ntiβ.


## 2.2.4. White's Method

This method is mainly based on the cumulative distribution function presented in equation (2) to formulate a simple linear slope model as follows:(25)Fti=1−e−ti/αβ,1−Fti=e−ti/αβ,11−Fti=eti/αβ.Equation (25) can be linearized by taking the logarithm twice on both sides:(26)ln−ln1−Fti=βlnti−βlnα.Now, we can write equation (26) as a linear regression model:(27)Yi=a+bTi,where(28)Yi=ln−ln1−Fti,Ti=lnti,a=−βlnα,andb=β.Using the ordinary least squares method (OLSM), we can estimate the parameters of the regression model as follows:(29)b^LS=∑i=1nTiYi−nT¯Y¯∑i=1nTi2−∑i=1nTi2,a^OLS=Y¯−b^LST¯,i.e.,(30)α^=e−a^LS,β^=b^LS.It can also be obtained by estimating the first order of the Taylor series for ascending order data as follows, assuming that(31)Zi=ln−ln1−pi,tΡi=β+αZi,∂t∂β=1,∂t∂α=Zi,fa+b=fa+h1f′a+h2f=a,∴fa=β0+α0Zi,Eti=β0+α0Zi+β−β0.1+θ−θ0Zi,=β0+α0Zi+β−β0+θZi−θ0Zi=β+θZi.By using the method of least squares, dependent on the matrix approach, we can estimate the parameters of the Weibull distribution by extracting the estimated values ofα,β from the following formula:(32)β^α^=HTH−1HTtZi,where(33)H=1z11z2⋮⋮1zn.


## 3. Results and Discussion

### 3.1. The Comparison

The efficiency of estimation methods can be compared by comparing the variance of estimation methods. For comparing the efficiency of the two estimatorsδ1∗,δ2∗, we need to calculate the variance of each one of them separately:(34)Varδ1∗=1θ+n+12Var∑i=1ntiβ+1λ,(35)Varδ2∗=Γθ+nΓθ+n+c2/cVar∑i=1ntiβ+1λ.The efficiency ofδ2∗ relative to δ1∗ is as follows:(36)effδ1∗,δ2∗=Γθ+n/Γθ+n+c2/cVar∑i=1ntiβ+1/λ1/θ+n+12Var∑i=1ntiβ+1/λ=Γθ+n/Γθ+n+c2/c1/θ+n+12.Whenc=1, it becomes(37)effδ1∗,δ2∗=θ+n+12θ+n2>1.Whenc=−1, it becomes(38)effδ1∗,δ2∗=θ+n+12θ+n−12>1.Since eff >1, the Bayes estimator under the quadratic loss functionδ1∗ is more efficient than δ2∗.To compare the efficiency of the estimatorsδ1∗,α^MLE, we need to calculate the variance of each estimator, i.e.,(39)Varα^MLE=Var∑i=1ntiβn1/β^=1n1/β2Var∑i=1ntiβ1/β.Letλ in equations (34) and (35) be constant and known then:(40)Varδ1∗=1θ+n+12Var∑i=1ntiβ,Varδ2∗=Γθ+nΓθ+n+c2/cVar∑i=1ntiβ,Varα^OLS=b^OLS2n2Var∑i=1nti.Thus, the efficiency ofδ1∗ relative to α^MLE equals(41)effα^MLE,δ1∗=1/θ+n+12Var∑i=1ntiβ1/n1/β2Var∑i=1ntiβ1/β>1,effα^MLE,δ2∗=Γθ+n/Γθ+n+c2/cVar∑i=1ntiβ1/n1/β2Var∑i=1ntiβ1/β>1.Since eff >1, the efficiency of the maximum likelihood estimatorα^MLE is more efficient than the estimators δ1∗andδ2∗.The efficiency ofα^MLE relative to α^OLS equals(42)effα^MLE,α^OLS=b^OLS2/n2Var∑i=1nti1/n1/β2Var∑i=1ntiβ1/β>1,and hence α^MLE is more efficient than α^OLS.From the above mathematical expressions, it can be concluded the maximum likelihood estimatorα^MLE is more efficient than δ1∗, δ2∗, and α^OLS in the scale parameter estimation for the Weibull distribution. Also, in the case of Bayes estimator under the quadratic loss function, δ1∗ is more efficient than δ2∗.


