Inference on a progressive type I interval-censored truncated normal distribution

Abstract

In this paper, we consider the problem of making statistical inference for a truncated normal distribution under progressive type I interval censoring. We obtain maximum likelihood estimators of unknown parameters using the expectation-maximization algorithm and in sequel, we also compute corresponding midpoint estimates of parameters. Estimation based on the probability plot method is also considered. Asymptotic confidence intervals of unknown parameters are constructed based on the observed Fisher information matrix. We obtain Bayes estimators of parameters with respect to informative and non-informative prior distributions under squared error and linex loss functions. We compute these estimates using the importance sampling procedure. The highest posterior density intervals of unknown parameters are constructed as well. We present a Monte Carlo simulation study to compare the performance of proposed point and interval estimators. Analysis of a real data set is also performed for illustration purposes. Finally, inspection times and optimal censoring plans based on the expected Fisher information matrix are discussed.

1. Introduction

The application of life testing experiments is very common in many practical problems arising in multiple branches of science and engineering including mortality analysis, clinical trials, studies related to industry, reliability estimation, etc. In general, such experiments are conducted under various constraints such as cost and time limits, unintentional breakage of units, drop out of subjects from a study and so on. Thus it may not be possible to record exact failure times of all items placed on a test and in sequel, recorded observations appear as censored in nature. Censoring is very common in such studies and indicates situations where failure times of test units are available only for a portion of the total items. In such cases, only partial information in the form of some bound is known on failure times of units that had not failed. In literature, various censoring methodologies have been proposed to obtain censored data. Type I and type II censoring are probably the most commonly used methods in this regard. In type I censoring failure times are recorded up to a specified time point and failures occurring afterward this fixed time are not recorded. On the other hand in type II censoring a test continues until a prespecified number of failure times have been recorded. These two methods share one common feature that live units can be withdrawn only at the termination point of the test. In progressive censoring items put on a test can be withdrawn during the experimentation as well and hence this censoring is more flexible in nature compared to the other two basic schemes. Progressive censoring has been widely studied in lifetime analysis by various researchers. One may refer to Balakrishnan and Cramer [7] for an exhaustive list of references on this topic and also for a detailed discussion on its applications in life testing experiments. Sometimes it is relatively difficult to record exact failure times of items due to lack of continuous monitoring of subjects under study. In such situations, observations are often recorded in intervals and corresponding censoring is referred to as the interval censoring. Aggarwala [1] initially discussed progressive type I interval censoring in literature and studied an exponential distribution using this censoring. Since then this censoring has attracted some attention among researchers. Progressive type I interval censoring can briefly be described as follows. Suppose that a total of n identical units is placed on a life test experiment at time $t_0=0$. In this experiment units are inspected at m prespecified time points $t_1,t_2,\dots ,t_{m-1}$ such that $t_0<t_1<t_2<\cdots <t_{m-1}<t_m$ where $t_m$ denotes the time at which experiment terminates. Now let $X_i$ number of failure times are observed within the interval $(t_{i-1},t_i]$ and also $r_i$ number of live units are randomly withdrawn from the experiment at the inspected time $t_i$, $i=1,2,\dots ,m$. We further note that the number of surviving units $Y_i$ at each inspection time $t_i$ is random in nature and therefore $r_i$ should not be greater than $Y_i$. Usually $r_i$ is obtained using the prescribed percentage $p_i$ of the remaining surviving units at each $t_i$ for $i=1,2,\dots ,m-1$ with $p_m=1$ and equivalently, we have $r_i=\lfloor p_i\ast Y_i\rfloor$ where $\lfloor a\rfloor$ denotes the greatest integer less than or equal to a. In fact, $r_i$ can also be prespecified non-negative integers. In such situations, the actual number of surviving units removed at $t_i$ is given by $r_i^{obs}$ = min($r_i$, number of surviving units at inspection time $t_i$), $i=1,2,\dots ,m-1$ and $r_m^{obs}$= number of surviving units at inspection time $t_m$. In this paper, we denote the observed progressive type I interval censoring scheme as $(r_1,r_2,\dots ,r_m)=(r_1^{obs},r_2^{obs},\dots ,r_m^{obs})$. We denote the corresponding progressive type I interval-censored data as $(X_i,r_i,t_i)$, $i=1,2,\dots ,m$ where $n=\sum _{i=1}^m{X}_i+r_i$. Note that this censoring scheme corresponds to the traditional type I censoring for the case $r_i=0,i=1,2,\dots ,m-1$. At recent past this censoring scheme has received some attention among various researchers. Huang and Wu [17] obtained reliability sampling plans for exponential lifetimes under some cost constraint. Amin [4] computed MLEs and Bayes estimates of unknown log-normal parameters under progressive type I interval censoring. Further Lin et al. [22] established optimally spaced inspection times for this distribution. Ng and Wang [23] studied the Weibull distribution using different estimation methods. Chen and Lio [11] analyzed a generalized exponential distribution and computed estimates for unknown parameters using different estimation procedures. Cheng et al. [14] provided a useful algorithm to derive MLEs of model parameters under progressive type I interval censoring. Lin and Lio [21] computed Bayes estimates of parameters of generalized exponential and Weibull distributions. Chen et al. [12] further considered interval estimation for the generalized exponential distribution. Kus et al. [19] studied optimal plans for the Pareto distribution under some budget limit, see also Ismail [18]. Ahmadi and Yousefzadeh [2] studied the problem of estimating unknown parameters of a generalized half-normal distribution. Wu and Lin [33,34] obtained inference for lifetime performance index assuming the exponential and Weibull distributions. Singh and Tripathi [27] studied an inverse Weibull distribution under this censoring scheme. Recently Budhiraja et al. [10] obtained some interesting results for the maximum likelihood estimates under this censoring. Wang et al. [29] analyzed the mixed generalized exponential distribution based on progressive type I interval censoring. Wu et al. [35] and Wu [30] obtained interesting results for lifetime performance index when product lifetime follows Rayleigh and Chen distributions respectively. Budhiraja and Pradhan [8,9] established optimal plans under progressive type I interval censoring based on some cost function. Zhao and Bordes [36] obtained optimal censoring for step-stress models. Arabi Belaghi et al. [5] considered the problem of estimating unknown parameters of a Burr XII distribution from frequentist and Bayesian contexts. Azizi et al. [6] studied a competing risk model under progressive type I interval censoring assuming Weibull distributions. Du et al. [16] derived inference for entropy by considering the log-logistic distribution. Wu and Hsieh [32] obtained inference for the lifetime performance index for the Gompertz distribution. Recently, Malevich and Muller [24] established optimal design plans of inspection times and Roy and Pradhan [26] obtained Bayesian life testing plans under this censoring scheme.

In this paper, we have considered making inference for a truncated normal distribution under progressive type I interval censoring. The density function of this distribution is given by

$$f(x;\mu ,\tau )=\frac{e^{-(1/2\tau ){(x-\mu )}^2}}{\sqrt{2\pi \tau }\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)},x>0,\mu >0,\tau >0,$$ (1)

where μ and τ denote unknown parameters and $\mathrm{\Phi }(.)$ is the standard normal distribution function. We denote this distribution as $TN(\mu ,\tau )$. The parameter μ denotes the mode of the distribution. The corresponding cumulative distribution function (CDF) and the survival function (SF) are obtained as, respectively,

$$F(x;\mu ,\tau )=1-\frac{1-\mathrm{\Phi }\left(\frac{x-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)},x>0,$$ (2)

and

$$S(x;\mu ,\tau )=\frac{1-\mathrm{\Phi }\left(\frac{x-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)},x>0.$$ (3)

To our knowledge this distribution has not been studied yet using the progressive type I interval censoring. The purpose of this paper is two-fold. We first obtain some classical and Bayes estimates of unknown parameters μ and τ. Both point and interval estimation are considered. The maximum likelihood estimates (MLEs) of unknown parameters are derived using an expectation-maximization algorithm. We also obtain corresponding estimates using midpoint approximation and probability plot methods. The asymptotic intervals are constructed using the observed Fisher information matrix. Bayes estimates are derived under the squared error and linex loss functions using both proper and improper prior distributions. In sequel, highest posterior density (HPD) intervals of unknown parameters are also constructed. The second aim of the paper is to present inspection times and optimal plans under progressive type I interval censoring. Optimal plans have become extremely popular in reliability and life testing experiments in recent years.

