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Likelihood inference for pollutant loading under type I censoring

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Abstract

Exposure to toxic contaminants in the environment harms human and animal health and disturbs the integrity and function of the impacted ecosystem. The impact could be local, regional, and global. The concentration of a toxic substance below or above detection limits or thresholds in environmental samples is frequently recorded as non-detect. We discuss inferences based on exact and modified likelihood methods for the location-scale family with values below the detection limit, and as a special case for the normal distribution with a comparison between the methods. We demonstrate the procedure using Niagara River monitoring data.

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Acknowledgments

We would like to acknowledge the time and effort devoted by the editors and the anonymous reviewers to improve the quality of the work. We thank two anonymous reviewers for the provided helpful comments and suggestions on earlier drafts of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt

    Hossam M. Hassan

  2. Department of Statistics, Faculty of Economics and Political Science, Cairo University, Cairo, Egypt

    Abdel H. El-Shaarawi

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  1. Hossam M. Hassan
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  2. Abdel H. El-Shaarawi
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Correspondence to Abdel H. El-Shaarawi.

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Appendices

Appendix 1. Calculations for grouped censoring

EM algorithm

Let x1, x2, ...., xnbe a censored sample from a location-scale distribution where all {xi : d1 < xi < d2are censored; d1 and d2 are fixed detection limits. Assume that there are no non-detected (censored) values and r = n − no uncensored values. Let us define δi as follows

$$ {\delta}_i=\left\{\begin{array}{l}0\kern0.48em :{d}_1<{x}_i{d}_2\\ {}1\kern0.6em : otherwise,\end{array}\right. $$

in this case, the likelihood function will take the form

$$ L\left(\mu, \sigma \right)=\prod \limits_{i=1}^n\left(\frac{1}{0}f\left({z}_i\right)\right){\delta}_i{\left({\int}_{\xi_1}^{\xi_2}f(z) dz\right)}^{1-{\delta}_i} $$

where f(t) and F(t) are the probability density function and the cumulative distribution function for the standard location-scale parameter (free of μ and σ),\( {\xi}_j=\frac{d_j-\mu }{\sigma },j=1,2\; and\;{z}_i=\frac{x_i-\mu }{\sigma } \). Hence, the log-likelihood function is given by

$$ {\displaystyle \begin{array}{l}\ell \left(\mu, \sigma \right)=-r\;\log \sigma +\sum \limits_{i=1}^n\left[{\delta}_i\;\log\;f\left({z}_i\right)+\left(1-{\delta}_i\right)\;\log {\int}_{\xi_1}^{\xi_2}f(z) dz\right]\\ {}\kern1.92em =-r\;\log \sigma +\sum \limits_{i=1}^n{\delta}_i\;\log\;f\left({z}_i\right)+{n}_o\;\log \left(F\left({\xi}_2\right)-F\left({\xi}_1\right)\right)\end{array}} $$

Let \( C\left({\xi}_1,{\xi}_2\right)=\log {\int}_{\xi_1}^{\xi_2}f(z) dz; \) the censored part in the log-likelihood Eq. (4). Then we have the following

