A bivariate hierarchical Bayesian model for estimating cropland cash rental rates at the county level
Section 5. Conclusions and future work

• Previous
• Next

We use a bivariate HB model to obtain predictors of county-level cash rental rates for non-irrigated cropland in Iowa, Kansas, and Texas. The model incorporates auxiliary information related to land quality, commodity values, and farm characteristics. Significant correlations exist between the 2009 and 2010 model random effects at both the unit and county levels. As a consequence, using the information in the 2009 cash rent estimates reduces the posterior MSE relative to a univariate model. The analysis of the bivariate HB model provides support that a more refined approach than that of Berg et al. (2014) is possible. To incorporate unit-level covariates with unknown population means, we add a level to the hierarchical model that justifies adding a term to the posterior mean squared error to account for uncertainty in the unknown population means of the unit-level covariates. Unlike Berg et al. (2014), the proposed bivariate HB model allows variability to change over time and accounts for effects of benchmarking on the MSE.

The analysis of the residuals and the posterior predictive $\text{p}-$ values suggests that accounting for outliers may be an important way to substantially improve the model fit. One option is to consider a heavy-tailed distribution, such as a $t-$ distribution or a mixture of normal distributions, that may represent the observed responses more appropriately than the assumed normal distribution. An extension of Gershunskaya (2010) to bivariate framework and Bayesian estimation is one possible way to approach the issue of outliers.

Acknowledgements

The National Agricultural Statistics Service (NASS) of the United States Department of Agriculture (USDA) supported this work. The authors are grateful to Wendy Barboza, Dan Beckler, Angie Considine, Mark Harris, Sharyn Lavender, Joe Parsons, Scot Rumberg, Scott Shimmin, Curt Stock, and Linda Young from the National Agriculture Statistics Service. Further, the authors thank Bob Dobos from the National Resource Conservation Service and Rich Iovanna from the Farm Service Agency for their assistance in acquiring the National Commodity Crop Productivity Index. Without the generous assistance from these individuals, this research would have been impossible. The views expressed in this paper are those of the authors and do not necessarily represent the views of NASS or the USDA.

Appendix A

To specify the full conditional distributions for Gibbs sampling, we introduce notation. Let ${\Theta }_{\gamma }$ be the set of parameters except for the parameter denoted by $\gamma .$ Let $X_{ij}={\left(z_{ij\text{}\mathrm{,}09}\text{}\mathrm{,}{z}_{ij\text{}\mathrm{,}10}\right)}^{\prime }\text{},$ where $z_{ij\text{}\mathrm{,}09}\mathrm{<mo>=</mo>}{\left(x_{ij\text{}\mathrm{,}09}^{\prime }\mathrm{,}{0}_{p_{10}}^{\prime }\right)}^{\prime }\mathrm{,}$ and $z_{ij\text{}\mathrm{,}10}\mathrm{<mo>=</mo>}{\left(0_{p_{09}}^{\prime }\text{}\mathrm{,}{x}_{ij\text{}\mathrm{,}10}^{\prime }\right)}^{\prime }\text{}.$ Let $y_{ij}^{\prime }\mathrm{<mo>=</mo>}\left(y_{ij\text{}\mathrm{,}09}\text{}\mathrm{,}{y}_{ij\text{}\mathrm{,}10}\right)\mathrm{.}$ Let $A_i$ be the set of units (farm operators) in county $i$ that are in set 1, $B_{i\text{}\mathrm{,}09}$ be the set of units in county $i$ that are in set 2, and $B_{i\text{}\mathrm{,}10}$ be the set of units in county $i$ that are in set 3, where set 1, set 2, and set 3 are defined in Section 3. Full conditionals are as follows.

1. $\beta |\left({\Theta }_{\beta }\mathrm{,}y\right)\sim \text{N}\left({\Sigma }_{\beta \beta }{r}_{\beta }\text{}\mathrm{,}{\Sigma }_{\beta \beta }\right),$ where