We have organized the rest of this paper as follows. In Section 2, we compute maximum likelihood estimates of unknown parameters μ and τ using an EM algorithm. We further use this method to obtain asymptotic intervals from the observed Fisher information matrix. In this section, we also provide midpoint and probability plot estimates of both parameters under progressively type I interval-censored data. Bayes estimates of unknown parameters are derived with respect to squared error and linex loss functions in Section 3 using the importance sampling procedure. We further obtain highest posterior density (HPD) intervals using the importance sampling. A simulation study is conducted in Section 4 to compare the performance of proposed methods of estimation. A real data set is analyzed in Section 5 for illustration purposes. In Section 6, we discuss inspection times and optimal censoring plans under progressive type I interval censoring. Some concluding remarks are given in Section 7.

2. Likelihood estimation

In this section, we compute maximum likelihood estimates of unknown parameters of a $TN(\mu ,\tau )$ distribution under progressive type I interval censoring. Some more classical estimates are obtained using midpoint approximation and probability plot method of estimation. Asymptotic intervals are obtained from the observed Fisher information matrix.

2.1. Maximum likelihood estimates

Suppose that $\{X_i,r_i,t_i\},i=1,2,\dots ,m$ denotes a progressive type I interval censored sample of size m from a $TN(\mu ,\tau )$ distribution as defined in (1). The likelihood function of μ and τ based on this observed data can be written as

$$\begin{array}{rl}L(\mu ,\tau \mid data)& \propto {\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}\prod _{i=1}^m{[\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)]}^{X_i}\\ & \times {[1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)]}^{r_i},\end{array}$$ (4)

where $\mathrm{\Phi }(.)$ is the standard normal cumulative distribution function. The log-likelihood function $l(\mu ,\tau \mid data)$ is then given by

$$\begin{array}{rl}l(\mu ,\tau \mid data)& \propto -n\log \left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)+\sum _{i=1}^m{X}_i\log (\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))\\ & +\sum _{i=1}^m{r}_i\log (1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)),\end{array}$$ (5)

and likelihood equations of μ and τ are then obtained as

$$\begin{array}{rl}& -\frac{n\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\sqrt{\tau }\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\frac{1}{\sqrt{\tau }}\sum _{i=1}^m{X}_i\left(\frac{\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right)\\ & +\frac{1}{\sqrt{\tau }}\sum _{i=1}^m\frac{r_i\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}=0,\end{array}$$ (6)

and

$$\begin{array}{rl}& \frac{n\mu }{2{\tau }^{3|2}}\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\frac{1}{2{\tau }^{\frac{3}{2}}}\sum _{i=1}^m{X}_i\left(\frac{(t_{i-1}-\mu )\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)-(t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right)\\ & +\frac{1}{2{\tau }^{3|2}}\sum _{i=1}^m\frac{r_i(t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}=0.\end{array}$$ (7)

We observe that above likelihood equations have no analytical solutions for the MLEs $\hat{\mu }$ and $\hat{\tau }$ of unknown parameters μ and τ, respectively. In fact, these equations are nonlinear in nature and we can solve them numerically by making use of some iterative technique. Here we obtain desired MLEs using an EM algorithm. This method is initially discussed in Dempster et al. [15] and is very useful for computing MLEs particularly in situations where observed data is censored in nature. Suppose that ${\xi }_{ij}$, $j=1,2,\dots ,X_i$ denotes the lifetimes of jth item failed within the interval $(t_{i-1},t_i]$ and ${\xi }_{ij}^{\ast }$, $j=1,2,\dots ,r_i$ denotes the lifetimes for those items withdrawn at time $t_i$ for $i=1,2,\dots ,m$. Then, the log-likelihood function of μ and τ under the complete data set $C=({\xi }_{ij},{\xi }_{ij}^{\ast })$ is obtained as

$$\begin{array}{rl}{l}_c(C;\mu ,\tau )& \propto -\frac{n}{2}\log \left(\tau \right)-\frac{n{\mu }^2}{2\tau }-n\log \left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)+\frac{\mu }{\tau }\sum _{i=1}^m(\sum _{j=1}^{X_i}{\xi }_{ij}+\sum _{j=1}^{r_i}{\xi }_{ij}^{\ast })\\ & -\frac{1}{2\tau }\sum _{i=1}^m(\sum _{j=1}^{X_i}{\xi }_{ij}^2+\sum _{j=1}^{r_i}{{\xi }_{ij}^{\ast }}^2).\end{array}$$ (8)

In the E-step of the EM algorithm we compute the pseudo log-likelihood function and this is obtained as

$$\begin{array}{rl}{L}_s(\mu ,\tau )& \propto -\frac{n}{2}\log \left(\tau \right)-\frac{n{\mu }^2}{2\tau }-n\log \left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)+\frac{\mu }{\tau }\sum _{i=1}^m(\sum _{j=1}^{X_i}E[{\xi }_{ij}|t_{i-1}\le {\xi }_{ij}<t_i]\\ & +\sum _{j=1}^{r_j}E[{\xi }_{ij}^{\ast }|{\xi }_{ij}^{\ast }>t_i])-\frac{1}{2\tau }\sum _{i=1}^m(\sum _{j=1}^{X_i}E[{\xi }_{ij}^2|t_{i-1}\le {\xi }_{ij}<t_i]\\ & +\sum _{j=1}^{r_j}E[{{\xi }_{ij}^{\ast }}^2|{\xi }_{ij}^{\ast }>t_i]).\end{array}$$ (9)

In order to evaluate the above expression we need to compute the involved conditional expectations which are given below. Noticing that ${\xi }_{ij}$ and ${\xi }_{ij}^{\ast }$ follow the truncated normal distribution with parameters μ and τ, we find that we have

$$\begin{array}{rl}E[{\xi }_{i1}|t_{i-1}\le {\xi }_{i1}<t_i]& =\mu -\sqrt{\tau }\left[\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right],\end{array}$$ (10)

$$\begin{array}{rl}E[{\xi }_{i1}^{\ast }|{\xi }_{i1}^{\ast }>t_i]& =\mu +\sqrt{\tau }\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)},\end{array}$$ (11)

$$\begin{array}{rl}E[{\xi }_{i1}^2|t_{i-1}\le {\xi }_{i1}<t_i]& ={\mu }^2+\tau -2\mu \sqrt{\tau }\left[\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right]\\ & +\sqrt{\tau }\left[\frac{(t_{i-1}-\mu )\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)-(t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right],\end{array}$$ (12)

and

$$E[{{\xi }_{i1}^{\ast }}^2|{\xi }_{i1}^{\ast }>t_i]={\mu }^2+\tau +2\mu \sqrt{\tau }\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}+\sqrt{\tau }(t_i-\mu )\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}.$$ (13)