$$ {\displaystyle \begin{array}{l}\frac{\partial }{\partial \mu }C\left({\xi}_1,{\xi}_2\right)=\frac{1}{\int_{\xi_1}^{\xi_2}f(z) dz}{\int}_{\xi_1}^{\xi_2}\frac{\partial f}{\partial \mu } dz\\ {}\kern5.5em =\frac{1}{\int_{\xi_1}^{\xi_2}f(z) dz}{\int}_{\xi_1}^{\xi_2}\frac{\partial \log f(z)}{\partial \mu }f(z) dz\\ {}\begin{array}{l}\kern5.5em =E\left[\frac{\partial \log f(z)}{\partial \mu}\left|{\xi}_1<z<\right.{\xi}_2\right]\\ {}\kern5.5em =-\frac{1}{\sigma }E\left[\frac{\partial \log f(z)}{\partial z}\left|{\xi}_1<z<\right.{\xi}_2\right]\\ {}\kern5.5em =-\frac{1}{\sigma}\frac{f\left({\xi}_2\right)-f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}\end{array}\end{array}} $$
(25)
$$ {\displaystyle \begin{array}{l}\frac{\partial^2}{\partial {\mu}^2}C\left({\xi}_1,{\xi}_2\right)=\frac{\partial }{\partial \mu}\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \mu } dz\\ {}\kern6em =\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial }{\partial \mu}\left[\frac{\partial \log f(z)}{\partial \mu }f(z)\right] dz-{\left(\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \mu } dz\right)}^2\\ {}\begin{array}{l}\kern6em =E\left[\left.\frac{\partial^2\log f(z)}{\partial {\mu}^2}\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.{\left(\frac{\partial \log f(z)}{\partial \mu}\right)}^2\right|{\xi}_1<z<{\xi}_2\right]-{\left(E\left[\left.\frac{\partial \log f(z)}{\partial \mu}\right|{\xi}_1<z<{\xi}_2\right]\right)}^2\\ {}\kern6em =\frac{1}{\sigma^2}\left(E\left[\left.\frac{\partial^2\log f(z)}{\partial {z}^2}\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.\left(\frac{\partial \log f(z)}{\partial z}\right)2\right|{\xi}_1<z<{\xi}_2\right]-{\left(E\left[\left.\frac{\partial \log f(z)}{\partial z}\right|{\xi}_1<z<{\xi}_2\right]\right)}^2\right)\\ {}\kern6em =\frac{1}{\sigma^2}\left(\frac{\frac{\partial f\left({\xi}_2\right)}{\partial {\xi}_2}-\frac{\partial f\left({\xi}_1\right)}{\partial {\xi}_1}}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}-{\left(\frac{f\left({\xi}_2\right)-f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}\right)}^2\right)\end{array}\end{array}} $$
(26)
$$ {\displaystyle \begin{array}{l}\frac{\partial }{\partial \sigma }C\left({\xi}_1,{\xi}_2\right)=\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{d}_2\\ {}{d}_1\end{array}\frac{\partial }{\partial \sigma}\left[\frac{1}{\sigma }f\left(\frac{x-\mu }{\sigma}\right)\right] dx\\ {}\kern5.5em =\frac{1}{\sigma }+\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial \log f(z)}{\partial \sigma }f(z) dz\\ {}\begin{array}{l}\kern5.5em =-\frac{1}{\sigma }+E\left[\left.\frac{\partial \log f(z)}{\partial \sigma}\right|{\xi}_1<z<{\xi}_2\right]\\ {}\kern5.5em =-\frac{1}{\sigma}\left(1+E\left[\left.\frac{z\partial \log f(z)}{\partial z}\right|{\xi}_1<z<{\xi}_2\right]\right)\\ {}\kern5.5em =-\frac{1}{\sigma}\frac{\xi_2f\left({\xi}_2\right)-{\xi}_1f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}\end{array}\end{array}} $$
(27)
$$ {\displaystyle \begin{array}{l}\frac{\partial^2}{\partial {\sigma}^2}C\left({\xi}_1,{\xi}_2\right)=\frac{1}{\sigma^2}+\frac{\partial }{\partial \sigma}\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{d}^2\\ {}{d}^1\end{array}\frac{1}{\sigma}\frac{\partial f\left(\frac{x-\mu }{\sigma}\right)}{\partial \sigma } dx\\ {}\kern6em =\frac{1}{\sigma^2}+\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{d}_2\\ {}{d}_1\end{array}\frac{\partial }{\partial \sigma}\left[\frac{1}{\sigma}\frac{\partial \log f\left(\frac{x-\mu }{\sigma}\right)}{\partial \sigma }f\left(\frac{x-\mu }{\sigma}\right)\right] dx-{\left(\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \sigma } dz\right)}^2\\ {}\begin{array}{l}\kern6em =\frac{1}{\sigma^2}-\frac{1}{\sigma }E\left[\left.