MLE for the parameters of bivariate interval-valued model

Abstract

With contemporary data sets becoming too large to analyze the data directly, various forms of aggregated data are becoming common. The original individual data are points, but after aggregation the observations are interval-valued (e.g.). While some researchers simply analyze the set of averages of the observations by aggregated class, it is easily established that approach ignores much of the information in the original data set. The initial theoretical work for interval-valued data was that of Le-Rademacher and Billard (J Stat Plan Infer 141:1593–1602, 2011), but those results were limited to estimation of the mean and variance of a single variable only. This article seeks to redress the limitation of their work by deriving the maximum likelihood estimator for the all important covariance statistic, a basic requirement for numerous methodologies, such as regression, principal components, and canonical analyses. Asymptotic properties of the proposed estimators are established. The Le-Rademacher and Billard results emerge as special cases of our wider derivations.

Appendices

Appendix A

Early work on interval data sometimes transformed the interval-valued variable into two variables, center and range (or, given their one-to-one correspondence equivalently into the end point values). Consider the Y values of the interval-valued data sets of Table 3. Let us denote the interval centers by $Y^c = (a + b)/2$, and the interval half-range by $Y^r = (b - a)/2$. Then the first and second columns of Table 4(a) give the sample variances of Y for the interval centers and ranges, respectively (calculated by classical results or as special cases of Eq. (2)). The third column shows the sum $(\hbox {Var}(Y^c) + \hbox {Var}(Y^r))$. This can be compared with the sample variance $\hbox {Var}(Y)$ of the intervals in the right-most column (from Eq. (2) and Bertrand and Goupil (2000)). Thus we see that sometimes the sum $(\hbox {Var}(Y^c) + \hbox {Var}(Y^r))$ is greater, and sometimes less, than the symbolic variance $\hbox {Var}(Y)$; this depends on the actual data. The fourth data set consists of classical values (with $a\equiv [a,a]$); in this case, the $\hbox {Var}(Y^r)=0$ and so $\hbox {Var}(Y^c) = \hbox {Var}(Y)$, as it should.

Table 3 Some Data Sets (Y, X)

Likewise, by using the centers and range values for both Y and X, we can calculate the classical covariances of the centers and of the ranges and the symbolic interval covariances, from Eq. (3), shown in Table  4(b). Again, the sum $(\hbox {Cov}(Y^c, X^c) + \hbox {Cov}(Y^r, X^r))$ can be greater, or smaller, than the symbolic covariance $\hbox {Cov}(Y, X)$; and for classical observations, this sum equals the symbolic covariance correctly as expected.

Table 4 Variances and Covariances

For a second aspect, suppose a data set consists of intervals all with the same center but different range values. Then, the variance-covariance terms for the centers are zero; and in contrast, if the data are such that the observations have different center values but all have the same range value, then the variance-covariance terms for the ranges are zero. Then for methods that rely on the relevant variance-covariance matrices, the methodology cannot be properly implemented, since, e.g., in regression that matrix is zero and for principal components the eigenvalues are zero.

The variance-covariance definition of Eq. (3) does not have these limitations.

Appendix B

The log likelihood function $\ln L_I$ from Eq. (12) and Eq. (13) is

$$\begin{aligned} \ln L_I&\propto -n \ln (\sigma _x)-n \ln (\sigma _y)-(n/2)\ln (1-\rho ^2) \\&~~~ -\frac{1}{2(1-\rho ^2)}\sum _{i=1}^n\bigg [ \frac{\left( \theta _{i1}^x -\mu _x\right) ^2}{\sigma ^2_x} + \frac{\left( \theta _{i1}^y -\mu _y\right) ^2}{\sigma ^2_y} -2\rho \frac{(\theta _{i1}^x-\mu _x)(\theta _{i1}^y-\mu _y)}{\sigma _x\sigma _y}\bigg ] \\&~~~ -\frac{n\nu }{2} \ln \left( \gamma _1\gamma _2-\gamma _3^2 \right) -\frac{\gamma _1\gamma _2}{2(\gamma _1\gamma _2-\gamma _3^2)} \left( \frac{1}{\gamma _1}\sum _{i=1}^{n} \theta _{i2}^{x} +\frac{1}{\gamma _2}\sum _{i=1}^{n} \theta _{i2}^{ y} -\frac{2\gamma _3 }{\gamma _1 \gamma _2} \sum _{i=1}^{n} \theta _{i2}^{ xy} \right) \end{aligned}$$