### 3.2. Simulation

The simulation technique was used to achieve the objective of the study, to compare the estimation methods experimentally (maximum likelihood, minimax, MLINEX, and White's method). This method is characterized by flexibility, and to achieve this purpose, random samples were generated for the parameters to be estimated using the Monte Carlo method theoretically, not practically, and without violating the accuracy of the results required according to the following steps:(1)
The assumed values for the real distribution parameters were determined by Table1 from four different models, including different values for the shape and scale parameters.(2)
The assumed four different sizes were identified for sample (n = 10, 50, 100, 150) and frequency of experiment (L = 500, 1000, 2000).∗C is the Jeffrey constant.(3)
The data are generated from the Weibull distribution for assumed values of parameters and specified sample sizes, i.e.,(1)
Generating random numbersVi that follow the uniform distribution on the interval 0,1.(43)Ui∼U0,1,i=1,2,…,n,whereUi is a continuous random variable.(2)
Conversion of the data generated in Step (1), which follows the uniform distribution to the Weibull distribution data using the cumulative distribution function, according to the following inverse conversion method:(44)Ft=1−e−ti/αβ,1−Ft=e−ti/αβ,1−U=e−ti/αβ,ln1−U=−tiαβ=−tβαβ,t=−αβln1−U1β,0≤U≤1.Table 1
Different values of the parameters used.
ModelαβλθC∗I0.50.610.51II0.70.810.5−1III1.00.90.512IV1.31.20.51−2


## 3.1. The Comparison

The efficiency of estimation methods can be compared by comparing the variance of estimation methods. For comparing the efficiency of the two estimatorsδ1∗,δ2∗, we need to calculate the variance of each one of them separately:(34)Varδ1∗=1θ+n+12Var∑i=1ntiβ+1λ,(35)Varδ2∗=Γθ+nΓθ+n+c2/cVar∑i=1ntiβ+1λ.The efficiency ofδ2∗ relative to δ1∗ is as follows:(36)effδ1∗,δ2∗=Γθ+n/Γθ+n+c2/cVar∑i=1ntiβ+1/λ1/θ+n+12Var∑i=1ntiβ+1/λ=Γθ+n/Γθ+n+c2/c1/θ+n+12.Whenc=1, it becomes(37)effδ1∗,δ2∗=θ+n+12θ+n2>1.Whenc=−1, it becomes(38)effδ1∗,δ2∗=θ+n+12θ+n−12>1.Since eff >1, the Bayes estimator under the quadratic loss functionδ1∗ is more efficient than δ2∗.To compare the efficiency of the estimatorsδ1∗,α^MLE, we need to calculate the variance of each estimator, i.e.,(39)Varα^MLE=Var∑i=1ntiβn1/β^=1n1/β2Var∑i=1ntiβ1/β.Letλ in equations (34) and (35) be constant and known then:(40)Varδ1∗=1θ+n+12Var∑i=1ntiβ,Varδ2∗=Γθ+nΓθ+n+c2/cVar∑i=1ntiβ,Varα^OLS=b^OLS2n2Var∑i=1nti.Thus, the efficiency ofδ1∗ relative to α^MLE equals(41)effα^MLE,δ1∗=1/θ+n+12Var∑i=1ntiβ1/n1/β2Var∑i=1ntiβ1/β>1,effα^MLE,δ2∗=Γθ+n/Γθ+n+c2/cVar∑i=1ntiβ1/n1/β2Var∑i=1ntiβ1/β>1.Since eff >1, the efficiency of the maximum likelihood estimatorα^MLE is more efficient than the estimators δ1∗andδ2∗.The efficiency ofα^MLE relative to α^OLS equals(42)effα^MLE,α^OLS=b^OLS2/n2Var∑i=1nti1/n1/β2Var∑i=1ntiβ1/β>1,and hence α^MLE is more efficient than α^OLS.From the above mathematical expressions, it can be concluded the maximum likelihood estimatorα^MLE is more efficient than δ1∗, δ2∗, and α^OLS in the scale parameter estimation for the Weibull distribution. Also, in the case of Bayes estimator under the quadratic loss function, δ1∗ is more efficient than δ2∗.


