Bias Correction Method for Log-Power-Normal Distribution

Abstract

The log-power-normal distribution is a generalized version of the log-normal distribution. The maximum likelihood estimation method is the most popular method to obtain the estimates of the log-power-normal distribution parameters. In this article, we investigate the performance of the maximum likelihood estimation method for point and interval inferences. Moreover, a simple method that has less impact from the subjective selection of the initial solutions to the model parameters is proposed. The bootstrap bias correction method is used to enhance the estimation performance of the maximum likelihood estimation method. The proposed bias correction method is simple for use. Monte Carlo simulations are conducted to check the quality of the proposed bias correction method. The simulation results indicate that the proposed bias correction method can improve the performance of the maximum likelihood estimation method with a smaller bias and provide a coverage probability close to the nominal confidence coefficient. Two real examples about the air pollution and cement’s concrete strength are used for illustration.

1. Introduction

Log-normal (shortened by LN) distribution has been extensively used to characterize lifetime data in the applications of reliability analysis and quality control. Typical applications of the LN distribution can be found in modeling the fatigue failure and failure rate. In order to study composite materials subject to fatigue loading, Ratnaparkhi et al. [1] considered Yang’s differential equation for residual strength degradation to obtain the LN model for characterizing the fatigue life of a composite material. Considering the fatigue strength distribution as a function of the number of failure cycles, Murty et al. [2] derived this fatigue strength distribution via the S–N curve representing fatigue as follows. The number of cycles to failure at two levels of stress is used to obtain the S–N curve, and the numbers of cycles to failure are in turn assigned to follow a LN distribution. The LN distribution has a close relationship with the normal distribution. Let random variable, Y, follow a LN distribution with location parameter $\xi$ and scale parameter $\sigma$, then $log\left(Y\right)$ follows a normal distribution with mean $\xi$ and standard deviation $\sigma$.

Similar to the relationship between the LN distribution and the normal distribution, a close relationship can be found between the log-power-normal (LPN) distribution and power-normal (PN) distribution. The PN distribution is a generalized version of the normal distribution. Goto and Inoue [3] introduced the PN family of distributions, investigated its statistical properties and presented the moment function in terms of infinite series in 1980. Later, Goto et al. [4] discussed some applications through using PN distribution and [5] studied the maximum likelihood estimation method for the normalized PN distribution. Hereafter, the “MLE method” is used to denote the maximum likelihood estimation method for simplicity. Freeman and Modarres [6] used the inverse Box–Cox power transformation to obtain another expression for the PN distribution. They also obtained the mathematical forms of the mean and variance for the PN distribution. Castillo et al. [7] studied the statistical properties of the truncated PN by truncating the PN distribution at zero. They also studied the probabilistic properties based on the MLE and moments estimation methods. Maruo and Goto [8] used the delta method to propose several estimation methods to obtain the confidence interval of the percentile of the PN distribution. Gupta and Gupta [9] investigated the closeness of the PN and skew-normal distributions and then proposed estimation procedures to model skewed data via using the PN distribution. Maruo et al. [10] used simulation methods to evaluate the asymptotic impact of the truncation on the parameter estimation for the PN distribution. Assume that random variable Y follows a LPN distribution, then the logarithm transformation $X=log\left(Y\right)$ follows a PN distribution. Martiner-Florez et al. [11] studied the MLE method for the LPN distribution and used the LPN distribution to characterize air contamination data. Zhu et al. [12] used the PN distribution for quality control applications.

In the real world, the lifetime distribution often does not have a symmetric shape. A generalized version of the LN distribution can be helpful for reliability analysis and quality control applications. The log-skew-normal distribution could be an option because the logarithm transformation of random variable inside the skew-normal distribution can follow the shape of the skew-normal distribution. More information can be seen from [13]. The skew-normal distribution is not the best model to capture high degrees of kurtosis in data. Moreover, the quality of the estimation results based on the skew-normal distribution could not be reliable if the skewness parameter is close to zero. The PN distribution could be a better model for the higher kurtosis data, which have a narrow range of asymmetry, than the normal and skew-normal distributions. More information about the aforementioned distributions can be seen in [14,15]. In this study, we emphasize on suggesting a bias correction method for the LPN distribution.

2. Motivation and Organization

Let X follow the PN distribution with parameter $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)=(\xi ,\sigma ,\gamma )$. The probability density function (PDF) and cumulative density function (CDF) of the PN distribution are defined, respectively, by

$$\begin{array}{c}\hfill f_X\left(x\right|\Theta )=\frac{\gamma }{\sigma }{\left[\Phi \left(\frac{x-\xi }{\sigma }\right)\right]}^{\gamma -1}\varphi \left(\frac{x-\xi }{\sigma }\right),x,\xi \in R,\gamma ,\sigma >0\end{array}$$

(1)

and

$$\begin{array}{c}\hfill F_X\left(x\right|\Theta )={\left[\Phi \left(\frac{x-\xi }{\sigma }\right)\right]}^{\gamma },x,\xi \in R,\gamma ,\sigma >0,\end{array}$$

(2)

where $\xi$, $\sigma$ and $\gamma$ are the location, scale and shape parameters, respectively. Let $X=log\left(Y\right)$, Martiner-Florez et al. [11] presented the PDF and CDF of Y by

$$\begin{array}{c}\hfill f\left(y\right|\Theta )=\frac{\gamma }{y\sigma }{\left[\Phi \left(\frac{log\left(y\right)-\xi }{\sigma }\right)\right]}^{\gamma -1}\varphi \left(\frac{log\left(y\right)-\xi }{\sigma }\right),y>0\end{array}$$

(3)

and

$$\begin{array}{c}\hfill F\left(y\right|\Theta )={\left[\Phi \left(\frac{log\left(y\right)-\xi }{\sigma }\right)\right]}^{\gamma },y>0,\end{array}$$

(4)

respectively. The distribution of Y, named LPN, is defined by Equations (3) and (4). Symbolically, it is denoted by $Y\sim LPN(\Theta )$. The LPN distribution reduces to the LN distribution when $\gamma =1$. The survival function and hazard function at $y_0>0$ can be obtained, respectively, by

$$\begin{array}{c}\hfill S\left(y_0\right|\Theta )=1-F\left(y_0\right|\Theta )=1-{\left[\Phi \left(\frac{log\left(y_0\right)-\xi }{\sigma }\right)\right]}^{\gamma }\end{array}$$

(5)

and

$$\begin{array}{c}\hfill h\left(y_0\right|\Theta )\equiv \frac{f\left(y_0\right|\Theta )}{S\left(y_0\right|\Theta )}.\end{array}$$

(6)

The pth quantile function of $LPN(\Theta )$ can be presented by

$$\begin{array}{c}\hfill q_p\equiv q_p\left(p\right|\Theta )=exp\{\xi +\sigma {\Phi }^{-1}\left(p^{\frac{1}{\gamma }}\right)\},0<p<1.\end{array}$$

(7)

Figure 1 displays the PDFs of the standard LPN distribution with various shape parameters. Using the MLE method and Fisher information matrix to obtain reliable point and interval estimates has been the most popular procedure for the parametric modeling in the real world. Martiner-Florez et al. [11] investigated the MLE method for the LPN distribution. Zhu et al. [12] studied the evaluation of the process performance index via using PN distribution. They mentioned that using the Fisher information matrix to obtain the confidence interval of the process performance index could be problematic and recommended using bootstrap methods instead. Even if the parametric bootstrap method could be used to obtain the confidence interval of the model parameter, using reliable MLEs to generate high-quality bootstrap samples is the key factor for guaranteeing the estimation performance quality. However, the MLE method is sensitive to the initial solutions of the model parameters. Moreover, the log-likelihood function with several model parameters could not be convex. This fact makes the MLE method more sensitive to the initial solution when the dimension of the model parameter over two. We are motivated to propose methods to improve the quality of MLEs with the two aspects:

• For reducing the influence of subjective assigning the initial solutions of $\xi$, $\sigma$ and $\gamma$ in the LPN distribution, we propose a simple least square (LS) estimation method to obtain reliable initial solutions of $\xi$, $\sigma$ and $\gamma$ for MLE estimation.
• Propose a bias-correction MLE method to enhance the estimation performance of interval inference.