Next we maximize the pseudo-log-likelihood function with respect to unknown parameters in the M-step of the EM algorithm. If $({\mu }_{(k)},{\tau }_{(k)})$ be an estimate of $(\mu ,\tau )$ at the kth stage of iteration then we can compute the $(k+1)$th stage estimate $({\mu }_{(k+1)},{\tau }_{(k+1)})$ of unknown parameters by maximizing $L_s(\mu ,\tau )$ as given in (9). We further observe that ${\mu }_{(k+1)}$ can be obtained by solving the equation

$$n{\mu }_{(k+1)}+n\sqrt{{\tau }_{(k)}}\frac{\varphi \left(\frac{{\mu }_{(k+1)}}{\sqrt{{\tau }_{(k)}}}\right)}{\mathrm{\Phi }\left(\frac{{\mu }_{(k+1)}}{\sqrt{{\tau }_{(k)}}}\right)}-\sum _{i=1}^m(X_i{A}^{(k)}+r_i{B}^{(k)})=0.$$ (14)

Then updated estimate ${\tau }_{(k+1)}$ of τ can be computed as follows

$${\tau }_{(k+1)}=\frac{1}{n}\sum _{i=1}^m((X_i{C}^{(k)}+r_i{D}^{(k)})-{\mu }_{(k+1)}(X_i{A}^{(k)}+r_i{B}^{(k)})),$$ (15)

where $A^{(k)}=E[{\xi }_{i1}|t_{i-1}\le {\xi }_{i1}<t_i]$, $B^{(k)}=E[{\xi }_{i1}^{\ast }|{\xi }_{i1}^{\ast }>t_i]$, $C^{(k)}=E[{\xi }_{i1}^2|t_{i-1}\le {\xi }_{i1}<t_i]$, and $D^{(k)}=E[{{\xi }_{i1}^{\ast }}^2|{\xi }_{i1}^{\ast }>t_i]$ denote conditional expectations at the kth iteration. We perform this iterative process till a desired convergence is achieved. Next we obtain confidence intervals of unknown parameters μ and τ using asymptotic property of the maximum likelihood estimates. In this connection, we first obtain the observed Fisher information matrix of MLEs of μ and τ which is given by

$${\hat{I}}_n=\left(\begin{array}{cc}{v}_{11}& v_{12}\\ v_{21}& v_{22}\end{array}\right)={\left(\begin{array}{cc}-\frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }{\mu }^2}& -\frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }\mu \mathrm{\partial }\tau }\\ -\frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }\tau \mathrm{\partial }\mu }& -\frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }\tau \mathrm{\partial }\tau }\end{array}\right)}^{-1},$$

and involved expressions are reported in Appendix. Subsequently $100(1-p)\mathrm{\%}$ asymptotic confidence intervals for μ and τ can be obtained as $\hat{\mu }\pm Z_{p/2}\sqrt{v_{11}}$ and $\hat{\tau }\pm Z_{p/2}\sqrt{v_{22}}$, respectively, where $Z_{p/2}$ is the upper $(p/2)$th percentile of the standard normal distribution.

2.2. Midpoint approximation method

In this section, we estimate unknown parameters of a $TN(\mu ,\tau )$ distribution using the mid point approximation method. We assumed that $X_i$ number of failures is observed at the center $m_i=(t_{i-1}+t_i)/2$ of ith interval $(t_{i-1},t_i]$ and also $r_i$ number of units are censored at the inspection time $t_i$, $i=1,2,\dots ,m$. The likelihood function of μ and τ is then obtained as

$$L^{\ast }(\mu ,\tau \mid data)\propto \prod _{i=1}^m{[f(m_i)]}^{X_i}{[1-F(t_i)]}^{r_i}.$$

The desired estimates of μ and τ can be obtained by maximizing the following log-likelihood function

$$\begin{array}{rl}{l}^{\ast }(\mu ,\tau \mid data)& \propto \sum _{i=1}^m[X_i\log (f(m_i))+r_i\log (1-F(t_i))]\\ & \propto -\frac{1}{2}\sum _{i=1}^m{X}_i\log (\tau )-n\log \left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)-\frac{1}{2\tau }\sum _{i=1}^m{X}_i{(m_i-\mu )}^2\\ & +\sum _{i=1}^m{r}_i\log (1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)).\end{array}$$ (16)

Subsequently, we need to solve the following system of equations to obtain the midpoint estimates of unknown parameters:

$$-n\sqrt{\tau }\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\sum _{i=1}^m{X}_i(m_i-\mu )+\sqrt{\tau }\sum _{i=1}^m{r}_i\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}=0,$$ (17)

and

$$-\tau \sum _{i=1}^m{X}_i+n\mu \sqrt{\tau }\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\sum _{i=1}^m{X}_i{(m_i-\mu )}^2+\sqrt{\tau }\sum _{i=1}^m{r}_i\frac{(t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}=0.$$ (18)

Likelihood Equations (17) and (18) cannot be solved analytically due to their nonlinear nature. Here we have used the EM algorithm to obtain the respective estimates of μ and τ.

2.3. Estimation based on the probability plot

Let $(X_i,r_i,t_i)$, $i=1,2,\dots ,m$ with $n=\sum _{i=1}^m{X}_i+r_i$ denotes a progressive type I interval censored sample from a $TN(\mu ,\tau )$ distribution. Based on this sample the cumulative distribution function at time $t_i$ can be estimated as

$$\hat{F}\left(t_i\right)=1-\prod _{j=1}^i(1-{\hat{p}}_j),i=1,2,\dots ,m,$$ (19)

where

$${\hat{p}}_1=\frac{X_1}{n};{\hat{p}}_j=\frac{X_j}{n-\sum _{k=1}^{j-1}(X_k+r_k)};j=2,3,\dots ,m.$$

We also note that

$$t=\mu +\sqrt{\tau }\left[{\mathrm{\Phi }}^{-1}(1-\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)(1-F\left(t\right)))\right].$$ (20)

Now minimizing the expression $\sum _{i=1}^m[t_i-\mu -\sqrt{\tau }({\mathrm{\Phi }}^{-1}(1-\mathrm{\Phi }(\mu /\sqrt{\tau })(1-F(t_i)))){]}^2$ with respect to μ and τ leads to estimates of unknown parameters of the TN distribution based on probability plot method. We mention that these estimates can be computed numerically using some nonlinear optimization technique.

3. Bayesian estimation

Here we obtain different Bayes estimates of unknown parameters of a $TN(\mu ,\tau )$ distribution under symmetric and asymmetric loss functions. In many Bayes estimation problems squared error loss function is applied which penalizes equally to under- and over-estimation. In many practical studies consequences of over- and under-estimation are not symmetric in nature. For example in reliability and life testing studies over-estimation may be treated more serious than under-estimation. Generally in survival analysis over-dispersed data are studied more carefully than under-dispersed data. In such situations, an asymmetric loss function can be used to derive inference upon unknown quantities. Here we obtain Bayes estimates of μ and τ under squared error and linex loss functions. The squared error loss $L_1$ is given by

$$L_1(\tilde{\phi }(\eta ),\phi (\eta ))={(\tilde{\phi }(\eta )-\phi (\eta ))}^2,$$

where $\tilde{\phi }(\eta )$ denotes an estimator of $\phi (\eta )$, a function of the parameter η. In this case, the posterior mean ${\tilde{\phi }}_{SB}(\eta )$ of $\phi (\eta )$ denotes its Bayes estimate. In literature, linex is one of the most commonly used asymmetric loss function (see  [28]) and it is defined as

$$L_2(\tilde{\phi }(\eta ),\phi (\eta ))=e^{c\delta }-c\delta -1,c\ne 0,$$

where $\delta =\tilde{\phi }(\eta )-\phi (\eta )$. The corresponding estimate of $\phi (\eta )$ is given by

$${\tilde{\phi }}_{LB}(\eta )=-\frac{1}{c}ln\left[E_{\eta }\left(e^{-c\phi (\eta )}|\mathrm{data}\right)\right].$$