## 3.2. Simulation

The simulation technique was used to achieve the objective of the study, to compare the estimation methods experimentally (maximum likelihood, minimax, MLINEX, and White's method). This method is characterized by flexibility, and to achieve this purpose, random samples were generated for the parameters to be estimated using the Monte Carlo method theoretically, not practically, and without violating the accuracy of the results required according to the following steps:(1)
The assumed values for the real distribution parameters were determined by Table1 from four different models, including different values for the shape and scale parameters.(2)
The assumed four different sizes were identified for sample (n = 10, 50, 100, 150) and frequency of experiment (L = 500, 1000, 2000).∗C is the Jeffrey constant.(3)
The data are generated from the Weibull distribution for assumed values of parameters and specified sample sizes, i.e.,(1)
Generating random numbersVi that follow the uniform distribution on the interval 0,1.(43)Ui∼U0,1,i=1,2,…,n,whereUi is a continuous random variable.(2)
Conversion of the data generated in Step (1), which follows the uniform distribution to the Weibull distribution data using the cumulative distribution function, according to the following inverse conversion method:(44)Ft=1−e−ti/αβ,1−Ft=e−ti/αβ,1−U=e−ti/αβ,ln1−U=−tiαβ=−tβαβ,t=−αβln1−U1β,0≤U≤1.Table 1
Different values of the parameters used.
ModelαβλθC∗I0.50.610.51II0.70.810.5−1III1.00.90.512IV1.31.20.51−2


## 4. Comparison of Estimation Methods

The mean square error (MSE) was considered as a basis for comparison between estimation methods, which takes the following form:(45)MSEα^=∑i=1Lα^i−α2L,where L is the frequency of each experiment.


## 5. Results

The results of simulation experiments were presented and interpreted to find the best estimate of distribution parameters by applying equations (13), (19), (23), and (30). The experiment was carried out on values generated from equation (44) to find the efficiency of the estimators using MATLAB 15 in the generation process, as shown in Table 2.Table 2
The MSE values for all methods and sample sizes used in simulations whenL = 500.
ModelNMSEPriority order of methodsBestMLINEXMOMMLEOLSMLINEXMOMMLEOLSI100.09040.05310.07311.89713124MOM500.03910.03550.01380.95123214MLE1000.03060.02940.00640.82143214MLE1500.02970.02780.00430.09843214MLEII100.14460.06390.08692.14573124MOM500.04320.03170.01390.87423214MLE1000.03410.02990.00790.06573214MLE1500.03240.02830.00530.74213214MLEIII100.09290.02650.08651.69713124MOM500.04010.03820.01840.81453214MLE1000.03480.03140.00960.09853214MLE1500.03110.02960.00610.07453214MLEIV100.20490.08180.11540.87423124MOM500.05040.03670.02410.09543214MLE1000.03810.03120.01160.05123214MLE1500.03430.03000.00810.04513214MLETable2 shows that the maximum likelihood method was the best and more efficient in estimating the scale parameter for Weibull distribution, with the lowest MSE for all models and sample sizes used, compared with other methods, except small sample sizes n = 10. On the other hand, the minimax method based on the quadratic loss function was the most efficient for all models; it can be observed from the results of the last column of Table 2. Hence, the maximum likelihood method was the best in 12 out of the 16 estimations for the scale parameter.Also, it can be observed that the MSE values decreased with the increase in the sample size for all models. It corresponds to the statistical theory, which confirms the validity of the theoretical side in this paper about the behavior of this function. The simulation results are similar, although the experiment was repeated 500, 1000, and 2000 times, which shows the estimators' stability and efficiency.The simulation results showed that the MOM method ranked second in the efficiency in the case of medium and large samples and for all models and sizes of samples. The MLINEX ranked third in terms of efficiency, and OLS ranked fourth with the lowest efficiency. Likewise, we obtained the same results in Tables3 and 4.Table 3
The MSE values for all methods and sample sizes used in simulations whenL = 1000.
ModelNMSEPriority order of methodsBestMLINEXMOMMLEOLSMLINEXMOMMLEOLSI100.09460.05720.07312.16453124MOM500.03910.03380.01421.12453214MLE16000.03240.02880.00740.95423214MLE1500.02940.02830.00460.84213214MLEII100.13410.05900.08022.25413124MOM500.04100.03380.01461.19873214MLE1000.07410.03920.02820.87413214MLE1500.03110.02840.00560.04153214MLEIII100.09490.07800.08892.02453124MOM500.04010.03850.01931.79513214MLE1000.03260.03180.00991.62813214MLE1500.03150.02880.00690.07823214MLEIV100.21120.07440.11902.35473124MOM500.09380.03720.02291.84163214MLE1000.03970.03240.01180.68743214MLE1500.03440.02910.00790.07463214MLETable 4
The MSE values for all methods and sample sizes used in simulations whenL = 2000.
ModelNMSEPriority order of methodsBestMLINEXMOMMLEOLSMLINEXMOMMLEOLSI100.09080.06100.07532.45123124MOM500.03900.03210.01511.25713214MLE1000.03200.03080.00671.05783214MLE1500.03060.02930.00520.07433214MLEII100.13210.05830.07841.98453124MOM500.04320.03240.01561.12343214MLE1000.03470.02990.00871.04513214MLE1500.03100.02910.00530.65483214MLEIII100.09270.07400.08731.82573124MOM500.04030.03730.02051.08713214MLE1000.03410.03240.00990.82463214MLE1500.03110.02980.00740.06873214MLEIV100.20910.08040.12091.12483124MOM500.09230.03820.02251.08353214MLE1000.03770.03180.01250.54133214MLE1500.03440.02890.00810.05793214MLETable3 also shows that the maximum likelihood method was the best and more efficient in estimating the scale parameter for Weibull distribution, with the lowest MSE for all models and sample sizes used, compared with other methods, except small sample sizes n=10. The minimax method based on the quadratic loss function was the most efficient for all models. It is clear in the results of the last column of Table 3. Also, it is observed that the MSE values decreased with the increase in the sample size for all models. This, however, corresponds to the statistical theory. The simulation results are similar, although the experiment was repeated 500, 1000, and 2000 times, and this shows the stability of the estimator's efficiency.The simulation results showed that the MOM method ranked second in the efficiency in the case of medium and large samples and for all models and sizes of samples. The MLINEX ranked third in terms of efficiency, and OLS ranked fourth with the lowest efficiency.Likewise, Table4 also shows that the maximum likelihood method was the best and more efficient in estimating the scale parameter for Weibull distribution, with the lowest MSE for all models and sample sizes used, compared with other methods except small sample sizes n = 10.The minimax method based on the quadratic loss function was the most efficient for all models. It is clear in the results of the last column of Table4, and hence the maximum likelihood method was the best in 12 of the total 16 estimations for the scale parameter.Also, it can be observed that the MSE values decreased with the increase in the sample size for all models. The simulation results are similar, although the experiment was repeated 500, 1000, and 2000 times, and this shows the stability of the efficiency of the estimators. The simulation results showed that the MOM method ranked second in the efficiency in the case of medium and large samples and for all models and sizes of samples. The MLINEX ranked third in terms of efficiency, and OLS ranked fourth with the lowest efficiency.