In this study, we focus on studying the performance for estimating the survival function at different quantiles. The survival function is an important metric to measure the lifetime quality of products in reliability analysis and quality control applications. For simplicity, we use BcLS-MLE method here and after to denote the proposed bias-correction MLE method with using the LS estimates as initial solutions of $\xi$, $\sigma$ and $\gamma$.

The rest of this article is organized as follows. In Section 3, we investigate the MLE method, exact and observed Fisher information matrices. The confidence interval of a general differentiable function of $\Theta$ in the LPN distribution is constructed via using the observed Fisher information matrix and delta method. A regression method based on using the quantile function of LPN is used to obtain reasonable initial solutions for the LPN distribution parameters. The bootstrap bias-correction method is proposed to improve the inference performance of the MLE-based methods. Monte Carlo simulations are conducted in Section 4 to evaluate the performance of the proposed methods in terms of the measures of relative bias (rbias) and relative square root mean squared error (rsqMSE). We discuss the superiority of the proposed methods over the typical MLE method as well. In Section 5, two examples about the air pollution and cement’s concrete strength are used to illustrate the applications of the proposed estimation procedures. Finally, some concluding remarks are given in Section 6.

3. Parameter Estimation Methods

In this section, we investigate the typical MLE method and propose a bias-correction MLE method. Moreover, we use the quantile function of $LPN(\Theta )$ to propose a simple method to obtain a reasonable initial solution of $\Theta$.

3.1. Maximum Likelihood Estimation Method

Let $\mathit{Y}=(Y_1,Y_2,\dots ,Y_n)$ be a random sample taken from $LPN(\Theta )$. The likelihood and log-likelihood functions can be presented, respectively, by

$$\begin{array}{ccc}\hfill L(\Theta |\mathit{Y})& =& \prod _{i=1}^{n}f\left(y_i\right|\Theta )\hfill \\ & =& {\left(\frac{\gamma }{\sigma }\right)}^n\prod _{i=1}^n\frac{1}{y_i}{\left[\Phi \left(\frac{log\left(y_i\right)-\xi }{\sigma }\right)\right]}^{\gamma -1}\varphi \left(\frac{log\left(y\right)-\xi }{\sigma }\right)\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill \ell \equiv \ell (\Theta |\mathit{Y})& =& n(log\left(\gamma \right)-log\left(\sigma \right))-\sum _{i=1}^{n}log\left(y_i\right)\hfill \\ & +& (\gamma -1)\sum _{i=1}^{n}log(\Phi \left(z_i\right))+\sum _{i=1}^{n}log\left(\varphi \left(z_i\right)\right),\hfill \end{array}$$

(9)

where $z_i=\frac{log\left(y_i\right)-\xi }{\sigma }$ follows the standard PN distribution, $i=1,2,\dots ,n$. Martiner-Florez et al. [11] suggested the process to obtain the MLE of $\Theta$ and the observed Fisher information matrix. The first derivative of the log-likelihood function can be expressed by

$$\begin{array}{ccc}\hfill \frac{\partial \ell }{\partial {\theta }_1}& =& \frac{\partial \ell }{\partial \xi }=\frac{n}{\sigma }\left\{\overline{z}-(\gamma -1)\overline{w}\right\},\hfill \\ \hfill \frac{\partial \ell }{\partial {\theta }_2}& =& \frac{\partial \ell }{\partial \sigma }=-\frac{n}{\sigma }\left\{1-\frac{{\sum }_{i=1}^n{z}_i^2}{n}+(\gamma -1)\frac{{\sum }_{i=1}^n{z}_i{w}_i}{n}\right\},\hfill \\ \hfill \frac{\partial \ell }{\partial {\theta }_3}& =& \frac{\partial \ell }{\partial \gamma }=n\left\{\frac{1}{\gamma }+\overline{v}\right\}\hfill \end{array}$$

where $w_i=\frac{\varphi \left(z_i\right)}{\Phi \left(z_i\right)}$, $v_i=log(\Phi \left(z_i\right))$, $i=1,2,\dots ,n$. Denote the MLE of $\Theta$ by $\widehat{\Theta }=({\widehat{\theta }}_1,{\widehat{\theta }}_2,{\widehat{\theta }}_3)=(\widehat{\xi },\widehat{\sigma },\widehat{\gamma })$. $\widehat{\Theta }$ is the solution of the likelihood functions of $\frac{\partial \ell }{\partial {\theta }_i}=0,i=1,2,3$. The second order derivatives of the log-likelihood function can be expressed by

$$\begin{array}{c}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_1^2}=\frac{{\partial }^2\ell }{\partial {\xi }^2}=-\frac{1}{{\sigma }^2}\left\{n+(\gamma -1)\left(\sum _{i=1}^n{w}_i^2+\sum _{i=1}^n{z}_i{w}_i\right)\right\},\end{array}$$

(10)

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_1{\theta }_2}& =& \frac{{\partial }^2\ell }{\partial {\theta }_2{\theta }_1}=\frac{{\partial }^2\ell }{\partial \xi \partial \sigma }\hfill \\ & =& -\frac{1}{\sigma }\left\{2\sum _{i=1}^n{z}_i+(\gamma -1)\left(\sum _{i=1}^n{z}_i{w}_i^2+\sum _{i=1}^n{z}_i^2{w}_i-\sum _{i=1}^n{w}_i\right)\right\},\hfill \end{array}$$

(11)

$$\begin{array}{c}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_1{\theta }_3}=\frac{{\partial }^2\ell }{\partial {\theta }_3{\theta }_1}=\frac{{\partial }^2\ell }{\partial \xi \partial \gamma }=-\frac{1}{\sigma }\sum _{i=1}^n{w}_i,\end{array}$$

(12)

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_2^2}& =& \frac{{\partial }^2\ell }{\partial {\sigma }^2}\hfill \\ & =& -\frac{1}{{\sigma }^2}\left\{3\sum _{i=1}^n{z}_i^2-n-(\gamma -1)\left(2\sum _{i=1}^n{z}_i{w}_i-\sum _{i=1}^n{z}_i^3{w}_i-\sum _{i=1}^n{z}_i^2{w}_i^2\right)\right\},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_2{\theta }_3}& =& \frac{{\partial }^2\ell }{\partial {\theta }_3{\theta }_2}=\frac{{\partial }^2\ell }{\partial \sigma \partial \gamma }=-\frac{1}{\sigma }\sum _{i=1}^n{z}_i{w}_i,\hfill \end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial {\theta }_3^2}& =& \frac{{\partial }^2\ell }{\partial {\gamma }^2}=-\frac{n}{{\gamma }^2}.\hfill \end{array}$$

(15)

Let ${\eta }_{ij}=E\left(\frac{{\partial }^2\ell }{\partial {\theta }_i{\theta }_j}\right)$, for $i=1,2,3$ and $j=1,2,3$ and $a_{st}=E\left(Z^s{W}^t\right)$, for $s=0,1,2,3$ and $t=1,2$. We can show that

$$\begin{array}{ccc}\hfill a_{st}& =& {\int }_{-\infty }^{\infty }{z}^s{\left(\frac{\varphi \left(z\right)}{\Phi \left(z\right)}\right)}^t\gamma \varphi \left(z\right){[\Phi \left(z\right)]}^{\gamma -1}dz\hfill \\ & =& \gamma {\int }_{-\infty }^{\infty }{z}^s{\left(\varphi \left(z\right)\right)}^{t+1}{(\Phi \left(z\right))}^{\gamma -t-1}dz\hfill \end{array}$$