$$\begin{array}{ll}{\Sigma }_{\beta \beta }\hfill & \mathrm{<mo>=</mo>}{\left[{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in A_i}}{X}_{ij}^{\prime }{D}_{wij}^{\text{0}\text{.5}}{\Sigma }_{ee}^{-1}{D}_{wij}^{\text{0}\text{.5}}{X}_{ij}+{10}^{-6}{I}_{p_{09}+p_{10}}+\Omega \right]}^{-1}(\text{A}.1)\hfill \\ \Omega \hfill & \mathrm{<mo>=</mo>}\text{block-diag}\left({\tau }_{e\text{}\mathrm{,}09}^{-2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in B_{i\text{}\mathrm{,}09}}}{w}_{ij\text{}\mathrm{,}09}{x}_{ij\text{}\mathrm{,}09}{x}_{ij\text{}\mathrm{,}09}^{\prime }\mathrm{,}{\tau }_{e\text{}\mathrm{,}10}^{-2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in B_{i\text{}\mathrm{,}10}}}{w}_{ij\text{}\mathrm{,}10}{x}_{ij\text{}\mathrm{,}10}{x}_{ij\text{}\mathrm{,}10}^{\prime }\right)\hfill \\ r_{\beta }\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in A_i}}{X}_{ij}^{\prime }{D}_{wij}^{\text{0}\text{.5}}{\Sigma }_{ee}^{-1}{D}_{wij}^{\text{0}\text{.5}}\left(y_{ij}-v_i+r_{\beta 2}\right)\mathrm{,}\hfill \end{array}$$

1. and

$$r_{\beta 2}\mathrm{<mo>=</mo>}\left(\begin{array}{c}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D{\displaystyle {\sum }_{j\in B_{i\text{}\mathrm{,}09}}{\tau }_{e\text{}\mathrm{,}09}^{-2}{w}_{ij\text{},09}{x}_{ij\text{}\mathrm{,}09}\left(y_{ij\text{}\mathrm{,}09}-{\nu }_{i\text{}\mathrm{,}09}\right)}}\\ {\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D{\displaystyle {\sum }_{j\in B_{i\text{}\mathrm{,}10}}{\tau }_{e\text{}\mathrm{,}10}^{-2}{w}_{ij\text{}\mathrm{,}10}{x}_{ij\text{}\mathrm{,}10}\left(y_{ij\text{}\mathrm{,}10}-{\nu }_{i\text{}\mathrm{,}10}\right)}}\end{array}\right)\mathrm{.}(\text{A}.2)$$

2. ${\Sigma }_{ee}|\left({\Theta }_{{\Sigma }_{ee}}\text{}\mathrm{,}y\right)\sim \text{Inverse-Wishart}\left(A_e\text{}\mathrm{,}{d}_e\right),$ where $d_e\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D}\left|A_i\right|+\text{0}\text{.001,}$ and

$$A_e\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in A_i}}{D}_{w_{ij}}^{\text{0}\text{.5}}\left(y_{ij}-{\nu }_i-X_{ij}\beta \right){\left(y_{ij}-{\nu }_i-X_{ij}\beta \right)}^{\prime }{D}_{w_{ij}}^{\text{0}\text{.5}}.(\text{A}.3)$$

3. ${\Sigma }_{\nu \nu }|\left(\Theta {\text{Q}}_{{\Sigma }_{vv}}\text{}\mathrm{,}y\right)\sim \text{Inverse-Wishart}\left(A_{\nu }\text{}\mathrm{,}{d}_{\nu }\right),$ where

$$d_{\nu }\mathrm{<mo>=</mo>}D+\text{0}\text{.001}\mathrm{,}(\text{A}.4)$$

• and

$$A_{\nu }\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\nu }_i{\nu }_i^{\prime }\mathrm{.}(\text{A}.5)$$

${\tau }_{e\text{}\mathrm{,}t}^2|\left({\Theta }_{{\tau }_{e\text{},t}^2}\text{}\mathrm{,}y\right)\sim \text{Inverse-Gamma}\left(a_{et}\text{}\mathrm{,}{d}_{et}\right),$ where

$$d_{et}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}\left|B_{i\text{}\mathrm{,}t}\right|+\text{0}\text{.001}\mathrm{,}(\text{A}.6)$$

• and

$$a_{et}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in B_{it}}}{D}_{w{i}_j}{\left(y_{ij\text{}\mathrm{,}t}-{\nu }_{i\text{}\mathrm{,}t}-x_{ij\text{}\mathrm{,}t}{\beta }_t\right)}^2\text{}\mathrm{.}(\text{A}.7)$$

4. ${\nu }_i|\left({\Theta }_{{\nu }_i}\text{}\mathrm{,}y\right)\sim \text{N}\left({\mu }_{\nu \nu }\mathrm{,}{M}_i^{-1}\right),$ where

$$M_i\mathrm{<mo>=</mo>}{\left({\Sigma }_{\nu \nu }^{-1}+{\Sigma }_{ee\text{}\mathrm{,}wi}^{-1}+{\Omega }_{ee\text{}\mathrm{,}wi}^{-1}\right)}^{-1}\text{}\mathrm{,}(\text{A}.8)$$