\frac{\partial \log f(z)}{\partial \sigma}\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.\frac{\partial \log f(z)}{\partial {\sigma}^2}\right|{\xi}_1<z<{\xi}_2\right]\\ {}\kern6em +E\left[\left.{\left(\frac{\partial \log f(z)}{\partial \sigma}\right)}^2\right|{\xi}_1<z<{\xi}_2\right]-{\left(E\left[\left.\frac{\partial \log f(z)}{\partial \sigma}\right|{\xi}_1<z<{\xi}_2\right]\right)}^2\\ {}\begin{array}{l}\kern6em =\frac{2}{\sigma^2}\left(1+E\left[\left.\frac{z\partial \log f(z)}{\partial z}\right|{\xi}_1<z{\xi}_2\right]\right)\\ {}\kern6em +\frac{1}{\sigma^2}\left(E\left[\left.{z}^2\frac{\partial^2\log f(z)}{\partial {z}^2}\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.{\left(z\frac{\partial \log f(z)}{\partial z}\right)}^2\right|{\xi}_1<z<{\xi}_2\right]-{\left(E\left[\left.z\frac{\partial \log f(z)}{\partial z}\right|{\xi}_1<z<{\xi}_2\right]\right)}^2\right)\\ {}\kern6em =\frac{2}{\sigma^2}\frac{\xi_2f\left({\xi}_2\right)-{\xi}_1f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}+\frac{1}{\sigma^2}\left(\frac{\xi \begin{array}{c}2\\ {}2\end{array}\frac{\partial f\left({\xi}_2\right)}{\partial {\xi}_2}-\xi \begin{array}{c}2\\ {}1\end{array}\frac{\partial f\left({\xi}_1\right)}{\partial {\xi}_1}}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}-{\left(\frac{\xi_2f\left({\xi}_2\right)-{\xi}_1f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}\right)}^2\right)\end{array}\end{array}\end{array}} $$
(28)
$$ {\displaystyle \begin{array}{l}\frac{\partial^2}{\partial \mu \partial \sigma }C\left({\xi}_1,{\xi}_2\right)=\frac{\partial }{\partial \mu}\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \sigma } dz\\ {}\begin{array}{l}\kern6.75em =\frac{1}{\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial }{\partial \mu}\left[\frac{\partial \log f(z)}{\partial \sigma }f(z)\right] dz-\frac{1}{{\left(\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}f(z) dz\right)}^2}\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \mu } dz\int \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}\frac{\partial f(z)}{\partial \sigma } dz\\ {}\kern6.75em =E\left[\left.\frac{\partial \log f(z)}{\partial \sigma}\frac{\partial \log f(z)}{\partial \mu}\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.\frac{\partial^2\log f(z)}{\partial \sigma \partial \mu}\right|{\xi}_1<z<{\xi}_2\right]\\ {}\kern6.75em -E\left[\left.\frac{\partial \log f(z)}{\partial \sigma}\right|{\xi}_1<z<{\xi}_2\right]E\left[\left.\frac{\partial \log f(z)}{\partial \mu}\right|{\xi}_1<z<{\xi}_2\right]\end{array}\\ {}\begin{array}{l}\kern6.75em =\frac{1}{\sigma^2}\left(E\left[\left.z{\left(\frac{\partial \log f(z)}{\partial z}\right)}^2\right|{\xi}_1<z<{\xi}_2\right]+E\left[\left.z\frac{\partial^2\log f(z)}{\partial {z}^2}\right|{\xi}_1<z<{\xi}_2\right]-E\left[\left.z\frac{\partial \log f(z)}{\partial z}\right|{\xi}_1<z<{\xi}_2\right]E\left[\left.\frac{\partial \log f(z)}{\partial z}\right|{\xi}_1<z<{\xi}_2\right]\right)\\ {}\kern6.75em =\frac{1}{\sigma^2}\left[\frac{f\left({\xi}_2\right)-f\left({\xi}_1\right)}{F\left({\xi}_2\right)-F\left({\xi}_1\right)}+\frac{\left(F\left({\xi}_2\right)-F\left({\xi}_1\right)\right)\left({\xi}_2\frac{\partial f\left({\xi}_2\right)}{\partial {\xi}_2}-{\xi}_1\frac{\partial f\left({\xi}_1\right)}{\partial {\xi}_1}\right)-\left(f\left({\xi}_2\right)-f\left({\xi}_1\right)\right)\left({\xi}_2f\left({\xi}_2\right)-{\xi}_1f\left({\xi}_1\right)\right)}{{\left(F\left({\xi}_2\right)-F\left({\xi}_1\right)\right)}^2}\right]\end{array}\end{array}} $$
(29)