(16)

for $s=0,1,2,3$ and $t=1,2$. The value of $a_{st}$ can be evaluated via using numerical computation methods. Martiner-Florez et al. [11] obtained ${\eta }_{ij}$. We express ${\eta }_{ij}$ as a function of $a_{st}$ by

$$\begin{array}{ccc}\hfill {\eta }_{11}& =& -\frac{n}{{\sigma }^2}\left\{1+(\gamma -1)(a_{02}+a_{11})\right\},\hfill \\ \hfill {\eta }_{12}& =& {\eta }_{21}=-\frac{n}{\sigma }\left\{2a_{10}+(\gamma -1)(a_{12}+a_{21}-a_{01})\right\},\hfill \\ \hfill {\eta }_{13}& =& {\eta }_{31}=-\frac{n{a}_{01}}{\sigma },\hfill \\ \hfill {\eta }_{22}& =& -\frac{n}{{\sigma }^2}\left\{3a_{20}-1-(\gamma -1)(2a_{11}-a_{31}-a_{22})\right\},\hfill \\ \hfill {\eta }_{23}& =& {\eta }_{32}=-\frac{n{a}_{11}}{\sigma },\hfill \\ \hfill {\eta }_{33}& =& -\frac{n}{{\gamma }^2}.\hfill \end{array}$$

The Fisher information can be defined by

$$\begin{array}{c}\hfill \mathit{I}(\Theta )=[-{\eta }_{ij}].\end{array}$$

(17)

Because not all entries in $\mathit{I}(\Theta )$ have an analytic form, numerical integral methods are needed to obtain the $\mathit{I}(\Theta )$ based on data. A simple method is to approximate $\mathit{I}(\Theta )$ by the observed Fisher information matrix,

$$\begin{array}{c}\hfill \mathit{i}(\Theta )=\left[-\frac{{\partial }^2\ell }{\partial {\theta }_i\partial {\theta }_j}{|}_{\Theta =\widehat{\Theta }}\right].\end{array}$$

(18)

Let $g:R^3⟶R$ be a differentiable function and ${\mathit{I}}^{-1}(\Theta )=\left[{\eta }^{ij}(\Theta )\right]$, where ${\eta }^{ij}(\Theta )$ is the $(i,j)$th element of the inverse Fisher information matrix. The $(1-\alpha )\times 100\%$ confidence interval of $g(\Theta )$ can be obtained by

$$\begin{array}{c}\hfill g\left(\widehat{\Theta }\right)\pm z_{\alpha /2}\sqrt{\nabla g\left(\widehat{\Theta }\right){\mathit{I}}^{-1}\left(\widehat{\Theta }\right){\left(\nabla g\left(\widehat{\Theta }\right)\right)}^T},\end{array}$$

(19)

where $\nabla g\left(\widehat{\Theta }\right)={\left(\frac{\partial g}{\partial {\theta }_1},\frac{\partial g}{\partial {\theta }_2},\frac{\partial g}{\partial {\theta }_3}\right)}_{\Theta =\widehat{\Theta }}^T$ is the gradient of $g(\Theta )$ at $\Theta =\widehat{\Theta }$.

Please note that the terms of $a_{ij}$ could not be a finite number. If $\gamma -t-1<0$, $a_{st}$ is infinite as $z⟶-\infty$. Therefore, it could be difficult to use Fisher information matrix to obtain an approximate confidence interval of $g(\Theta )$. The observed Fisher information matrix, defined in Equation (18), is easy to be evaluated. The quality of the confidence interval of $g(\Theta )$ based on using the observed Fisher information matrix needs to be verified. In this study, we focus on studying the performance for estimating the survival function at different quantiles. We will conduct intensive simulations to evaluate the performance of the proposed interval inference method and typical MLE method in Section 4.

3.2. The Least Square Method for Initial Solutions

There are three parameters in $LPN(\Theta )$, it is not easy to simultaneously set up the initial solutions of $\xi$, $\sigma$ and $\gamma$ to implement the MLE method. A high-quality initial solution ${\Theta }_0$ is a key factor to obtain a reliable MLE of $\Theta$. We suggest a simple process to obtain a proper initial solution of $\Theta$. From the quantile Equation of (7), we obtain the linear equation

$$\begin{array}{c}\hfill log\left(q_p\right)=\xi +\sigma {\Phi }^{-1}\left(p^{\frac{1}{\gamma }}\right).\end{array}$$

(20)

Let $\gamma ={\gamma }_0$, the order statistics of the random sample $\mathit{Y}$ be $Y_{\left(1\right)}\le Y_{\left(2\right)},\dots ,Y_{\left(n\right)}$ and $l{q}_i=log\left(q_{p_i}\right)$, where $q_{p_i}\simeq Y_{\left(i\right)}$, $p_i=\frac{i-0.5}{n}$ for $i=1,2,\dots ,n$. For each ${\gamma }_0$, the initial solutions of $\xi$ and $\sigma$ can be lease square (LS) estimates of the intercept and slope of the regression line,

$$\begin{array}{c}\hfill {\widehat{lq}}_i=\dot{\xi }+\dot{\sigma }{\Phi }^{-1}\left(p^{\frac{1}{\gamma }}\right),i=1,2,\dots ,n.\end{array}$$

(21)

The LS initial solutions of $\xi$, $\sigma$ and $\gamma$ can be denoted by $\dot{\xi }$, $\dot{\sigma }$ and ${\gamma }_0$, respectively; that is, we only need to set up the initial value of $\gamma$ to be ${\gamma }_0$, then use Equation (21) and ${\gamma }_0$ to obtain the LS initial solutions of $\xi$ and $\sigma$. We denote the obtained MLEs via using the LS initial solutions $\dot{\xi }$, $\dot{\sigma }$ and ${\gamma }_o$ by $\tilde{\Theta }=(\tilde{\xi },\tilde{\sigma },\tilde{\gamma })$.

Table 1. The rbias and rsqMSE of the MLEs of $\xi$, $\sigma$ and $\sigma$ based on 1000 repetitions.

		rbias			rsqMSE
$\mathit{\gamma }$	$\mathit{n}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$
1.5	30	−0.0835	0.0921	5.4927	0.2391	0.5666	10.9186
	50	−0.0342	0.0047	3.0918	0.1947	0.4729	7.7778
	100	−0.0382	0.044	2.1881	0.1652	0.3936	5.7547
	200	−0.0226	0.0379	1.4381	0.1446	0.3259	4.5906
3	30	−0.0503	0.0168	3.3806	0.196	0.4635	6.2849
	50	−0.0182	−0.0315	2.154	0.1747	0.4198	4.7815
	100	−0.0359	0.0405	1.9895	0.158	0.3528	4.5396
	200	−0.0398	0.066	1.5186	0.13	0.2851	3.6875
5	30	−0.0385	0.0095	2.2604	0.1719	0.412	3.9232
	50	−0.0037	−0.0504	1.4851	0.157	0.3723	3.4087
	100	−0.1169	0.206	4.3107	0.2112	0.485	6.6299
	200	−0.0367	0.0557	1.2671	0.117	0.2416	2.6537
7	30	−0.0218	−0.0173	1.5653	0.1501	0.356	2.8169
	50	0.0001	−0.0507	1.1676	0.1445	0.3401	2.6136
	100	−0.205	0.5069	4.2835	0.7688	2.595	6.1334
	200	−0.0368	0.0568	1.1545	0.1062	0.213	2.2886
9	30	−0.0052	−0.0452	1.0662	0.1388	0.344	2.0683
	50	−0.005	−0.0361	1.0684	0.1352	0.3223	2.1908
	100	−0.0967	0.194	2.4719	0.384	1.1917	3.8185
	200	−0.0251	0.0337	0.8484	0.0961	0.1903	1.754

reports the MLEs for $LPN(\Theta )$ with $(\xi ,\sigma )=(5,0.6)$ and $\gamma =1.5,3,5,7,9$. We use the measures of rbias and rsqMSE to evaluate the performance of the MLE method. Let the estimate of parameter $\theta$ is $\widehat{\theta }$, the rbias is defined by

$$\begin{array}{c}\hfill \mathrm{rbias}=\frac{\mathrm{Bias}}{\theta },\end{array}$$

(22)

where $\mathrm{Bias}=\frac{1}{B}{\sum }_{i=1}^n{\widehat{\theta }}_i-\theta$ and B is the number of repetitions in simulations. The rsqMSE is defined by

$$\begin{array}{c}\hfill \mathrm{rsqMSE}=\frac{\sqrt{\mathrm{MSE}}}{\theta },\end{array}$$

(23)

where $\mathrm{MSE}=\frac{1}{B}{\sum }_{i=1}^n{({\widehat{\theta }}_i-\theta )}^2$. The measures of rbias and rsqMSE are free of scale.