• ${\mu }_{\nu \nu }\mathrm{<mo>=</mo>}{M}_i^{-1}\left(r_{i_1}+r_{i_2}\right),{\Sigma }_{ee\text{}\mathrm{,}wi}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in A_i}}{D}_{w_{ij}}^{\text{0}\text{.5}}{\Sigma }_{ee}^{-1}{D}_{w_{ij}}^{\text{0}\text{.5}}\text{},$

$$\begin{array}{ll}{\text{W}}_{ee\text{}\mathrm{,}wi}\hfill & \mathrm{<mo>=</mo>}\text{diag}\left({\tau }_{e\text{}\mathrm{,}09}^{-2}{\displaystyle \sum _{j\in B_{i09}}}{w}_{ij\text{}\mathrm{,}09}\text{}\mathrm{,}{\tau }_{e\text{}\mathrm{,}10}^{-2}{\displaystyle \sum _{j\in B_{i10}}}{w}_{ij\text{}\mathrm{,}10}\right)\mathrm{,}(\text{A}.9)\hfill \\ r_{i_1}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in A_i}}{D}_{w_{ij}}^{\text{0}\text{.5}}{\Sigma }_{ee}^{-1}{D}_{w_{ij}}^{\text{0}\text{.5}}\left(y_{ij}-X_{ij}^{\prime }\beta \right)\mathrm{,}\hfill \end{array}$$

• and

$$r_{i_2}\mathrm{<mo>=</mo>}\left(\begin{array}{c}{\displaystyle {\sum }_{j\in B_{i\text{}\mathrm{,}09}}{w}_{ij\text{}\mathrm{,}09}\left(y_{ij\text{}\mathrm{,}09}-x_{ij\text{}\mathrm{,}09}^{\prime }{\beta }_{09}\right)}{\tau }_{e\text{}\mathrm{,}09}^{-2}\\ {\displaystyle {\sum }_{j\in B_{i\text{}\mathrm{,}10}}{w}_{ij\text{}\mathrm{,}10}\left(y_{ij\text{}\mathrm{,}10}-x_{ij\text{}\mathrm{,}10}^{\prime }{\beta }_{10}\right)}{\tau }_{e\text{}\mathrm{,}10}^{-2}\end{array}\right)\mathrm{.}(\text{A}.10)$$

Appendix B

We define an estimator of the diagonal elements of $V\left\{{\overline{x}}_{wi10}|{\overline{x}}_{N_i\text{}\mathrm{,}10}\right\}:=V_{xxi\text{}\mathrm{,}10}$ corresponding to unit-level covariates, $x_{ijk10}$ for $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}{p}_{10}.$ The variance estimator is based on a working assumption that a probability proportional to size with replacement (PPSWR) sample is a reasonable approximation for the cash rent survey design. As discussed in Cochran (1977), use of a PPSWR approximation is often reasonable if the sampling fraction is less than 10%. Suppose the draw probability for element $j$ in area $i$ for the PPSWR design is $p_{ij}\mathrm{<mo>=</mo>}{n}_{i10}^{-1}{w}_{ij\text{}\mathrm{,}10}^{-1}.$ Because $n_{i10}\le 1$ for some counties, we define the estimator of the diagonal elements of $V_{xxi10}$ corresponding to unit level covariates as a convex combination of a direct estimator of the within-area variance and a variance estimator that pools information across all counties in a state. For area $i$ with $n_{i10}\mathrm{<mo>></mo>\; 1},$ the estimate of the within-area variance of $x_{ijk10}$ under the assumed PPSWR design (Särndal, Swensson and Wretman, 1992) is given by

$$S_{ik10}^2\mathrm{<mo>=</mo>}\frac{n_{i10}^2}{{\left({\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{n_{i10}}{w}_{ij\text{}\mathrm{,}10}}\right)}^2\left(n_{i10}-1\right)}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_{i10}}}{w}_{ij\text{}\mathrm{,}10}^2{\left(x_{ijk10}-{\overline{x}}_{wik10}\right)}^2\text{}\mathrm{,}$$

where ${\overline{x}}_{wik10}$ is the $k^{\text{th}}$ element of ${\overline{x}}_{wi10}.$ The pooled estimator of the variance is defined by

$$S_{pk10}^2\mathrm{<mo>=</mo>}\frac{1}{w_{\mathrm{..10}}^2\left({\tilde{n}}_{10}-{\tilde{D}}_{10}\right)}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}\left(n_{i10}^2{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_{i10}}}{w}_{ij\text{}\mathrm{,}10}^2{\left(x_{ijk10}-{\overline{x}}_{wik10}\right)}^2\right)I\left[n_{i10}\mathrm{<mo>></mo>\; 1}\right]\mathrm{,}$$