Modified likelihood

The censored term in the log-likelihood equation may be modified using the Taylor series using the quadratic approximation.

We may approximate \( C\left({\xi}_1,{\xi}_2\right)=\log \int {\displaystyle \begin{array}{c}{\xi}_2\\ {}{\xi}_1\end{array}}f(z) dz \) as the following

$$ {\displaystyle \begin{array}{c}C\left({\xi}_1,{\xi}_2\right)\approx C\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)+{C}_{\xi_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\left({\xi}_1-{\hat{\xi}}_1\right)+{C}_{\xi_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\left({\xi}_2-{\hat{\xi}}_2\right)\\ {}+\frac{1}{2!}\left[{C}_{\xi_1{\xi}_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right){\left({\xi}_1-{\hat{\xi}}_1\right)}^2+2{C}_{\xi_1{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\left({\xi}_1-{\hat{\xi}}_1\right)\left({\xi}_2-{\hat{\xi}}_2\right)+{C}_{\xi_2{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right){\left({\xi}_2-{\hat{\xi}}_2\right)}^2\right]\end{array}} $$

which can be written as

$$ C\left({\xi}_1,{\xi}_2\right)\approx C+ $$
$$ {\displaystyle \begin{array}{c}C=C\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_1{C}_{\xi_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_2{C}_{\xi_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)+\frac{1}{2}\hat{\xi}\begin{array}{c}2\\ {}1\end{array}{C}_{\xi_1{\xi}_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)+\frac{1}{2}\hat{\xi}\begin{array}{c}2\\ {}2\end{array}{C}_{\xi_2{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)+{\hat{\xi}}_1{\hat{\xi}}_2{C}_{\xi_1{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\\ {}{\leftthreetimes}_1={C}_{\xi_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_1{C}_{\xi_1{\xi}_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_2{C}_{\xi_1{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\\ {}\begin{array}{c}{\leftthreetimes}_2={C}_{\xi_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_2{C}_{\xi_2{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)-{\hat{\xi}}_1{C}_{\xi_1{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\\ {}{\leftthreetimes}_3=\frac{1}{2}{C}_{\xi_1{\xi}_1}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\\ {}\begin{array}{c}{\leftthreetimes}_4=\frac{1}{2}{C}_{\xi_2{\xi}_2}\left({\hat{\xi}}_1,{\hat{\xi}}_2\right)\\ {}{\leftthreetimes}_5={C}_{\xi_1{\xi}_2}\left({\hat{\xi}}_1{\hat{\xi}}_2\right)\end{array}\end{array}\end{array}} $$

where \( {\hat{\xi}}_1 \) and \( {\hat{\xi}}_2 \) may be estimated using the proportion of censored values as \( {\hat{\xi}}_1={F}^{-1}\left(\frac{n_2-{n}_0}{n}\right),{\hat{\xi}}_2={F}^{-1}\left(\frac{n_2}{n}\right),{n}_2=\sum {\displaystyle \begin{array}{c}n\\ {}i=1\end{array}}{\delta}_{2i}, \)

$$ {\delta}_{2i}=\left\{\begin{array}{c}1:{z}_i<{\xi}_2\\ {}0:\mathrm{otherwise}\end{array}\right. $$

and

$$ \int {\displaystyle \begin{array}{c}{\xi}_2\\ {}{\hat{\xi}}_1\end{array}}f(z) dz=\frac{n_o}{n} $$

So we can write the log-likelihood function as

$$ \ell \left(\mu, \sigma \right)=-r\log \sigma +\sum \limits_{i=1}^n{\delta}_i\log f\left({z}_i\right)+{n}_o\left(C+{\leftthreetimes}_1{\xi}_1+{\leftthreetimes}_2{\xi}_2+{\leftthreetimes}_3\xi \begin{array}{c}2\\ {}1\end{array}+{\leftthreetimes}_4\xi \begin{array}{c}2\\ {}2\end{array}+{\leftthreetimes}_5{\xi}_1{\xi}_2\right) $$
(30)