In practical applications, we do not have enough knowledge to set up all three initial solutions in $\Theta$. We consider using ${\xi }_0=1$, ${\sigma }_0=1$ and ${\gamma }_0=1$ as initial solutions to obtain the MLEs of $\xi$, $\sigma$ and $\gamma$, respectively. The rbias and rsqMSE of each MLE are evaluated using 1000 repetitions. In view of Table 1, we find the rbias of $\gamma$ is significantly larger than the rbias of $\xi$ and $\sigma$. We note that the initial solution of ${\Theta }_0=({\xi }_0,{\sigma }_0,{\gamma }_0)=(1,1,1)$ is a proper initial solution for the simulation cases with $\gamma =1.5,3$ and 5. When the value of $\gamma$ increases, the initial solution of ${\Theta }_0=(1,1,1)$ becomes an improper initial solution. We find unreasonable results that the rbias and rsqMSE increase as the sample size increases from 50 to 100 for $\gamma =5,7$ and 9. These results indicate that the MLE method for $LPN(\Theta )$ is very sensitive to the initial solution of $\Theta$ when the sample size is large. Moreover, we also find that the reqMSE of $\gamma$ is significantly larger than the reqMSEs of $\xi$ and $\sigma$. We will use Monte Carlo simulation method to study the impact of sample size on the performance of MLE method in Section 4. Based on our findings from the MLE method, a bias correction method is provided in this study to improve the estimation performance of $\gamma$.

3.3. The Bias Correction MLE Method

MLE often is a biased estimate. The Firth method, which was studied in [16], and the bootstrap method are two widely used methods to reduce the bias of MLE. Because we cannot guarantee the existence of the expected values of the second and third derivatives of the log-likelihood function for $LPN(\Theta )$, the Firth method is unable to be used in this study. The bootstrap bias correction method can be a potential method to reduce the bias of MLEs. Let the bias-correction estimate of $\gamma$ be ${\tilde{\gamma }}_B$. The bootstrap bias correction method can be implemented as the following steps to obtain ${\tilde{\gamma }}_B$.

Bootstrap Bias Correction Method

Initial Step A: 

Obtain a working sample $\mathit{y}=(y_1,y_2,\dots ,y_n)$.

Step A1: 

Using LS estimate $\dot{\Theta }=(\dot{\xi },\dot{\sigma },{\gamma }_0)$ as the initial solution of $\Theta$ to obtain the MLE, $\tilde{\Theta }=(\tilde{\xi },\tilde{\sigma },\tilde{\gamma })$, based on the working random sample $\mathit{y}$.

Step A2: 

Generate a bootstrap random sample ${\mathit{y}}^{*}$ from $LPN\left(\tilde{\Theta }\right)$ and transform ${\mathit{y}}^{*}$ into a standardized PN sample ${\mathit{z}}^{*}=(z_1^{*},z_2^{*},\dots ,z_n^{*})$, where $z_i^{*}=\frac{log\left(y_i^{*}\right)-\tilde{\xi }}{\tilde{\sigma }}$, $i=1,2,\dots ,n$. We can treat ${\mathit{z}}^{*}$ as a random sample that is taken from $PN\left(\tilde{\gamma }\right)$, where $PN\left(\tilde{\gamma }\right)$ is the standardized PN distribution with parameter $\gamma =\tilde{\gamma }$. Obtain the MLE of $\gamma$ based on ${\mathit{z}}^{*}$ and $PN\left(\tilde{\gamma }\right)$ and denote the obtained MLE by ${\tilde{\gamma }}^{*}$.

Step A3: 

Repeat Step A2 $B_c$ times, where $B_c$ is a huge positive integer. Let ${\tilde{\gamma }}_j^{*}$ denote the MLE of $\gamma$ based on jth bootstrap sample for $j=1,2,\dots ,B_c$. Efron and Tibshirani [17] showed that the estimated bias of $\tilde{\gamma }$ can be obtained by

$${\tilde{\gamma }}_{Bias}^{*}=\frac{1}{B_c}\sum _{j=1}^{B_c}{\tilde{\gamma }}_j^{*}-\tilde{\gamma }.$$

(24)

The bootstrap bias correction MLE of $\gamma$ can be presented by

$${\tilde{\gamma }}_B=\tilde{\gamma }-{\tilde{\gamma }}_{Bias}^{*}=2\tilde{\gamma }-\frac{1}{B_c}\sum _{j=1}^{B_c}{\tilde{\gamma }}_j^{*}.$$

(25)

4. Monte Carlo Simulations

4.1. Performance Comparison

In this section, we use Monte Carlo simulation method to verify the performance of the MLE method with the initial solution ${\Theta }_0=({\xi }_0,{\sigma }_0,{\gamma }_0)=(1,1,1)$ and LS initial solutions with using ${\gamma }_0=1$. Because the moment functions of $LPN(\Theta )$ are complicated, it is difficult to use the first three moment functions to obtain the moment estimates as the initial solutions of $\xi$, $\sigma$ and $\gamma$. Setting each parameter by 1 can be an optional initial solution to obtain the MLEs of parameters if the user does not have good knowledge to set up a quality initial solution for each model parameter.

Parameter combinations of $\xi =5,\sigma =0.6$, $\gamma =1.5,3,5,7,9$; $\xi =8,10$, $\sigma =0.6$, $\gamma =3$; and $\xi =5$, $\sigma =$ 1, 3, $\gamma =3$ are considered for the Monte Carlo simulations. The measures of rbias and rsqMSE are used to evaluate the performance of the MLE and BcLS-MLE methods based on $B=5000$ repetitions. When $\gamma =1.5,3,5$, the initial solution ${\Theta }_0=({\xi }_0,{\sigma }_0,{\gamma }_0)=(1,1,1)$ is a good initial solution for $\Theta$. ${\Theta }_0$ becomes an improper initial solution for $\gamma$ as the true value of $\gamma$ increases. Please note that the initial solution ${\xi }_0=1$ and ${\sigma }_0=1$ are still good initial solutions. We will show that the MLE method is very sensitive to the initial values even only one parameter without using good initial solution to obtain the MLE of $\Theta$. We use $B_c=$ 1000 bootstrap samples for bias correction in the simulation study. The survival function is an important metric in the applications of the reliability analysis and quality control. Let

$$\begin{array}{c}\hfill g(\Theta |y_0)=S\left(y_0\right|\Theta )=1-{\left[\Phi \left(\frac{log\left(y_0\right)-\xi }{\sigma }\right)\right]}^{\gamma },\end{array}$$

(26)

where $y_0$ can be determined by $q_p$ in Equation (7). The coverage probability (CP) for estimating $g(\Theta |y_0)(=S\left(y_0\right|\Theta ))$ is obtained based on 5000 repetitions for using the MLE and BcLS-MLE methods. In this study, we consider the quantile with $p=0.1,0.3$ and 0.5 to determine the value of $y_0$.

Replacing the Fisher information matrix by the observed Fisher information matrix, the approximate $(1-\alpha )\times 100\%$ confidence interval of $g(\Theta )$ can be presented by

$$\begin{array}{c}\hfill g\left(\widehat{\Theta }\right|y_0)\pm z_{\alpha /2}\sqrt{\nabla g\left(\widehat{\Theta }\right|y_0){\mathit{i}}^{-1}\left(\widehat{\Theta }\right){\left(\nabla g\left(\widehat{\Theta }\right|y_0)\right)}^T}\end{array}$$

(27)