We obtain Bayes estimates of both the unknown parameters using informative and non-informative prior distributions. The informative prior distribution π of μ and τ is considered as

$$\pi (\mu ,\tau )={\pi }_2\left(\tau \right){\pi }_1\left(\mu |\tau \right),$$

where ${\pi }_1(\mu |\tau )$ is assumed to follow a $TN(a,\tau /b)$ distribution and ${\pi }_2(\tau )$ has an inverse gamma $IG(p,q/2)$ distribution. Also a, b, p and q denote hyper-parameters and reflect prior knowledge about the unknown parameters. We see that the joint prior distribution of μ and τ can be written as

$$\begin{array}{rl}\pi (\mu ,\tau )& \propto \frac{1}{\mathrm{\Phi }\left(\frac{a\sqrt{b}}{\sqrt{\tau }}\right)}{\left(\frac{1}{\tau }\right)}^{p+(3/2)}{e}^{-(1/2\tau )[b{(\mu -a)}^2+q]},\\ & a>0,b>0,p>0,q>0,\mu >0,\tau >0.\end{array}$$ (21)

Using (5) and (21), the joint posterior distribution of μ and τ turns out to be

$$\begin{array}{rl}\pi (\mu ,\tau |\mathrm{data})& =K\frac{{\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}}{\mathrm{\Phi }((a\sqrt{b}/\sqrt{\tau }))}{\left(\frac{1}{\tau }\right)}^{(p+(\sum _{i=1}^m{X}_i+1/2))+1}\\ & \times e^{-(1/2\tau )[b{(\mu -a)}^2+q+\sum _{i=1}^m{X}_i{(m_i-\mu )}^2]}\\ & \times \prod _{i=1}^m{(1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right))}^{r_i},\mu >0,\tau >0,\end{array}$$ (22)

where K is the normalizing constant such that

$$\begin{array}{rl}{K}^{-1}& ={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{{\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}}{\mathrm{\Phi }\left(\frac{a\sqrt{b}}{\sqrt{\tau }}\right)}{\left(\frac{1}{\tau }\right)}^{(p+(\sum _{i=1}^m{X}_i+1)/2)+1}\\ & \times e^{-(1/2\tau )[b{(\mu -a)}^2+q+\sum _{i=1}^m{X}_i{(m_i-\mu )}^2]}\\ & \times \prod _{i=1}^m{(1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right))}^{r_i}\mathrm{d}\mu \mathrm{d}\tau .\end{array}$$

Now Bayes estimate of $h=h(\mu ,\tau )$, a function of μ and τ, under the loss function $L_1$ is obtained as

$${\tilde{h}}_{SB}={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}h(\mu ,\tau )\pi (\mu ,\tau |\mathrm{data})\mathrm{d}\mu \mathrm{d}\tau .$$ (23)

Similarly for the $L_2$ loss, we have

$${\tilde{h}}_{LB}=-\frac{1}{c}ln\left[E\left(e^{-ch(\mu ,\tau )}|\mathrm{data}\right)\right],$$ (24)

where $E(e^{-ch(\mu ,\tau )}|\mathrm{data})={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-ch(\mu ,\tau )}\pi (\mu ,\tau |\mathrm{data})\mathrm{d}\mu \mathrm{d}\tau$.

We note that both Bayes estimators appear as the ratio of two integrals which are difficult to simplify in closed forms. So some approximation technique is required to evaluate such integrals. Here we make use of an importance sampling technique to compute Bayes estimators of unknown parameters of a $TN(\mu ,\tau )$ distribution. The highest posterior density (HPD) intervals of μ and τ are also obtained using progressive type I interval-censored samples. The posterior distribution in Equation (22) is of the form

$$\begin{array}{rl}\pi (\mu ,\tau |\mathrm{data})& \propto \frac{{\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}}{\mathrm{\Phi }\left(\frac{a\sqrt{b}}{\sqrt{\tau }}\right)}{\left(\frac{1}{\tau }\right)}^{(p+(\sum _{i=1}^m{X}_i+1)/2)+1}\\ & \times e^{-(1/2\tau )[b{(\mu -a)}^2+q+\sum _{i=1}^m{X}_i{(m_i-\mu )}^2]}\\ & \times \prod _{i=1}^m(1-\mathrm{\Phi }{\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}^{r_i}).\end{array}$$

We re-express this distribution as

$$\begin{array}{rl}\pi (\mu ,\tau |\mathrm{data})& \propto I{G}_{\tau }(\frac{\sum _{i=1}^m{X}_i}{2}+p,\frac{1}{2}(\sum _{i=1}^m{X}_i^2{m}_i^2+a^{2}b+q-\frac{{(\sum _{i=1}^m{X}_i{m}_i+ab)}^2}{\sum _{i=1}^m{X}_i+b}))\\ & \times T{N}_{\mu |\tau }(\frac{\sum _{i=1}^m{X}_i{m}_i+ab}{\sum _{i=1}^m{X}_i+b},\frac{\tau }{\sum _{i=1}^m{X}_i+b})M(\mu ,\tau ),\end{array}$$ (25)

where

$$M(\mu ,\tau )=\frac{\mathrm{\Phi }\left(\frac{\sum _{i=1}^m{X}_i{m}_i+ab}{\sqrt{\tau (\sum _{i=1}^m{X}_i+b)}}\right)}{\mathrm{\Phi }\left(\frac{a\sqrt{b}}{\sqrt{\tau }}\right)}{\left(\frac{1}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}\right)}^n\prod _{i=1}^m{(1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right))}^{r_i}.$$

We use the following steps to generate samples from the posterior distribution $\pi (\mu ,\tau |\mathrm{data})$.

1. Generate τ from $I{G}_{\tau }((\sum _{i=1}^m{X}_i)/2+p,\frac{1}{2}(\sum _{i=1}^m{X}_i^2{m}_i^2+a^{2}b+q-(\sum _{i=1}^m{X}_i{m}_i+\$\$a{b}^2\phantom{\sum _{i=1}^m})/(\sum _{i=1}^m{X}_i+b)))$.

2. Generate μ from $T{N}_{\mu |\tau }(((\sum _{i=1}^m{X}_i{m}_i+ab)/\sum _{i=1}{)}^m{X}_i+b,({\tau }_1/\sum _{i=1}{)}^m{X}_i+b)$.

3. Repeat step 1 and step 2, t times to obtain samples $({\mu }_1,{\tau }_1)$, $({\mu }_2,{\tau }_2)$, $\dots$, $({\mu }_t,{\tau }_t)$.