And so the first and the second derivatives with respect to μ and σ are:

$$ \frac{\partial \ell }{\partial \mu }=-\frac{1}{\sigma}\left[\sum \limits_{i=1}^n{\delta}_i\frac{\partial \log f\left({z}_i\right)}{\partial {z}_i}+{n}_o\left(\left({\leftthreetimes}_1+{\leftthreetimes}_2\right)+\left(2{\leftthreetimes}_3+{\leftthreetimes}_5\right){\xi}_1+\left(2{\leftthreetimes}_4+{\leftthreetimes}_5\right){\xi}_2\right)\right] $$
(31)
$$ \frac{\partial \ell }{\partial \sigma }=-\frac{r}{\sigma }-\frac{1}{\sigma}\left[\sum \limits_{i=1}^n{\delta}_i{z}_i\frac{\partial^2\log f\left({z}_i\right)}{\partial {z}_i}+{n}_o\left({\leftthreetimes}_1{\xi}_1+{\leftthreetimes}_2{\xi}_2+2{\leftthreetimes}_3\xi \begin{array}{c}2\\ {}1\end{array}+2{\leftthreetimes}_4\xi \begin{array}{c}2\\ {}2\end{array}+2{\leftthreetimes}_5{\xi}_1{\xi}_2\right)\right] $$
(32)
$$ \frac{\partial^2\ell }{\partial {\mu}^2}=\frac{1}{\sigma^2}\left[\sum \limits_{i=1}^n{\delta}_i\frac{\partial^2\log f\left({z}_i\right)}{\partial z\begin{array}{c}2\\ {}i\end{array}}+2{n}_o\left({\leftthreetimes}_3+{\leftthreetimes}_4+{\leftthreetimes}_5\right)\right] $$
(33)
$$ \frac{\vartheta^2\ell }{\vartheta {\sigma}^2}=\frac{r}{\sigma^2}+\frac{1}{\sigma^2}\left[\sum \limits_{i=1}^n{\delta}_i{z}_i\frac{\partial \log f\left({z}_i\right)}{\partial {z}_i}+\sum \limits_{i=1}^n{\delta}_iz\begin{array}{c}2\\ {}i\end{array}\frac{\partial \log f\left({z}_i\right)}{\partial z\begin{array}{c}2\\ {}i\end{array}}\right]+\frac{2}{\sigma^2}\left[{n}_o\left({\leftthreetimes}_1{\xi}_1+{\leftthreetimes}_2{\xi}_2+3{\leftthreetimes}_3\xi \begin{array}{c}2\\ {}1\end{array}+3{\leftthreetimes}_4\xi \begin{array}{c}2\\ {}2\end{array}+3{\leftthreetimes}_5{\xi}_1{\xi}_2\right)\right] $$
(34)
$$ \frac{\vartheta^2\ell }{\vartheta \sigma \vartheta \mu}=\frac{1}{\sigma^2}\left[\sum \limits_{i=1}^n{\delta}_i\frac{\partial \log f\left({z}_i\right)}{\partial {z}_i}+\sum \limits_{i=1}^n{\delta}_i{z}_i\frac{\partial^2\log f\left({z}_i\right)}{\partial z\begin{array}{c}2\\ {}i\end{array}}\right]+\frac{1}{\sigma^2}\left[{\mathrm{n}}_o\left(\left({\leftthreetimes}_1+{\leftthreetimes}_2\right)+\left(4{\leftthreetimes}_3+2{\leftthreetimes}_5\right){\xi}_1+\left(4{\leftthreetimes}_4+2{\leftthreetimes}_5\right){\xi}_2\right)\right] $$
(35)

Appendix 2. R function

An R function is written to estimate the mean and the standard deviation using the modified likelihood method for a vector of normal data Y, d the detection limit, and all values below d are censored.

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Hassan, H.M., El-Shaarawi, A.H. Likelihood inference for pollutant loading under type I censoring. Environ Monit Assess 192, 225 (2020). https://doi.org/10.1007/s10661-020-8178-5

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