Based on Equation (26), we can show that $\frac{\partial g(\Theta |y_0)}{\partial {\theta }_i}=-\frac{\partial \delta }{\partial {\theta }_i}$, where $\delta ={\left[\Phi \left(\frac{log\left(y_0\right)-{\theta }_1}{{\theta }_2}\right)\right]}^{{\theta }_3}$. Taking logarithm transformation to the both sides of $\delta ={\left[\Phi \left(\frac{log\left(y_0\right)-{\theta }_1}{{\theta }_2}\right)\right]}^{{\theta }_3}$, we obtain $log\left(\delta \right)={\theta }_{3}log\left[\Phi \left(\frac{log\left(y_0\right)-{\theta }_1}{{\theta }_2}\right)\right]$. It can be shown that $\frac{\partial \delta }{\partial {\theta }_i}=\delta \times \frac{\partial }{\partial {\theta }_i}\left\{{\theta }_{3}log\left[\Phi \left(\frac{log\left(y_0\right)-{\theta }_1}{{\theta }_2}\right)\right]\right\}$. We can obtain the following results:

$$\begin{array}{ccc}& & \left(i\right)\frac{\partial \delta }{\partial \xi }=\frac{\partial \delta }{\partial {\theta }_1}=-\frac{{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3-1}\varphi \left(z_0\right),\hfill \\ & & \left(ii\right)\frac{\partial \delta }{\partial \sigma }=\frac{\partial \delta }{\partial {\theta }_2}=-\frac{z_0{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3-1}\varphi \left(z_0\right),\hfill \\ & & \left(iii\right)\frac{\partial \delta }{\partial \gamma }=\frac{\partial \delta }{\partial {\theta }_3}={[\Phi \left(z_0\right)]}^{{\theta }_3}log[\Phi \left(z_0\right)],\hfill \end{array}$$

where $z_0=\frac{log\left(y_0\right)-{\theta }_1}{{\theta }_2}$. Based on the results of (i), (ii), (iii) and the equation of $\frac{\partial g(\Theta |y_0)}{\partial {\theta }_i}=-\frac{\partial \delta }{\partial {\theta }_i}$, $i=1,2,3$, we obtain

$$\begin{array}{ccc}\hfill \frac{\partial g(\Theta |y_0)}{\partial \xi }& =& \frac{\partial g(\Theta |y_0)}{\partial {\theta }_1}=\frac{{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3-1}\varphi \left(z_0\right)=\frac{{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3}\times w_0,\hfill \\ \hfill \frac{\partial g(\Theta |y_0)}{\partial \sigma }& =& \frac{\partial g(\Theta |y_0)}{\partial {\theta }_2}=\frac{z_0{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3-1}\varphi \left(z_0\right)=\frac{z_0{\theta }_3}{{\theta }_2}\times {[\Phi \left(z_0\right)]}^{{\theta }_3}\times w_0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \frac{\partial g(\Theta |y_0)}{\partial \gamma }& =& \frac{\partial g(\Theta |y_0)}{\partial {\theta }_3}=-{[\Phi \left(z_0\right)]}^{{\theta }_3}log[\Phi \left(z_0\right)]=-{[\Phi \left(z_0\right)]}^{{\theta }_3}{v}_0,\hfill \end{array}$$

where $w_0=\frac{\varphi \left(z_0\right)}{\Phi \left(z_0\right)}$ and $v_0=log[\Phi \left(z_0\right)]$. The gradient of g at $\Theta =\widehat{\Theta }$ can be resented by

$$\begin{array}{c}\hfill \nabla g\left(\widehat{\Theta }\right)={\left({[\Phi \left(z_0\right)]}^{{\theta }_3}\left(\frac{w_0{\theta }_3}{{\theta }_2},\frac{z_0{w}_0{\theta }_3}{{\theta }_2},-v_0\right)\right)}_{\Theta =\widehat{\Theta }}^T.\end{array}$$

(28)

The $(1-\alpha )\times 100\%$ approximate confidence interval of $g(\Theta )$ can be obtained using Equations (27) and (28).

Each parameter combination was run with a $LPN(\Theta )$ random sample that has a size $n=30,50,100,150$ and 200 in the simulation study. All simulation results based on the MLE method with the initial solution of ${\Theta }_0=({\xi }_0,{\sigma }_0,{\gamma }_0)=(1,1,1)$ and the BcLS-MLE method with ${\gamma }_0=1$, and the CP of $S\left(y_0\right|\Theta )$ with $y_0=q_{0.1},q_{0.3}$ and $q_{0.5}$ are reported in

Table 2. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =0.6,\gamma =1.5$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
$\mathit{n}$	$\mathit{Method}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
30	MLE	−0.085	0.088	5.648	0.239	0.558	11.002	0.893	0.917	0.917
30	BcLS-MLE	−0.052	0.001	5.143	0.232	0.55	11.595	0.906	0.917	0.903
50	MLE	−0.046	0.026	3.469	0.201	0.48	8.196	0.908	0.927	0.920
50	BcLS-MLE	−0.042	0.015	3.490	0.201	0.479	8.684	0.917	0.930	0.916
100	MLE	−0.040	0.043	2.316	0.168	0.398	6.219	0.926	0.927	0.923
100	BcLS-MLE	−0.028	0.017	1.927	0.157	0.377	5.975	0.933	0.929	0.921
150	MLE	−0.032	0.043	1.629	0.145	0.342	4.912	0.931	0.925	0.919
150	BcLS-MLE	−0.020	0.014	1.155	0.312	0.312	4.187	0.941	0.928	0.924
200	MLE	−0.022	0.036	1.440	0.146	0.327	4.604	0.919	0.920	0.900
200	BcLS-MLE	−0.014	0.010	0.730	0.107	0.265	2.806	0.946	0.932	0.919

,

Table 3. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =0.6,\gamma =3$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
$\mathit{n}$	$\mathit{Method}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
30	MLE	−0.055	0.030	3.493	0.198	0.464	6.337	0.874	0.912	0.886
30	BcLS-MLE	−0.022	−0.053	3.124	0.202	0.486	6.576	0.911	0.924	0.924
50	MLE	−0.024	−0.018	2.280	0.174	0.414	5.012	0.917	0.931	0.926
50	BcLS-MLE	−0.021	−0.026	2.454	0.179	0.422	5.692	0.927	0.934	0.936
100	MLE	−0.039	0.043	2.176	0.162	0.359	4.886	0.935	0.933	0.939
100	BcLS-MLE	−0.022	0.003	1.864	0.151	0.342	5.067	0.936	0.938	0.935
150	MLE	−0.032	0.047	1.613	0.158	0.332	3.963	0.939	0.934	0.940
150	BcLS-MLE	−0.018	0.009	1.272	0.127	0.285	3.910	0.944	0.938	0.940
200	MLE	−0.043	0.071	1.619	0.134	0.289	3.677	0.941	0.940	0.954
200	BcLS-MLE	−0.016	0.011	0.929	0.112	0.251	2.975	0.946	0.936	0.942

,

Table 4. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =0.6,\gamma =5$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
$\mathit{n}$	$\mathit{Method}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
30	MLE	−0.032	−0.006	2.164	0.169	0.407	3.945	0.822	0.898	0.769
30	BcLS-MLE	0.007	−0.103	1.807	0.180	0.447	4.105	0.916	0.920	0.885
50	MLE	−0.006	−0.046	1.513	0.155	0.368	3.400	0.874	0.911	0.833
50	BcLS-MLE	−0.006	−0.052	1.808	0.383	0.38	4.183	0.918	0.923	0.900
100	MLE	−0.119	0.209	4.533	0.221	0.526	6.952	0.875	0.916	0.856
100	BcLS-MLE	−0.014	−0.010	1.580	0.141	0.307	4.027	0.938	0.927	0.926
150	MLE	−0.037	0.058	1.325	0.126	0.251	2.687	0.920	0.935	0.925
150	BcLS-MLE	−0.014	0.001	1.190	0.123	0.262	3.271	0.948	0.942	0.941
200	MLE	−0.037	0.056	1.308	0.117	0.242	2.753	0.938	0.931	0.932
200	BcLS-MLE	−0.014	0.007	0.953	0.110	0.232	2.790	0.954	0.941	0.950

,

Table 5. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =0.6,\gamma =7$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
$\mathit{n}$	$\mathit{Method}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
30	MLE	−0.024	−0.014	1.588	0.150	0.361	2.825	0.774	0.878	0.681
30	BcLS-MLE	0.014	−0.109	1.361	0.170	0.421	3.093	0.894	0.912	0.853
50	MLE	0.003	−0.058	1.110	0.145	0.342	2.582	0.827	0.902	0.753
50	BcLS-MLE	0.001	−0.058	1.476	0.156	0.360	3.393	0.890	0.918	0.841
100	MLE	−0.219	0.560	4.111	0.890	2.962	6.035	0.749	0.874	0.695
100	BcLS-MLE	−0.010	−0.015	1.325	0.135	0.29	3.263	0.896	0.916	0.842
150	MLE	−0.033	0.548	0.899	0.237	10.023	1.464	0.815	0.906	0.786
150	BcLS-MLE	−0.010	−0.005	1.024	0.116	0.244	2.755	0.884	0.913	0.847
200	MLE	−0.028	0.041	1.005	0.105	0.212	2.127	0.817	0.892	0.774
200	BcLS-MLE	−0.012	0.004	0.897	0.106	0.219	2.536	0.889	0.915	0.861