## 6. Discussion

The present analysis is commenced to select an appropriate parameter estimation method among several estimation methods for two-parameter Weibull distribution. Generally, the maximum likelihood method is used for parameter estimation when the shape parameter is known. However, we estimated mathematically the scale parameter of the two-parameter Weibull distribution. The relative efficient estimation method is identified based on minimum MSE. The theoretical outcomes demonstrated that the efficiency of the obtained estimatorα^MLE from MLE is more efficient than the estimators δ1∗,δ2∗, and α^OLS applied in estimation for the parameters. Further, the results demonstrated that the estimator obtained from MLE is best and more efficient in estimating the scale parameter than other associated methods. Furthermore, increasing the sample size reduced the MSE. Further, the acquired information validated that simulation results are providing the same information as hypothetical results.


## 7. Conclusion

The theoretical results showed that the efficiency of the maximum likelihood estimatorα^MLE is more efficient than the estimators δ1∗,δ2∗, and α^OLS used in estimation for the parameters of Weibull distribution. Further, the results show that the maximum likelihood estimator is best and more efficient in estimating the scale parameter than the other methods. Moreover, the mean squares error (MSE) was reduced with the increase in the size of the sample. Furthermore, the theoretical results are validated by the simulation results. The obtained information validates that simulation results are providing the same information as hypothetical results. However, the experiment was repeated 500, 1000, and 2000 times to show the stability of the efficiency of the estimators. The maximum likelihood method ranked first, the minimax method ranked second, MLINEX ranked third, and OLS ranked fourth in terms of efficiency.



---

*Source: 5806068-2021-11-02.xml*