Bayes estimates of parameters μ and τ under squared loss and linex loss functions are now obtained as, respectively,

$${\hat{\mu }}_{SB}^{\ast }=\frac{\sum _{i=1}^t{\mu }_{i}M({\mu }_i,{\tau }_i)}{\sum _{i=1}^{m}M({\mu }_i,{\tau }_i)},{\hat{\tau }}_{SB}^{\ast }=\frac{\sum _{i=1}^t{\tau }_{i}M({\mu }_i,{\tau }_i)}{\sum _{i=1}^{m}M({\mu }_i,{\tau }_i)},$$ (26)

and

$${\hat{\mu }}_{LB}^{\ast }=-\frac{1}{c}\log \left[\frac{\sum _{i=1}^t{e}^{-c{\mu }_i}M({\mu }_i,{\tau }_i)}{\sum _{i=1}^{m}M({\mu }_i,{\tau }_i)}\right],{\hat{\tau }}_{LB}^{\ast }=-\frac{1}{c}\log \left[\frac{\sum _{i=1}^t{e}^{-c{\tau }_i}M({\mu }_i,{\tau }_i)}{\sum _{i=1}^{m}M({\mu }_i,{\tau }_i)}\right].$$ (27)

Next we construct HPD intervals of unknown parameters of the $TN(\mu ,\tau )$ distribution using the method of Chen and Shao [13]. For the sake of completeness, we illustrate HPD interval of the parameter μ using this method. Let $\pi (\mu \mid data)$ and $\mathrm{\Pi }(\mu \mid data)$ respectively denote posterior density function and posterior distribution function of μ. Also let $a_p$ be such that $P[\mu \le a_p\mid data]=p$, 0<p<1. We first obtain a simulation consistent estimate of the pth quantile $a_p$ of the posterior distribution of μ. Assume that

$${\omega }_i=\frac{M({\mu }_i,{\tau }_i)}{\sum _{i=1}^{t}M({\mu }_i,{\tau }_i)},i=1,2,\dots ,t.$$

Arrange $\{({\mu }_1,{\tau }_1),({\mu }_2,{\tau }_2),\dots ,({\mu }_t,{\tau }_t)\}$, as $\{({\mu }_{(1)},{\tau }_{(1)}),({\mu }_{(2)},{\tau }_{(2)}),\dots ,({\mu }_{(t)},{\tau }_{(t)})\}$ where ${\mu }_{(1)}<{\mu }_{(2)}<\cdots <{\mu }_{(t)}$, and ${\omega }_i$ is associated with ${\mu }_{(i)}$ for $i=1,2,\dots ,t$. Then a simulation consistent estimate of $a_p$ is ${\hat{a}}_p={\mu }_{(Z_p)}$ where $Z_p$ is given by

$$\sum _{i=1}^{Z_p}{\omega }_{(i)}\le p<\sum _{i=1}^{Z_p+1}{\omega }_{(i)}.$$

Now $100(1-p)\mathrm{\%}$ credible intervals of μ are obtained as $({\hat{a}}_{\kappa },{\hat{a}}_{\kappa +1-\alpha })$ for $\kappa ={\omega }_{(1)},{\omega }_{(1)}+{\omega }_{(2)},\dots ,\sum _{i=1}^{Z_{1-\alpha }}{\omega }_{(i)}$. Then HPD interval of μ is given by $({\hat{a}}_{{\kappa }^{\ast }},{\hat{a}}_{{\kappa }^{\ast }+1-\alpha })$ where ${\kappa }^{\ast }$ is such that ${\hat{a}}_{{\kappa }^{\ast }+1-\alpha }-{\hat{a}}_{{\kappa }^{\ast }}\le {\hat{a}}_{\kappa +1-\alpha }-{\hat{a}}_{\kappa }$ for all κ. The HPD interval of the parameter τ can be obtained similarly.

Remark 3.1

We have also obtained Bayes estimates of unknown parameters of the $TN(\mu ,\tau )$ distribution under a non-informative prior distribution of μ and τ given as

$${\pi }_1(\mu ,\tau )\propto \frac{1}{\tau },\mu >0,\tau >0.$$

The corresponding joint posterior distribution $(\mu ,\tau )$ given the progressive type I interval censored data turns out to be

$$\begin{array}{rl}{\pi }_1(\mu ,\tau |\mathrm{data})& =k_1{\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}{\left(\frac{1}{\tau }\right)}^{((\sum _{i=1}^m{X}_i)/2+1)}{e}^{-(1/2\tau )\sum _{i=1}^m{X}_i{(m_i-\mu )}^2}\\ & \times \prod _{i=1}^m{(1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right))}^{r_i}\end{array}$$

where $k_1^{-1}$ is the normalizing constant such that

$$\begin{array}{rl}{k}_1^{-1}& ={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\left(\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)\right)}^{-n}{\left(\frac{1}{\tau }\right)}^{(\left(\left(\sum _{i=1}^m{X}_i\right)/2\right)+1)}{e}^{-(1/2\tau )\sum _{i=1}^m{X}_i{(m_i-\mu )}^2}\\ & \times \prod _{i=1}^m{(1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right))}^{r_i}\mathrm{d}\mu \mathrm{d}\tau .\end{array}$$

We have used importance sampling to obtain desired estimates of both unknown parameters under squared error and linex loss functions. For the sake of conciseness, we have not presented the detailed calculations which is quite similar to the informative prior case.

4. Simulation study

Here we perform a Monte Carlo simulation study to compare the performance of proposed estimation methods discussed in the previous sections. We assess the behavior of all estimators in terms of their average estimates and mean square error (MSE) values. These values are computed on the basis of various progressive type I interval-censored samples drawn from a $TN(\mu ,\tau )$ distribution when true unknown parameters are $\mu =0.5$ and $\tau =1$. For a given parent sample of size n we first generate a progressive type I interval-censored sample $(X_i,r_i,t_i)$, $i=1,2,\dots ,m$ of size m using prespecified inspection time $t_i$ and censoring scheme $p_i$. We have generated required samples using the algorithm as discussed in Aggarwala [1]. Following this method we first generate $X_1$ from a binomial $Bin(n,F(t_1;\mu ,\tau ))$ distribution and given observation $X_1$ we calculate $r_1$ as $\lfloor p_1\times (n-X_1)\rfloor$. Further for $i=2,3,\dots ,m$, we have

$$X_i|(X_{i-1},r_{i-1},\dots ,X_1,r_1)\sim \mathrm{Bin}(n-\sum _{i=1}^{j-1}(X_j+r_j),\frac{F(t_i;\mu ,\tau )-F(t_{i-1};\mu ,\tau )}{1-F(t_{i-1};\mu ,\tau )}),$$

with $r_i=\lfloor p_i\times (n-\sum _{i=1}^{j-1}(X_j+r_j)-X_i)\rfloor$. We have obtained estimates using different inspection times $t_1=0.3$, $t_2=0.8$, $t_3=1.2$, $t_4=1.6$ and $t_5=2$ and censoring schemes $p_1=(0.25,0.25,0,0,1)$, $p_2=(0,0,0.25,0.25,1)$ and $p_3=(0,0,0,0,1)$. Here $p_3$ denotes the type I censoring scheme and other schemes can be interpreted similarly. We compute different estimates of μ and τ under these combinations when n is assigned values as 30 and 50. We obtain MLEs of unknown parameters using the EM algorithm. The true unknown parameters are considered as the initial guess in the EM algorithm. Similarly midpoint estimates are computed. Bayes estimates of μ and τ are computed using the importance sampling procedure. These estimates are obtained under non-informative (NIN) and informative (IN) prior distributions as discussed in Section 3. The average estimates and MSE of MLEs, Midpoint (MP) and probability plot (PP) estimates are reported in Table 1. It is seen that estimates obtained using the MP method perform quite good compared to the respective maximum likelihood estimates of μ and τ as far as average estimates and MSEs are concerned. The performance of probability plot method is also appreciated. In general, maximum likelihood estimates of both the parameters have an advantage over the other two methods. In general, mean square errors of proposed estimates of both the parameters tend to decrease as sample size increases. This holds for almost all the tabulated schemes. The traditional type I censoring scheme $p_3$ provide marginally better estimation results compared to the other two schemes. In Table 2, we present Bayes estimates along with MSEs for both the unknown parameters. These estimates are obtained under squared error and linex loss functions for n = 30 and n = 50 using the MH algorithm. Bayes estimates under linex loss are computed for arbitrarily selected c such as $-0.5$ and 0.5 indicating weights to under- and over-estimation, respectively. Bayes estimates under IN prior are computed when hyper-parameters are assigned as $a=0.001$, b = 2.5, p = 3 and q = 4. From this table, we observe that estimates obtained under IN prior show superior behavior than estimates obtained under NIN prior in terms of both average estimates and MSEs values. Bayes estimates computed under NIN prior distribution compete quite good with maximum likelihood, MP and PP estimates, see Table 1. We also observe that proper Bayes estimates of both the unknown parameters perform better than all the classical estimates tabulated in Table 1. Here also estimates obtained using the type I censored data produce better estimation results compared to the other two schemes $p_1$ and $p_2$. Moreover, we tend to get better Bayes estimates of unknown parameters with increasing sample sizes. In Table 3, we have presented the $95\mathrm{\%}$ asymptotic confidence and HPD interval estimates of unknown parameters μ and τ for proposed censoring schemes. We have presented coverage percentage and average interval length of both interval estimates. We observe that asymptotic intervals compete good with respective non-informative HPD intervals as far as coverage probability and interval length are concerned. The HPD intervals of both the unknown parameters obtained using the IN prior distribution show superior behavior in this respect. It is also observed that interval length tend to decrease with an increase in sample sizes. Coverage probabilities obtained from the censoring $p_3$ remain smaller than those obtained using the schemes $p_1$ and $p_2$.