,

Table 6. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =0.6,\gamma =9$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
$\mathit{n}$	$\mathit{Method}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
30	MLE	−0.024	−0.014	1.588	0.150	0.329	2.825	0.774	0.878	0.681
30	BcLS-MLE	0.014	−0.109	1.361	0.170	0.401	3.093	0.894	0.912	0.853
50	MLE	0.003	−0.058	1.110	0.145	0.319	2.582	0.827	0.902	0.753
50	BcLS-MLE	0.001	−0.058	1.476	0.156	0.348	3.393	0.890	0.918	0.841
100	MLE	−0.219	0.560	4.111	0.890	1.837	6.035	0.749	0.874	0.695
100	BcLS-MLE	−0.010	−0.015	1.325	0.127	0.271	3.263	0.896	0.916	0.842
150	MLE	−0.033	0.548	0.899	0.237	5.843	1.464	0.815	0.906	0.786
150	BcLS-MLE	−0.010	−0.005	1.024	0.111	0.229	2.755	0.884	0.913	0.847
200	MLE	−0.028	0.041	1.005	0.105	0.21	2.127	0.817	0.892	0.774
200	BcLS-MLE	−0.012	0.004	0.897	0.204	0.22	2.536	0.889	0.915	0.861

,

Table 7. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =8,\sigma =0.6,\gamma =3$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
30	MLE	−0.240	0.759	19.273	0.317	1.453	20.039	0.830	0.897	0.861
30	BcLS-MLE	−0.010	−0.068	2.900	0.124	0.484	6.363	0.915	0.926	0.912
50	MLE	−0.569	2.507	28.406	1.265	7.082	28.561	0.758	0.829	0.863
50	BcLS-MLE	−0.011	−0.034	2.315	0.110	0.417	5.479	0.923	0.929	0.928
100	MLE	−22.560	244.3	9.796	22.722	275.9	15.194	0.001	0.246	0.347
100	BcLS-MLE	−0.012	−0.003	1.785	0.093	0.338	4.912	0.936	0.936	0.933
150	MLE	−0.144	0.448	9.109	0.168	0.530	11.604	0.907	0.911	0.936
150	BcLS-MLE	−0.010	0.006	1.181	0.078	0.281	3.692	0.938	0.932	0.940
200	MLE	−0.172	0.539	12.389	0.202	0.677	15.230	0.903	0.893	0.913
200	BcLS-MLE	−0.009	0.008	0.861	0.068	0.247	2.844	0.944	0.933	0.942

,

Table 8. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =10,\sigma =0.6,\gamma =3$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
30	MLE	−0.693	4.379	29.92	1.621	13.58	30.06	0.724	0.822	0.841
30	BcLS-MLE	−0.009	−0.057	2.909	0.099	0.480	6.303	0.916	0.924	0.915
50	MLE	0.320	1108.6	−0.511	0.367	1108.9	0.511	<0.001	1.000	0.000
50	BcLS-MLE	−0.009	−0.030	2.253	0.088	0.415	5.363	0.935	0.932	0.925
100	MLE	−11.60	1559.5	10.04	24.47	1887.1	18.03	<0.001	0.412	0.293
100	BcLS-MLE	−0.009	−0.002	1.616	0.072	0.330	4.592	0.940	0.933	0.940
150	MLE	−0.391	2.077	25.12	0.775	5.094	26.29	0.789	0.800	0.813
150	BcLS-MLE	−0.008	0.006	1.121	0.061	0.278	3.591	0.942	0.936	0.939
200	MLE	−3.225	21.49	31.91	4.198	28.38	31.97	0.145	0.231	0.400
200	BcLS-MLE	−0.006	0.007	0.827	0.054	0.245	2.762	0.947	0.941	0.938

,

Table 9. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =1,\gamma =3$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
30	MLE	−0.136	0.096	3.926	0.336	0.464	6.384	0.866	0.911	0.904
30	BcLS-MLE	−0.024	−0.067	2.619	0.32	0.476	5.748	0.895	0.917	0.894
50	MLE	−0.139	0.118	3.768	0.326	0.448	6.345	0.887	0.935	0.934
50	BcLS-MLE	−0.043	−0.016	2.72	0.304	0.425	6.244	0.916	0.923	0.904
100	MLE	−0.236	0.291	5.718	0.678	1.22	10.046	0.894	0.903	0.88
100	BcLS-MLE	−0.039	0.01	1.69	0.244	0.332	4.407	0.922	0.931	0.91
150	MLE	−0.2	0.338	2.551	0.26	3.132	3.331	0.905	0.926	0.918
150	BcLS-MLE	−0.033	0.016	1.184	0.207	0.282	3.418	0.931	0.93	0.911
200	MLE	−0.081	0.085	1.532	0.213	0.277	3.239	0.922	0.924	0.914
200	BcLS-MLE	−0.03	0.018	0.924	0.183	0.247	2.882	0.935	0.929	0.916

and

Table 10. The rbias and rsqMSEs of the BcLS-MLE and MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =3,\gamma =3$, and $1-\alpha =0.95$.

		rbias			rsqMSE			CP
30	MLE	−1.473	0.549	13.51	1.585	0.631	15.13	0.832	0.911	0.867
30	BcLS-MLE	−0.076	−0.067	2.854	0.978	0.476	6.256	0.901	0.912	0.869
50	MLE	−2.059	0.796	23.434	2.198	0.963	24.01	0.841	0.898	0.842
50	BcLS-MLE	−0.113	−0.023	2.56	0.895	0.42	6.01	0.918	0.925	0.881
100	MLE	−8.22	33.8	8.472	14.96	39.63	16.66	0.035	0.705	0.253
100	BcLS-MLE	−0.101	0.003	1.704	0.732	0.332	4.742	0.923	0.928	0.89
150	MLE	−0.535	0.192	4.585	1.004	0.413	7.847	0.901	0.926	0.89
150	BcLS-MLE	−0.084	0.007	1.191	0.626	0.282	3.609	0.926	0.931	0.892
200	MLE	−0.932	0.356	8.286	1.278	0.505	11.92	0.897	0.91	0.854
200	BcLS-MLE	−0.063	0.006	0.805	0.537	0.244	2.596	0.934	0.929	0.889

. From these five tables, we find the following results.

