Control chart for monitoring zero-or-one inflated double-bounded environmental processes

Abstract

The Kumaraswamy distribution is widely used for modeling rates and proportions, such as relative humidity. Motivated by one environmental application, where the variable of interest is observed on the unit-interval and inflated at one, we propose Shewhart-type control charts based on the inflated Kumaraswamy distribution. In particular, we propose a control chart for monitoring the inflated Kumaraswamy median parameter. For highly asymmetric distributions or in the presence of outliers, the median is a more appropriate location parameter than the average. We proposed control charts considering two approaches: individual observations and non-individual observations. For the non-individual observations, three estimators were considered for the median parameter: sample median, maximum likelihood estimator, and the estimator proposed by Hodges and Lehmann (Ann Math Stat 34:598–611, 1963). The control chart performance is evaluated in terms of run length analysis. The numerical results show that the proposed control chart performed well in the two considered approaches.

Appendices

Appendix A

1.1 Maximum likelihood estimation

Let $y _1,\ldots ,y _n$, with $y _t\in [0,1)$ or $y _t\in (0,1]$, be a sample from the iKw distribution at c, where c is fixed for all observations ($c=0$ or $c=1$). The log-likelihood function for $\theta =(\gamma ,\mu ,\phi )$ is

$$\begin{aligned} \ell (\theta )=\ell _{1}(\gamma )+\ell _{2}(\mu ,\phi ), \end{aligned}$$

where

$$\begin{aligned} \ell _{1}(\gamma )=&\log \gamma \sum _{t=1}^{n}{1\!\text {l}}_{\{c\}}(y_t)+\log (1-\gamma )\Bigg [n-\sum _{t=1}^{n}{1\!\text {l}}_{\{c\}}(y_t)\Bigg ], \\ \ell _{2}(\mu ,\phi )=&\sum _{t=1}^{n}(1-{1\!\text {l}}_{\{c\}}(y_t))\left[ \log (\phi )\!+\log \left( \!\frac{\log (0.5)}{\log (1\!-\!{\mu }^\phi )}\!\right) \!+\!(\phi \!-\!1)\log ({y}_t) + \right. \\&\left( \!\frac{\log (0.5)}{\log (1\!-\!{\mu }^\phi )} \!-\!1\!\! \right) \log (1\!-\!{y}_t^\phi ) \Bigg ]. \end{aligned}$$

We note that inference on $(\mu ,\phi )$ can be performed separately from that carried out on $\gamma$. Therefore, it is possible to independently obtain the score functions for $\gamma$ and for $(\mu ,\phi )$.

The score function for $\gamma$ can be written as

$$\begin{aligned} U_\gamma (\gamma ) =\frac{\partial \ell _1(\gamma )}{\partial \gamma }= \sum \limits _{t=1}^{n} \frac{{1\!\text {l}}_{\{c\}}(y _t)-\gamma }{\gamma (1-\gamma )}. \end{aligned}$$

The score function for $\mu$ can be expressed as

$$\begin{aligned} U_\mu (\mu ,\phi )=\frac{\partial \ell _2(\mu ,\phi )}{\partial \mu } =\sum \limits _{t=1}^{n}(1-{1\!\text {l}}_{\{c\}}(y _t)) \phi \frac{{\mu }^{\phi -1}}{(1-{\mu }^\phi )\log (1-{\mu }^\phi )} \left( \delta \log (1-{y}_t^\phi )+1\right) . \end{aligned}$$

Let $a_t=\frac{{\mu }^{\phi -1}}{(1-{\mu }^\phi )\log (1-{\mu }^\phi )} \left( \delta \log (1-{y}_t^\phi )+1\right)$, thus

$$\begin{aligned} U_\mu (\mu ,\phi ) = \phi \sum \limits _{t=1}^{n} \left[ (1-{1\!\text {l}}_{\{c\}}(y_t)) a_t \right] . \end{aligned}$$

Finally, the score function for $\phi$ is given by

$$\begin{aligned} U_\phi (\mu ,\phi )=&\frac{\partial \ell _2(\mu ,\phi )}{\partial \phi }=\sum \limits _{t=1}^{n}(1-{1\!\text {l}}_{\{c\}}(y _t))\left\{ \frac{1}{\phi }+\log (y_t)+a_t \mu \log (\mu )- \right. \\&\left. (\delta -1)\frac{y_t^{\phi }\log (y_t)}{(1-y_t^{\phi })}\right\} . \end{aligned}$$

Thus, the score vector obtained by differentiating the log-likelihood function concerning the unknown parameters is $(U_\gamma (\gamma ),U_\mu (\mu ,\phi ),U_\phi (\mu ,\phi )).$

The maximum likelihood estimator (MLE) of $\gamma$ is obtained as a solution of the following equation: $U_\gamma (\gamma )=(0,0,0)$. The closed-form solution is given by ${\hat{\gamma }}=\frac{\sum \limits _{t=1}^{n} {1\!\text {l}}_{\{c\}}(y _t)}{n}$, i.e. the ${\hat{\gamma }}$ estimator represents the proportion of values equal 0 or 1 in the sample. Moreover, the MLE of $\mu$ and $\phi$ can be obtained by solving the nonlinear system $(U_{\mu }(\mu ,\phi ), U_{\phi }(\mu ,\phi ))=(0,0)$. These estimators cannot be expressed in closed-form. They can be computed numerically using a Newton or quasi-Newton nonlinear optimization algorithm.

Appendix B

1.1 Performance measures of the iKw control chart

This appendix presents all the figures of the numerical performance evaluation of the iKw control chart for non-individual observations. These numerical results are shonw in Tables 3, 4, 5, 6, 7, and 8.

Table 3 Performance, in terms of ARL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 1, 2 and 3

Table 4 Performance, in terms of ARL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 4, 5 and 6

Table 5 Performance, in terms of MRL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 1, 2 and 3

Table 6 Performance, in terms of MRL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 4, 5 and 6

Table 7 Performance, in terms of SDRL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 1, 2 and 3

Table 8 Performance, in terms of SDRL, of the iKw control chart considering SM, HL and ML estimators for the median—Scenario 4, 5 and 6