Table 1. Maximum likelihood estimates using EM, MP and PP algorithms.

		n = 30						n = 50
		μ			τ			μ			τ
C.S.		EM	MP	PP	EM	MP	PP	EM	MP	PP	EM	MP	PP
$p_1$	Avg	0.4839	0.5294	0.5543	1.0393	0.9167	0.8653	0.4884	0.5332	0.5147	1.0218	0.9366	0.9483
	MSE	0.0377	0.0482	0.0758	0.0847	0.0587	0.1745	0.0382	0.0341	0.0614	0.0581	0.0371	0.1159
$p_2$	Avg	0.4789	0.5234	0.5152	1.0381	0.9195	0.8987	0.4831	0.5366	0.4959	1.0283	0.9339	0.9460
	MSE	0.0404	0.0442	0.0711	0.0428	0.0572	0.1462	0.0368	0.0345	0.0534	0.0260	0.0409	0.1005
$p_3$	Avg	0.4853	0.5173	0.5183	1.0388	0.9445	0.9376	0.4892	0.5372	0.4918	1.0337	0.9658	0.9825
	MSE	0.0391	0.0393	0.0704	0.0775	0.0401	0.1222	0.0331	0.0212	0.0555	0.0033	0.0204	0.0721

Table 2. Bayesian estimates of parameters.

				IN Prior			NIN Prior
					Linex			Linex
C.S.	Method			SEL	c = −0.5	c = 0.5	SEL	c = −0.5	c = 0.5
$n=30$
$p_1$	MH	μ	Avg	0.48515	0.48643	0.48433	0.58530	0.42425	0.56615
			MSE	0.02577	0.02508	0.02665	0.06805	0.05029	0.04598
	MH	τ	Avg	1.01501	0.98316	0.99848	1.04190	0.95109	1.02778
			MSE	0.04957	0.05131	0.04810	0.09997	0.09050	0.09522
$p_2$	MH	μ	Avg	0.49790	0.49613	0.48004	0.56525	0.43131	0.55941
			MSE	0.02157	0.02177	0.02148	0.05884	0.06117	0.05664
	MH	τ	Avg	1.06997	0.96519	1.05565	0.90480	0.91288	1.11733
			MSE	0.05308	0.05419	0.05334	0.08143	0.08236	0.08142
$p_3$	MH	μ	Avg	0.51330	0.51042	0.50648	0.63131	0.46671	0.62616
			MSE	0.02282	0.02335	0.02239	0.04606	0.04780	0.04443
	MH	τ	Avg	0.99024	0.99385	1.01769	0.99934	0.99742	0.99292
			MSE	0.04808	0.05078	0.04483	0.07184	0.07352	0.07214
$n=50$
$p_1$	MH	μ	Avg	0.49100	0.48644	0.48577	0.54494	0.54910	0.54090
			MSE	0.01689	0.01694	0.01689	0.04395	0.04543	0.04253
	MH	τ	Avg	0.99550	1.00480	0.98708	0.83448	0.83927	0.82953
			MSE	0.04077	0.04285	0.03980	0.07314	0.07282	0.07353
$p_2$	MH	μ	Avg	0.52222	0.52609	0.52845	0.71314	0.71624	0.71011
			MSE	0.02052	0.02127	0.01981	0.06093	0.06242	0.05949
	MH	τ	Avg	0.97670	0.98197	0.97128	0.80626	0.80866	0.80376
			MSE	0.03184	0.03242	0.03127	0.07166	0.07112	0.07221
$p_3$	MH	μ	Avg	0.50334	0.50657	0.50021	0.63939	0.64215	0.63670
			MSE	0.02512	0.02593	0.02435	0.05475	0.05621	0.05333
	MH	τ	Avg	1.00301	1.00806	0.99785	0.88047	0.88328	0.87759
			MSE	0.03095	0.03158	0.03035	0.06589	0.06555	0.06625

Table 3. Coverage probabilities and average lengths of asymptotic and HPD intervals.

				HPD interval
		Asymptotic interval		NIN Prior		IN Prior
n	C.S.	μ	τ	μ	τ	μ	τ
30	$p_1$	0.8990	0.8780	0.8738	0.8553	0.9428	0.9362
		1.01997	1.41021	0.66736	0.87997	0.62102	0.76932
	$p_2$	0.8849	0.8637	0.8583	0.8436	0.9210	0.9187
		1.05020	1.42186	0.54208	0.73005	0.53197	0.71021
	$p_3$	0.8574	0.8431	0.8328	0.8384	0.9197	0.9084
		1.02473	1.36021	0.51030	0.68938	0.48957	0.57843
50	$p_1$	0.9186	0.8960	0.8977	0.8646	0.9555	0.9577
		0.98956	1.37480	0.51849	0.62099	0.44868	0.55834
	$p_2$	0.9175	0.8673	0.8883	0.8594	0.9436	0.9413
		1.09532	1.25834	0.44636	0.46539	0.43178	0.46537
	$p_3$	0.8872	0.8545	0.8664	0.8386	0.9296	0.9055
		0.96392	1.03141	0.36512	0.35433	0.33315	0.30214

5. Real data analysis

We analyze a real data set in support of proposed estimation methods. This data set is taken from https://openmv.net/info/unlimited-time-test which was uploaded on first March 2013. This data set consists of grades from a midterm exam, as well as the time taken by the student to write the exam. It was an ‘infinite’ time midterm, so there was no time pressure to finish within the allocated period. The exam time taken by all 80 students (after divided by 100) in minutes are reported as follows:

We consider this data set as a progressive type I interval censored with n = 80, m = 4, $X=(12,26,15,5)$, $r=(15,4,0,3)$, and $t_1=1.5$, $t_2=2$, $t_3=2.5$, $t_4=3$. We now perform goodness of fit test to verify whether a TN distribution is a suitable model for this data set. For comparison purposes, we take into account half-normal (HN) and folded-normal (FN) distributions as well. The negative log-likelihood (NL) and Kolmogorov–Smirnov (K–S) test statistic criteria are used to assess the goodness of fit for all the competing models. Maximum distance under the K–S test given as

$$D_n(F)={\sup }_{0\le t\le \mathrm{\infty }}|\hat{F}(t;\mu ,\tau )-F(t;\hat{\mu },\hat{\tau })|$$

denotes the distance between an empirical distribution $\hat{F}(t;\mu ,\tau )$ under the observed progressive type I censored data and the population distribution $F(t;\hat{\mu },\hat{\tau })$. The empirical CDF at each inspection time $t_i$ can be estimated using Equation (19). Based on the observed data we computed MLEs of unknown vector parameter $(\mu ,\tau )$ of TN distributions as $(1.9368,0.4438)$, the corresponding NL estimate is 17.9530 and K–S statistic value turns out to be 0.1046. For HN distribution MLE is 2.3799, NL estimate is 114.4501, and K–S test statistic value is obtained as 0.5191. Finally for FN distribution MLEs of unknown vector parameter is $(1.70428,0.20894)$, NL is $220.86289$ and K–S test statistic value is given by 0.56145. Note that smaller value of NL criterion and K–S test statistic indicate better fit to the given data set. From computed estimates we observe that the proposed TN distribution fits the data set reasonably good compare to the other two models. We next computed asymptotic confidence intervals of μ and τ as $(1.7796,2.0940)$ and $(0.1158,0.7717)$ respectively under observed data set. In addition, we report non-informative Bayes estimates of these parameters in Table 4. Estimates under the linex loss function are computed for two choices of c such as $-0.5$ and 0.5. It is seen that estimates obtained using different methods vary marginally from each other. Finally, the corresponding non-informative HPD credible intervals for μ and τ are obtained as $(1.6448,2.0046)$ and $(0.6568,1.2338)$ respectively.

Table 4. Bayes estimates of μ and τ for real data.

	μ	τ
SEL	1.8203	0.8888
Linex, c = −0.5	1.8226	0.8970
Linex, c = 0.5	1.8181	0.8809

6. Inspection times and optimal censoring

Here we provide optimal inspection times when data are observed using progressive type I interval censoring. Optimal design of censoring schemes is quite important in life testing experiments. These types of inference have received considerable attention in the literature. Huang and Wu [17] studied reliability sampling plans for progressive type I interval-censored life tests under the assumption of exponential distribution. Lin et al. [22] and [20] discussed A- and D-optimal schemes for log-normal and Weibull distributions respectively. Akdogan et al. [3] and Kus et. al. [19] obtained the optimal censoring schemes for the Burr XII and Pareto distributions using cost constraints. Wu and Huang [31] also studied optimal plans under competing risks set up. Singh and Tripathi [27] obtained various inspection times and optimal censoring schemes for the inverse Weibull distribution. One can also refer to Roy and Pradhan [25,26], Budhiraja et al. [10], Arabi Belaghi et al. [5], Zhao and Bordes [36], Malevich and Muller [24] for some more useful results on optimal plans under progressive type I interval censoring. Recall that under this scheme inspection times are prescribed before the start of an experiment using a priori information related to the test. Many practical studies of interest including life testing experiments require procedures to derive efficient estimates of unknown quantities of interest. In this regard, we mention that adequate inference on effective inspection times may, in turn, yield better estimation results for unknown parameters. So it is important to study the impact of various inspection times on the effectiveness of different estimation methods. In many situations, inspection times are considered of equal length which may not be appropriate for deriving inference upon data indicating monotonic hazard rate functions. To overcome this problem an equal probability (EP) inspection times can be taken into account where the probability of expected number of failures in each inspection interval remains the same. Notice that probability that a test unit fails in the interval $(0,t_1]$ is given by $[F(t_1)-F(0)]/[1-F(0)]=F(t_1)$ and accordingly the expected number of failures ${\zeta }_1$ in this duration becomes $E(X_1)$ which is $nF(t_1)$. Recall that $q_1$ per cent of units are removed at time $t_1$. Thus for a given ${\zeta }_1$, the expected number of removed units is ${\psi }_1=(n-{\zeta }_1)q_1$. Similarly for $i=2,3,\dots ,m$ we find that we have

$$\begin{array}{rl}{\zeta }_i& =E(X_i|X_{i-1},r_{i-1},\dots ,X_1,r_i){|}_{{\zeta }_{i-1},{\psi }_{i-1},\dots ,{\zeta }_1,{\psi }_1}\\ & =(n-\sum _{j=1}^{i-1}({\zeta }_i+{\psi }_i))\frac{F(t_i)-F(t_{i-1})}{1-F(t_{i-1})}\end{array}$$ (28)

and

$$\begin{array}{rl}{\psi }_i& =E(r_i|X_{i-1},r_{i-1},\dots ,X_1,r_i){|}_{{\zeta }_{i-1},{\psi }_{i-1},\dots ,{\zeta }_1,{\psi }_1}\\ & =(n-\sum _{j=1}^{i-1}({\zeta }_i+{\psi }_i)-{\zeta }_i)q_i.\end{array}$$ (29)

Thus time $t_i,i=1,2,\dots ,m,$ with ${\zeta }_1={\zeta }_2=\cdots ={\zeta }_m$ and $\sum _{i=1}^m{\zeta }_i=n(1-h)$ is the required inspection times where h represents the total degree of censoring. We have computed inspection times for various schemes using the algorithm as discussed in Singh and Tripathi [27]. We have performed these computations for varying choices of h such as $h=20(10)50\mathrm{\%}$. We present these results in Table 5 where a − represents the situation that experiment can be terminated only after the failure of all remaining units. Tabulated results suggest that inspection times under the type I interval censoring scheme tend to remain small which is followed by $p_2$ and $p_1$ censoring schemes respectively. We also observed that in general higher degree of censoring lead to smaller inspection times. These quantities can also be computed based on some optimality criteria where the goal may be to infer the inspection times which optimizes the considered criterion. Techniques such as genetic algorithm or simulated annealing algorithm can be used to obtain the desired inspection times. Now we compute optimal proportion $p=(p_1,p_2,\dots ,p_m)$ or equivalently $r=(r_1,r_2,\dots ,r_m)$ among all possible censoring schemes in a similar manner. Generally, the maximum possible censoring satisfying the relation $\sum _{i=1}^m{r}_i=\lfloor nh\rfloor$ is given by $(\genfrac{}{}{0ex}{}{\lfloor nh\rfloor +m-1}{m-1})$ for prescribed n and h where $r_i$'s are the positive integers. The statistical package ‘partitions’ in R software can be used to generate all solutions of the equation $\sum _{i=1}^m{r}_i=\lfloor nh\rfloor$. Computational complexity may occur for relatively large values of n and m. We propose the following algorithm to obtain optimal censoring schemes (see also  [27]).

1. Input $n,m,h,\mu ,\tau$ and inspection times $t_i=(t_1,t_2,\dots ,t_m)$.

2. Set deg = floor($n\ast h$); A = rep(deg, m); B = firstblockpart(A, deg); $r_j={\psi }_j=$ B, $j=1,2,\dots ,m$; schemes = choose((deg + m−1),(m−1)); count = 0 and k = 1.

3. If ($k\le$ schemes) then set i = 1 and go to Step 4 else go to Step 8.

4. If ($i\le m$) then calculate ${\zeta }_i$ from the above relation (??), else go to Step 6.

5. If ($\sum _{j=1}^i>(n-\mathrm{deg}-ϵ)$) then go to step 7 else set i = i + 1 and go to Step 4.

6. Set count = count + 1; If($(n-\mathrm{deg}-ϵ)\le \sum _{j=1}^m{\zeta }_j\le (n-\mathrm{deg}+ϵ)$) then calculate considered criterion value correspond to the values of r, ${\zeta }_j$, say $va{l}_{count}$.