• In most cells of Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10, the BcLS-MLE method can reduce the rbias of MLEs.
• Note that reducing the rbias of an estimate could inflate the rsqMSE. Some cells in Table 2, Table 3, Table 4, Table 5 and Table 6 show that the rsqMSE of BcLS-MLE is larger than the rsqMSE of MLE when the rbias of MLE is reduced. For example, the MLE method in Table 4 slightly outperforms the BcLS-MLE method in many cells in terms of the measures of rbias and rsqMSE for $\gamma =5$. Carefully checking the values of rbias and rsqMSE in Table 4, two estimation methods are competitive for $\gamma =5$. For Table 2, Table 3, Table 4, Table 5 and Table 6, the proposed BcLS-MLE significantly outperforms the MLE method in terms of the measures of rbias and rsqMSE in most cells.
• ${\Theta }_0=(1,1,1)$ is a proper initial solution to obtain the MLEs of $\Theta$ for $\gamma =1.5,3$ and 5. When the value of $\gamma$ increases, ${\Theta }_0=(1,1,1)$ or the LS estimates with ${\gamma }_0=1$ becomes improper initial solutions. From Table 5 and Table 6, we find that the proposed BcLS-MLE method is more reliable than the MLE method if the initial solution is improper. Even if the rbias and rsqMSEs of the BcLS-MLE and MLE in Table 5 and Table 6 are closed, The CP based on the proposed BcLS-MLE method is more closed to the nominal value 95% than the value of CP based on the MLE method. The CP of the MLE method could drop down under 70%, which seriously underestimates the nominal confidence coefficient.
• Based on the CPs from Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10, the proposed BcLS-MLE method can improve the interval estimation performance of the MLE method and provide a CP more closed to the nominal confidence coefficient.
• In practice, we can try different values of $\gamma$ to obtain the LS estimates $\tilde{\xi }$ and $\tilde{\sigma }$. Then, select the vector of $(\tilde{\xi },\tilde{\sigma },{\gamma }_0)$ which has the maximal log-likelihood as the initial values of $\xi$, $\sigma$ and $\gamma$ to obtain better estimation performance. In the simulation design, we only consider ${\gamma }_0=1$ for all parameter combinations. This consideration is based on the performance comparison between the proposed BcLS-MLE and MLE methods for the cases with proper and improper initial solutions.
• From Table 7, Table 8, Table 9 and Table 10, we find that the MLE method performs bad when the values of $\xi$ or $\sigma$ change from that in Table 2, Table 3, Table 4, Table 5 and Table 6. Many cells of bias and rsqMSEs in these four tables are unacceptably large. Moreover, some CPs in Table 8 and Table 10 are close to or equal to 0 and seriously underestimate the nominal confidence coefficient of 0.95. These findings indicate that the MLE method is unreliable. The proposed BcLS-MLE method outperform the MLE method in terms of the metrics of bias, rsqMSE and CP.
• Selecting a proper initial solution of $\Theta$ to implement the MLE method is important if the working sample is large.

4.2. Discussions

In our simulation experience for LPN or PN, the MLE method often fails to be implemented due to improper initial solutions were used. Even if the MLE method can be implemented to obtain a set of MLEs, the obtained MLEs could be improper MLEs. This is because the log-likelihood function easily falls into a local optimum with an improper initial solution. The practitioner struggles to simultaneously assign good initial solutions for $\xi$, $\sigma$ and $\gamma$ to implement the MLE method. The proposed LS initial solution method is feasible as an optional method because we only need to assign the initial solution of $\gamma$.

Carefully checking the structure of the log-likelihood function, we find that it is difficult to estimate the shape parameter $\gamma$ when the sample size is small. The terms contain $\gamma$ in the log-likelihood function can be expressed as a summation by

$$nlog\left(\gamma \right)+(\gamma -1)\sum _{i=1}^{n}log(\Phi \left(z_i\right))=nlog\left(\gamma \right)+(\gamma -1)\sum _{i=1}^n{v}_i.$$

(29)

There are two situations that the MLE of $\gamma$ could overestimate or underestimate its true value if the sample size is small.

Situation 1: 

When the log-likelihood function chooses $\gamma >1$ as a candidate, the term “$nlog\left(\gamma \right)$” is positive and the term “$(\gamma -1){\sum }_{i=1}^n{v}_i$” is negative due to $(\gamma -1)>0$ and ${\sum }_{i=1}^n{v}_i<0$. For small size sample, the summation term ${\sum }_{i=1}^n{v}_i$ often is small. Therefore, the log-likelihood function could favor a large $\gamma$ in (29) as the MLE to maximize the log-likelihood function. Therefore, $\gamma$ is easy to be overestimated in this situation.

Situation 2: 

When the log-likelihood function chooses $\gamma <1$ as a candidate, the term “$nlog\left(\gamma \right)$” is negative and the term “$(\gamma -1){\sum }_{i=1}^n{v}_i$” is positive due to $(\gamma -1)<0$ and ${\sum }_{i=1}^n{v}_i<0$. Because of the summation term ${\sum }_{i=1}^n{v}_i$ often is small for small size sample, the log-likelihood function could favor a small $\gamma$ toward 1 in (29) as the MLE to maximize the log-likelihood function. In most applications of reliability analysis and quality control, right-skewed data are more often than left-skewed data. Therefore, it is easy to underestimate $\gamma$ in this situation if the working data follow a right-skewed distribution.

On the basis of Situation 1 and Situation 2 for small size sample, we do not suggest using a small or big value as the initial value for $\gamma$ with a small size sample to protect the log-likelihood function falls into the trap of local maximum. Luckily, the shape parameter is at the power of the standardized normal distribution $\Phi \left(\frac{log\left(y\right)-\xi }{\sigma }\right)$, the true value of $\gamma$ should not be too large for most real data. A good idea is to search the initial values of $\gamma$ in a range of positive domain, and pick up the best one from all LS initial solutions to result in the maximum log-likelihood among all considered competitors. For example, we can figure out the domain of $\gamma$ of $\{2<\gamma <10\}$ into a discrete set of $\left\{\gamma \right|\gamma ={\gamma }_i,i=1,2,\dots ,m\}$, where ${\gamma }_1=2$, ${\gamma }_i={\gamma }_{i-1}+0.01$ for $2\le i\le (m-1)$ and ${\gamma }_m=10$ as candidates. Then, use the proposed LS initial solution method to find the LS initial solutions of $\xi$ and $\sigma$ corresponding to ${\gamma }_i$. Denote the initial solution set by $\{({\dot{\xi }}_i,{\dot{\sigma }}_i,{\gamma }_i),i=1,2,\dots ,m\}$. Then, select the one with the maximal log-likelihood based on the working sample as initial solutions to find the MLEs of $\xi$, $\sigma$ and $\gamma$.

We use Monte Carlo simulation to show the strength of the proposed BcLS-MLE method over the MLE method. Let $\xi =\sigma =5$ and $\gamma =1.5$, 5, 7, 9, 12. Consider the sample size $n=50$, 100 and 200 for the Monte Carlo simulation. The MLE method has a high failure rate if we use ${\xi }_0=1$, $\sigma =1$ and ${\gamma }_0$ as initial solutions, where ${\gamma }_0$ is randomly generated from the uniform distribution, $U(2,10)$. Instead of the MLE method, we can implement the proposed BcLS-MLE method to find the BcLS-MLEs of $\tilde{\xi }$, $\tilde{\sigma }$ and ${\tilde{\gamma }}_B$. The initial solutions of $\dot{\xi }$ and $\dot{\sigma }$ can be obtained via doing global searching for $\gamma$ over the domain of $\left\{\gamma \right|\gamma ={\gamma }_i,i=1,2,\dots ,m\}$, where ${\gamma }_1=2$, ${\gamma }_i={\gamma }_{i-1}+0.01$ for $2\le i\le (m-1)$ and ${\gamma }_m=10$. Then, select the set $({\dot{\xi }}_i^{*},{\dot{\sigma }}_i^{*},{\gamma }_i^{*})$ which has the maximal log-likelihood among m competitors as the initial solution of $(\xi ,\sigma ,\gamma )$. If more than one set reach same maximal log-likelihood, select the set with the smallest index of i as initial solution set to implement the proposed BcLS-MLE method. All simulation results are reported in

Table 11. The rbias and rsqMSE of BcLS-MLE, and the CP of $S\left(y_0\right|\Theta )$ for $LPN(\Theta )$ with $\xi =5,\sigma =5$, and $1-\alpha =0.95$ based on the LS initial solutions by searching $\gamma$ in [2,10].

		rbias			rsqMSE			CP
$\mathit{\gamma }$	$\mathit{n}$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	${\mathit{q}}_{\mathbf{0}.\mathbf{1}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{3}}$	${\mathit{q}}_{\mathbf{0}.\mathbf{5}}$
1.5	50	−0.281	−0.002	2.722	1.580	0.464	6.582	0.865	0.918	0.819
	100	−0.204	0.009	1.769	1.278	0.370	5.683	0.882	0.923	0.828
	200	−0.122	0.009	0.733	0.897	0.266	2.715	0.904	0.928	0.838
5	50	−0.056	−0.049	1.801	1.357	0.381	4.153	0.913	0.925	0.886
	100	−0.107	−0.011	1.385	1.141	0.303	3.505	0.926	0.929	0.893
	200	−0.08	−0.001	0.792	0.877	0.228	2.401	0.936	0.926	0.893
7	50	0.042	−0.069	1.356	1.283	0.360	3.199	0.92	0.93	0.881
	100	−0.040	−0.027	1.136	1.081	0.282	2.883	0.92	0.933	0.884
	200	−0.056	−0.007	0.748	0.857	0.216	2.175	0.931	0.932	0.892
9	50	0.114	−0.0808	1.034	1.2218	0.342	2.572	0.922	0.928	0.885
	100	−0.009	−0.032	0.927	1.02	0.265	2.404	0.931	0.928	0.892
	200	−0.03	−0.012	0.675	0.832	0.206	1.965	0.932	0.932	0.886
12	50	0.238	−0.104	0.637	1.177	0.331	1.904	0.924	0.929	0.884
	100	0.010	−0.051	0.615	0.97	0.252	1.843	0.927	0.925	0.886
	200	0.022	−0.021	0.503	0.777	0.19	1.598	0.926	0.927	0.889

. The rbias and rsqMSE are evaluated based on $B=5000$ repetitions and $B_c=1000$ bootstrap samples are used for bias correction.