where $w_{\mathrm{..10}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D}\left({\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{n_i}}{w}_{ij\text{}\mathrm{,}10}\right)I\left[n_i\mathrm{<mo>></mo>\; 1}\right],$ ${\tilde{n}}_{10}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D}{n}_{i10}I\left[n_{i10}\mathrm{<mo>></mo>\; 1}\right],$ and ${\tilde{D}}_{10}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D}I\left[n_{i10}\mathrm{<mo>></mo>\; 1}\right].$ The element of the diagonal covariance matrix $V_{xxi\text{}\mathrm{,}10}$ corresponding to the $k^{\text{th}}$ unit level covariate is then given by

$$\widehat{V}\left\{{\overline{x}}_{wik10}\right\}\mathrm{<mo>=</mo>}{n}_{i10}^{-1}{\widehat{S}}_{ik10}^{2}I\left[n_{i10}\ne 1\right]+n_{i10}^{-1}{S}_{pk10}^{2}I\left[n_{i10}\mathrm{<mo>=</mo>\; 1}\right]\mathrm{,}(\text{B}.1)$$

where

$${\widehat{S}}_{ik10}^2\mathrm{<mo>=</mo>}\frac{n_{i10}}{n_{i10}+1}{S}_{ik10}^2+\frac{1}{n_{i10}+1}{S}_{pk10}^2\mathrm{.}(\text{B}.2)$$

We provide a heuristic justification for the combination in (B.2), which is related to Haff (1980). Let $S^2\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{X}_i^2,$ where $X_i\sim \text{N}\left(0,{\sigma }^2\right).$ Assume ${\sigma }^2\sim \text{Inverse-Gamma}\left(\alpha \mathrm{,}\beta \right),$ where $E\left[{\sigma }^2\right]:=v\mathrm{<mo>=</mo>}\beta {\left(\alpha -1\right)}^{-1}\text{}.$ Then,

$$E\left[{\sigma }^2|S^2\right]\mathrm{<mo>=</mo>}\frac{2\left(\alpha -1\right)v}{n+2\left(\alpha -1\right)}+\frac{n{S}^2}{n+2\left(\alpha -1\right)}\mathrm{.}$$

In application to estimation of county-level cash rental rates, $S_{ik10}^2$ plays the role of $S^2$ and $S_{pk10}^2$ plays the role of $v.$ Taking $\alpha \mathrm{<mo>=</mo>}\text{1}\text{.5}$ gives the desired multiplier.

Appendix C

Data simulation from the posterior distributions

Consider the posterior samples for ${\beta }_{09}\text{}\mathrm{,}{\beta }_{10}\text{}\mathrm{,}{\Sigma }_{\nu \nu }$ and ${\Sigma }_{ee}\text{},$ denoted by ${\beta }_{09}^s\text{}\mathrm{,}{\beta }_{10}^s\text{},{\Sigma }_{\nu \nu }^s$ and ${\Sigma }_{ee}^s,$ respectively, for $s\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}S.$ Define

$${\Sigma }_{eeij}^s:=D_{wij}^{-\text{0}\text{.5}}{\Sigma }_{ee}^s{D}_{wij}^{-\text{0}\text{.5}}\mathrm{,}$$

for $s\mathrm{<mo>=</mo>\; 1,}\dots ,S.$ Draw replicates ${\nu }_{i09}^r\text{}\mathrm{,}{\nu }_{i10}^r\text{}\mathrm{,}{y}_{ij09}^r$ and $y_{ij10}^r,$ for $r\mathrm{<mo>=</mo>\; 1,}\dots ,R,$ following model (1-3) and properties of the multivariate conditional normal distribution as follows:

$$\begin{array}{ll}{\nu }_{i09}^r\hfill & \sim N\left(0\mathrm{,}{\Sigma }_{\nu \nu \text{}\mathrm{,}\left(11\right)}^r\right)\hfill \\ {\nu }_{i10}^r\hfill & \sim N\left({\left({\Sigma }_{\nu \nu \text{}\mathrm{,}\left(11\right)}^r\right)}^{-1}{\Sigma }_{\nu \nu \text{}\mathrm{,}\left(12\right)}^r{\nu }_{i09}^r\mathrm{,}{\left({\Sigma }_{\nu \nu \text{}\mathrm{,}\left(11\right)}^r\right)}^{-1}{\Sigma }_{\nu \nu \text{}\mathrm{,}\left(11\right)}^r{\Sigma }_{\nu \nu \text{}\mathrm{,}\left(22\right)}^r-{\left({\Sigma }_{\nu \nu \text{}\mathrm{,}\left(12\right)}^r\right)}^2\right)\mathrm{,}\hfill \\ {\mu }_{i09}^r\hfill & =x_{ij09}^{\prime }{\beta }_{09}^r\mathrm{,}\hfill \\ y_{ij09}^r\hfill & \sim N\left({\mu }_{i09}^r+{\nu }_{i09}^r\mathrm{,}{\Sigma }_{eeij\text{}\mathrm{,}\left(11\right)}^r\right)\hfill \\ {\mu }_{i10}^r\hfill & =x_{ij10}^{\prime }{\beta }_{10}^r\mathrm{,}\hfill \\ y_{ij10}^r\hfill & \sim N({\mu }_{i10}^r+{\nu }_{i10}^r+{\left({\Sigma }_{eeij\text{}\mathrm{,}\left(11\right)}^r\right)}^{-1}{\Sigma }_{eeij\text{}\mathrm{,}\left(12\right)}^r\left(y_{ij09}^r-{\mu }_{i09}^r-{\nu }_{i09}^r\right)\mathrm{,}\hfill \\ \hfill & {\left({\Sigma }_{eeij\text{}\mathrm{,}\left(11\right)}^r\right)}^{-1}\left({\Sigma }_{eeij\text{}\mathrm{,}\left(11\right)}^r{\Sigma }_{eeij\text{}\mathrm{,}\left(22\right)}^r-{\left({\Sigma }_{eeij\text{}\mathrm{,}\left(12\right)}^r\right)}^2\right).\hfill \end{array}$$

Although the number of posterior samples is $S\mathrm{<mo>=</mo>}$20,000, we construct $R\mathrm{<mo>=</mo>}$1,901 replicates, where $r$ is selected from the sequence 1,000 to $T$ by skipping every 10 samples.

References

Battese, G.E., Harter, R.M. and Fuller, W.A. (1988). An error-components model for prediction of county crop areas using survey and satellite data. Journal of the American Statistical Association, 83, 28-36.

Berg, E., Cecere, W. and Ghosh, M. (2014). Small area estimation for county-level farmland cash rental rates. Journal of Survey Statistics and Methodology, 2, 1-37.

Cochran, W.G. (1977). Sampling Techniques. 3^rd Edition, New York: John Wiley & Sons, Inc.

Datta, G.S., Day, B. and Maiti, T. (1998). Multivariate Bayesian small area estimation: An application to survey and satellite data. Sankhyā: The Indian Journal of Statistics, 60, 344-362.

Dhuyvetter, D., and Kastens, T. (2009). Kansas Land Values and Cash Rents at the County Level.

Dobos, R.R., Sinclair, H.R. and Robotham, M.P. (2012). User Guide for the National Commodity Crop Productivity Index (NCCPI), Version 2.0, NRCS/USDA publication.

Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2009). Bayesian Data Analysis: Second Edition, CRC Press.

Gershunskaya, J. (2010). Robust small area estimation using a mixture model. Proceedings of the Survey Research Methods Section, American Statistical Association.

Ghosh, M., and Steorts, R. (2013). Two-stage Bayesian benchmarking as applied to small area estimation. TEST, 22, 670-687.

Haff, L.R. (1980). Empirical Bayes estimation of the multivariate normal covariance matrix. Annals of Statistics, 586-597.

Kim, J.K., Park, S. and Lee, Y. (2017). Statistical inference using generalized linear mixed models under informative cluster sampling. Canadian Journal of Statistics, DOI: 10.1002/cjs.11339.

Lohr, S.L., and Prasad, N.G.N. (2003). Small area estimation with auxiliary survey data. Canadian Journal of Statistics, 31, 383-396.

Rao, J.N.K., and Molina, I. (2015). Small Area Estimation, New York: John Wiley & Sons, Inc.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling, New York: Springer-Verlag.

Woodard, S., Paulson, N., Baylis, K. and Woddard, J. (2010). Spatial analysis of Illinois agricultural cash rents. The Selected Works of Kathy Baylis. http://works.bepress.com/kathy baylis/29.

You, Y., and Rao, J.N.K. (2002). A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights. Canadian Journal of Statistics, 30, 431-439.

You, Y., Rao, J.N.K. and Dick, P. (2004). Benchmarking hierarchicial Bayes small area estimators with applications in census undercoverage estimation. Statistics in Transition, 6, 631-640.

• Previous
• Article main page
• Next