7. B = nextblockpart(B, A). Set k = k + 1 and go to Step 3.

8. Optimal censoring r corresponds to the optimum observation from all the $va{l}_{count}$.

Here we note that Steps 5 and 6 optimize the cost of computations as estimate of the proposed criterion can be obtained for only count number of selected censoring schemes satisfying the relation $(n-\mathrm{deg}-ϵ)\le \sum _{j=1}^m{\zeta }_j\le (n-\mathrm{deg}+ϵ)$ where ε may be smaller than 0.5 just to control the round off error. We have obtained two optimal censoring schemes based on minimizing and maximizing the trace of expected variance-covariance matrix of MLEs and the expected Fisher information matrix, referred to as Criterion I and Criterion II, respectively. We also note that the expected information matrix can be obtained by replacing $X_i$ by ${\zeta }_i$ and $r_i$ with ${\psi }_i$ in the second-order partial derivatives of log-likelihood function with respect to μ and τ as provided in Appendix.

In Table 6, we have reported optimal censoring schemes under progressive type I interval censoring. From this table, we observe that removal pattern of the live units is, in general, not affected by an increase in sample sizes. For instance when n is 30 or 50, and for h except h = 0.20, Criterion I suggests removal of units at the first stage and Criterion II suggests removal of items at the second stage of the experiment. Also optimal censoring schemes suggest removal of many units at the last stage of the test if h = 0.20. If interest lies in computing optimal proportion instead of optimal number of units then Equation (29) can be used to obtain the desired results.

Table 5. EP inspection times.

h	C.S.	$t_1$	$t_2$	$t_3$	$t_4$	$t_5$
0.20	$p_1$	0.29598	0.66793	1.05666	1.56058	–
	$p_2$	0.29598	0.57478	0.85963	1.29933	2.66053
	$p_3$	0.29598	0.57478	0.85963	1.17787	1.58802
0.30	$p_1$	0.26045	0.58637	0.99474	1.51440	–
	$p_2$	0.26045	0.50538	0.75063	1.10685	1.78224
	$p_3$	0.26045	0.50538	0.75063	1.01223	1.31534
0.40	$p_1$	0.22462	0.50538	0.88440	1.33155	2.09535
	$p_2$	0.22462	0.43600	0.64454	0.93466	1.39876
	$p_3$	0.22462	0.43600	0.64454	0.85963	1.09301
0.50	$p_1$	0.18843	0.42442	0.73476	1.07020	1.48896
	$p_2$	0.18843	0.36632	0.54005	0.77455	1.11613
	$p_3$	0.18843	0.36632	0.54005	0.71501	0.89687

Table 6. Optimal censoring schemes for m = 5.

n	h	Two associated optimal censoring schemes
30	0.20	(3, 0, 0, 0, 3)	(1, 0, 1, 2, 2)
	0.30	(3, 3, 0, 1, 2)	(1, 4, 2, 0, 2)
	0.40	(8, 1, 1, 0, 2)	(6, 4, 0, 0, 2)
	0.50	(10, 2, 1, 1, 1)	(3, 11, 0, 0, 1)
50	0.20	(1, 4, 0, 1, 4)	(1, 1, 2, 3, 3)
	0.30	(10, 1, 0, 0, 4)	(0, 9, 1, 3, 2)
	0.40	(15, 0, 2, 0, 3)	(4, 12, 1, 1, 2)
	0.50	(19, 2, 1, 1, 2)	(17, 5, 0, 1, 2)

7. Conclusion

We have considered estimation of the unknown parameters of the truncated normal distribution when data are observed using progressive type I interval censoring. We computed different classical estimates of parameters using various procedures such as maximum likelihood, midpoint and probability plot methods. These methods are quite popular in literature. Recently, Ahmadi and Yousefzadeh [2] applied these procedures for computing estimates of generalized half normal distribution based on progressive type I interval-censored data. We mention that sometimes probability plot method can also be used as initial guess for unknown model parameters. We also computed Bayes estimates of parameters using importance sampling technique with respect to squared error and linex loss functions. We conducted a Monte Carlo simulation study and compared the performance of proposed estimation methods under different censoring schemes. We observed that mid point and probability plot methods provide marginally satisfactory results. Bayes method provides better estimation results under proper prior information. We also proposed two different criteria to obtain inspection times and established optimal censoring plans as well. Finally, we mention that there are some issues we are not able to address in this paper. More work is required to study some decision-theoretic properties of probability plot and midpoint estimation methods. Also we have obtained optimal plans based on maximum likelihood estimates. Various estimates can also be computed using empirical Bayesian methods. Likewise Bayesian optimal plans may also be studied, as suggested by anonymous reviewers. These are some possible future works.

Appendix.

$$\begin{array}{rl}\frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }{\mu }^2}& =-\frac{n}{\tau }\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\frac{n}{\tau }{\left(\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}\right)}^2+\frac{1}{\tau }\sum _{i=1}^m{X}_i(\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-{\varphi }^{\mathrm{\prime }}\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\\ & -{\left(\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right)}^2)-\frac{1}{\tau }\sum _{i=1}^m{r}_i(\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}+{\left(\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}\right)}^2),\\ \frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }\mu \mathrm{\partial }\tau }& =\frac{n\mu }{2{\tau }^2}\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\frac{n}{2{\tau }^{3/2}}\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}-\frac{n\mu }{2{\tau }^2}{\left(\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}\right)}^2+\frac{1}{2{\tau }^2}\sum _{i=1}^m{X}_i\\ & \times (\frac{((t_i-\mu ){\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-(t_{i-1}-\mu ){\varphi }^{\mathrm{\prime }}\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))-\sqrt{\tau }(\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\\ & -\frac{(\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))((t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-(t_{i-1}-\mu )\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}{{(\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}^2})-\frac{1}{2{\tau }^2}\\ & \times \sum _{i=1}^m{r}_i((t_i-\mu )\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}+\sqrt{\tau }\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}+(t_i-\mu ){\left(\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}\right)}^2),\\ \frac{{\mathrm{\partial }}^{2}l}{\mathrm{\partial }{\tau }^2}& =-\frac{n{\mu }^2}{4{\tau }^3}\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}-\frac{3n\mu }{4{\tau }^{5/2}}\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}+\frac{n{\mu }^2}{4{\tau }^3}{\left(\frac{\varphi \left(\frac{\mu }{\sqrt{\tau }}\right)}{\mathrm{\Phi }\left(\frac{\mu }{\sqrt{\tau }}\right)}\right)}^2+\frac{1}{4{\tau }^3}\sum _{i=1}^m{X}_i\\ & \times (\frac{({(t_i-\mu )}^2{\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-{(t_{i-1}-\mu )}^2{\varphi }^{\mathrm{\prime }}\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\\ & +\frac{3\sqrt{\tau }((t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-(t_{i-1}-\mu )\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\\ & -{\left(\frac{((t_i-\mu )\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-(t_{i-1}-\mu )\varphi \left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right))}{\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)-\mathrm{\Phi }\left(\frac{t_{i-1}-\mu }{\sqrt{\tau }}\right)}\right)}^2)\\ & -\frac{1}{4{\tau }^3}\sum _{i=1}^m{r}_i({(t_i-\mu )}^2\frac{{\varphi }^{\mathrm{\prime }}\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}+3\sqrt{\tau }(t_i-\mu )\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}\\ & +{\left((t_i-\mu )\frac{\varphi \left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}{1-\mathrm{\Phi }\left(\frac{t_i-\mu }{\sqrt{\tau }}\right)}\right)}^2).\end{array}$$

Funding Statement

Yogesh Mani Tripathi gratefully acknowledges the partial financial support for this research work under a grant EMR/2016/001401 Science and Engineering Research Board (SERB), India.

Disclosure statement

No potential conflict of interest was reported by the authors.AUGRPThe disclosure statement has been inserted. Please correct if this is inaccurate.