Table 11 shows that we can obtain reliable MLEs via using the proposed BcLS-MLE method even if we do not have enough knowledge to assign the initial values of $\xi$, $\sigma$ and $\gamma$. The LS initial solution method is feasible to assign proper initial solutions of the model parameters through searching an given interval of $\gamma$. Because the confidence interval using observed Fisher information matrix is conservative, the CP of the survival function slightly underestimates the nominal confidence coefficient if p is small. When p increases, the CP could be underestimated over the expectation. From the last column of Table 2 to Table 11, we find that the CP could drop below 90% from the nominal value of 95% when the survival function at the median is evaluated. The evaluation of low pth quantile is more important than the evaluation of upper pth quantile in lifetime data analysis. Therefore, the proposed BcLS-MLE method can be used to obtain a reliable MLE of $g(\Theta )$ and provide an approximate confidence interval for $g(\Theta )$ through using the observed Fisher information matrix.

5. Examples

5.1. Example 1

The first example is related to air pollution in the city of New York, USA. In this dataset, the daily measurements of ozone concentration (in ppb D ppm×1000) in the atmosphere of New York city during May–September 1973. All measurements were from the New York State Department of Conservation. This dataset was previously analyzed by Martiner-Florez et al. [11], Nadarajah [18] and Leiva et al. [19]. It is usually assumed that the air pollutant data are uncorrelated and independent. For easy reference, we report this data set with 116 measurements in Appendix A.

Nadarajah [18] has shown that the LPN distribution is a best model, which outperforms over the LN, log-skew-normal, Birnbaum–Saunders distribution(BS) distributions. Martiner-Florez et al. [11] reported the MLEs of the LPN distribution parameters as $\widehat{\xi }=4.986$, $\widehat{\sigma }=0.146$ and $\widehat{\gamma }=0.012$ with the log-likelihood −540.266. We got BcLS-MLEs to be $\widehat{\xi }=4.9354$, $\widehat{\sigma }=0.1789$ and $\widehat{\gamma }=0.0196$ with the log-likelihood −540.297. The improvement of the BcLS-MLEs over the MLEs is insignificant for this data set and the values of log-likelihood based on the MLE and the proposed BcLS-MLE method are closed. In this example, the MLE method and the proposed BcLS-MLE method can perform equally well.

5.2. Example 2

A dataset of cement’s concrete strength (CS) is used to illustrate the proposed BcLS-MLE and interval inference methods. The quality of concrete is mostly evaluated on the basis of its strength. CS is used to measure the ability of concrete to withstand loading. A cement dataset which has 1030 observations, each observation has 9 features, can be obtained from the link of https://www.kaggle.com/prathamtripathi/regression-with-neural-networking (accessed on 17 February 2022). In this dataset, the target feature is the CS per kiloNewton. The CS with the feature age at 28 are selected and used for illustrating the proposed methods. The feature of age in the dataset stands for the time before it needs repairing. Subsequently, total 425 CS observations at age 28 are taken from the original data set and reported in Appendix A. The histogram of the logarithm transformation of CS and its density curve are displayed in Figure 2.

In view of Figure 2, we find that the data set of log(CS) is left-skewed. Moreover, we find that the sample skewness and kurtosis coefficients are −0.493 and 0.275, respectively. We use $LPN(\xi ,\sigma ,\gamma )$ with the initial solution $({\xi }_0,{\sigma }_0,{\gamma }_0)=(1,1,1)$ and ${\gamma }_0=1$ to implement the MLE method and the proposed BcLS-MLE method, respectively. We obtain the MLEs $\widehat{\xi }=6.442$, $\widehat{\sigma }=0.223$ and $\widehat{\gamma }=0.422$, and the BcLS-MLEs $\tilde{\xi }=4.191$, $\tilde{\sigma }=0.143$ and $\tilde{\gamma }=0.056$. The log-likelihood based on the obtained MLEs and BcLS-MLEs are −2109.504 and −1726.389, respectively. As per our expectation, we favor the BcLS-MLE method on the basis of the maximal log-likelihood.

The quantile-to-quantile plot of using $LPN(\tilde{\xi },\tilde{\sigma },\tilde{\gamma })$ as the probability model of the CS data set is displayed in Figure 3. The p-value of Kolmogorov–Smirnov test for the CS dataset based on the $LPN(\tilde{\xi },\tilde{\sigma },\tilde{\gamma })$ is 0.8404. We can find a good straight fit in Figure 3. Moreover, the p-value of the Kolmogorov–Smirnov test also supports that the LPN distribution can be a good candidate for fitting the CS dataset. The BS distribution, Weibull distribution and LN distribution are used to compare with the LPN distribution for fitting the cement’s CS dataset.

Table 12. Maximum likelihood estimates using cement concrete strength data set.

Distribution	$\mathit{\xi }$	$\mathit{\sigma }$	$\mathit{\gamma }$	$\mathit{\alpha }$	$\mathit{\beta }$	Log-Likelihood	AIC	BIC
BS	−	−	−	0.437	33.544	−1736.55	3477.09	3485.20
Weibull				2.670	41.390	−1734.26	3472.51	3480.62
LN	3.52	0.425	−	−	−	−1735.36	3474.72	3482.82
LPN	4.19	0.143	0.056	−	−	−1726.39	3458.78	3470.93

shows that the LPN distribution is the best model for fitting the cement’s CS dataset based on the values of maximal log-likelihood, minimal Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) among all competitors under considered.

We use the observed Fisher information and delta method to find the confidence interval of the survival function at $y_0=q_p$ for $p=0.1,0.3$ and 0.5. After algebra computation, the 95% confidence interval of the survival function is $(0.878,0.922)$, $(0.666,0.734)$ and $(0.467,0.533)$ for the survival function at $p=0.1$, 0.3 and 0.5, respectively. The resulting confidence interval covers the true survival function of 0.9, 0.7 and 0.5, respectively. Moreover, the lengths of these three confidence intervals are 0.044, 0.068 and 0.066, respectively. All the lengths are short.

6. Conclusions

In this article, we propose a parametric bootstrap bias correction MLE method, called the BcLS-MLE method, to improve the performance of the typical MLE method. Moreover, a LS method based on the quantile function is used to obtain proper initial solutions for the LPN distribution parameters. Monte Carlo simulation method is used to evaluate the performance of the proposed bootstrap bias correction MLE method. From simulation results, we find that the proposed bootstrap bias correction MLE method is easy to use to obtain reliable MLEs for the LPN distribution parameters.

Compared with the typical MLE method, the merit of the proposed BcLS-MLE method is that practitioners only need to set up the initial value of the shape parameter and then use the proposed LS method to obtain reliable initial solutions for the LPN distribution parameters to implement the proposed BcLS-MLE method. Based on the proposed BcLS-MLE method, the delta method based on the observed Fisher information matrix can be used to obtain an approximate $(1-\alpha )\times 100\%$ confidence interval of the survival function at pth quantile. The delta method can be used for any differentiable function other than the survival function. A real data set about cement’s CS is used to illustrate the proposed methods.

We find that the CP of the survival function at the quantile of median could be less satisfied because of underestimation. Other parameter estimation methods could be helpful to fill this gap. This topic can be studied in the future.