Then successively differentiating $\ln L_I$ with respect to each of the eight parameters in ${{\varvec{\tau }}}$, we obtain

$$\begin{aligned} \frac{\partial \ln L_I}{\partial \mu _x}&= \frac{1}{(1-\rho ^2) } \left( \frac{1}{\sigma ^2_x} \sum _{i=1}^n\left( \theta _{i1}^x-\mu _x\right) -\frac{\rho }{\sigma _x \sigma _y}\sum _{i=1}^n\left( \theta _{i1}^y-\mu _y\right) \right) ,\nonumber \\ \frac{\partial \ln L_I}{\partial \mu _y}&= \frac{1}{(1-\rho ^2)} \left( \frac{1}{\sigma ^2_y} \sum _{i=1}^n\left( \theta _{i1}^y-\mu _y\right) -\frac{ \rho }{\sigma _x \sigma _y}\sum _{i=1}^n\left( \theta _{i1}^x-\mu _x\right) \right) ,\nonumber \\ \frac{\partial \ln L_I}{\partial \sigma _x}&= \frac{-n}{\sigma _x} + \frac{1}{2(1-\rho ^2)}\left( \sum _{i=1}^n\frac{2\left( \theta _{i1}^x-\mu _x\right) ^2}{\sigma ^3_x} - 2\rho \sum _{i=1}^n\frac{\left( \theta _{i1}^x-\mu _x\right) \left( \theta _{i1}^y-\mu _y\right) }{\sigma _x^2\sigma _y}\right) , \nonumber \\ \frac{\partial \ln L_I}{\partial \sigma _y}&= \frac{-n}{\sigma _y} + \frac{1}{2(1-\rho ^2)}\left( \sum _{i=1}^n\frac{2\left( \theta _{i1}^y-\mu _y\right) ^2}{\sigma ^3_y} - 2\rho \sum _{i=1}^n\frac{\left( \theta _{i1}^x-\mu _x\right) \left( \theta _{i1}^y-\mu _y\right) }{\sigma _x\sigma _y^2}\right) , \nonumber \\ \frac{\partial \ln L_I}{\partial \rho }&= \frac{n\rho }{(1-\rho ^2)} - \frac{\rho }{(1-\rho ^2)^2} \left( \sum _{i=1}^n \frac{ \left( \theta _{i1}^x-\mu _x\right) ^2}{\sigma ^2_x} + \sum _{i=1}^n \frac{ \left( \theta _{i1}^y-\mu _y \right) ^2}{\sigma ^2_y} \right) \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \frac{1+ \rho ^2}{(1-\rho ^2)^2} \sum _{i=1}^n\frac{(\theta _{i1}^x-\mu _x)(\theta _{i1}^y-\mu _y)}{\sigma _x\sigma _y}, \nonumber \\ \frac{\partial \ln L_I}{\partial \gamma _1}&= -\frac{n\nu \gamma _2}{2G} + \frac{\gamma _2^2}{2G^2}\sum _{i=1}^n\theta _{i2}^x - \frac{G-\gamma _1 \gamma _2}{2G^2}\sum _{i=1}^n\theta _{i2}^y - \frac{\gamma _2\gamma _3}{G^2} \sum _{i=1}^n\theta _{i2}^{xy},\nonumber \\ \frac{\partial \ln L_I}{\partial \gamma _2}&= -\frac{n\nu \gamma _1}{2G} - \frac{G-\gamma _1 \gamma _2}{2G^2}\sum _{i=1}^n\theta _{i2}^x + \frac{\gamma _1^2}{2G^2}\sum _{i=1}^n\theta _{i2}^y - \frac{\gamma _1\gamma _3}{G^2} \sum _{i=1}^n\theta _{i2}^{xy}, \nonumber \\ \frac{\partial \ln L_I}{\partial \gamma _3}&= \frac{n\nu \gamma _3}{ G} - \frac{ \gamma _2 \gamma _3}{G^2}\sum _{i=1}^n\theta _{i2}^x - \frac{ \gamma _1 \gamma _3}{G^2}\sum _{i=1}^n\theta _{i2}^y + \frac{ G+ 2\gamma _3^2}{ G^2}\sum _{i=1}^n \theta _{i2}^{xy} \end{aligned}$$

(42)

where $G=\gamma _1\gamma _2 - \gamma _3^2$.

Then, substituting the relevant maximum likelihood estimator and setting the derivatives to zero, we can obtain the maximum likelihood estimators $\hat{{{\varvec{\tau }}}}_{xy} = (\hat{\mu }_x, \hat{\mu }_y, \hat{\sigma }_x^2, \hat{\sigma }_y^2, \hat{\rho }, \hat{\gamma }_1,\hat{\gamma }_2,\hat{\gamma }_3)$ for ${{\varvec{\tau }}}_{xy} = (\mu _x, \mu _y, \sigma _x^2, \sigma _y^2, \rho , \gamma _1,\gamma _2,\gamma _3)$ to be as given by Eqs. (19–22). We also note that instead of solving the partial derivative in Eq. (42) for the derivation of the estimator $\hat{\rho }$, we can more easily obtain the result of Eq. (42) by following, e.g., (Casella and Berger (2002), p.358) who suggest using a partially maximized likelihood function.