The ordering of future observations from multiple groups

Abstract

There are many situations where comparison of different groups is of great interest. Considering the ordering of the efficiency of some treatments is an example. We present nonparametric predictive inference (NPI) for the ordering of real-valued future observations from multiple independent groups. The uncertainty is quantified using NPI lower and upper probabilities for the event that the next future observations from these groups are ordered in a specific way. Several applications of these NPI lower and upper probabilities are explored, including multiple groups inference, diagnostic accuracy and ranked set sampling.

Keywords:

• Nonparametric predictive inference
• Lower and upper probabilities
• Ordering
• Multiple groups inference
• Ranked set sampling
• Diagnostic accuracy

Acknowledgements

The author would like to thank Prof. Balakrishnan for the stimulating discussions during his visit to Durham in November 2018, and acknowledges receipt of a Durham University Global Engagement Travel Grant supporting this visit. The author would like to thank the two anonymous reviewers whose suggestions helped improve this paper.

Notes

1 If the probability mass $1/(n_y+1)$ is assigned to the right-end of the intervals $(y_{j-1},y_j),j=1,\dots ,n_y+1,$ that is to the data y observations and to $y_{n_y+1}=\infty ,$ then ${\sum }_{i=1}^{n_x+1}{\sum }_{k=1}^{n_z+1}I(x_i<y_{n_y+1}<z_{k-1})=0$ and thus ${\sum }_{i=1}^{n_x+1}{\sum }_{j=1}^{n_y+1}{\sum }_{k=1}^{n_z+1}I(x_i<y_j<z_{k-1})={\sum }_{i=1}^{n_x+1}{\sum }_{j=1}^{n_y}{\sum }_{k=1}^{n_z+1}I(x_i<y_j<z_{k-1}).$ Similarly, if the probability mass $1/(n_y+1)$ is assigned to the left-end of the intervals $(y_{j-1},y_j),j=1,\dots ,n_y+1,$ that is, it will be assigned to $y_0=-\infty$ and to the data y observations, then ${\sum }_{i=1}^{n_x+1}{\sum }_{k=1}^{n_z+1}I(x_i<y_0<z_{k-1})=0$ and thus ${\sum }_{i=1}^{n_x+1}{\sum }_{j=1}^{n_y+1}{\sum }_{k=1}^{n_z+1}I(x_i<y_{j-1}<z_{k-1})={\sum }_{i=1}^{n_x+1}{\sum }_{j=1}^{n_y}{\sum }_{k=1}^{n_z+1}I(x_i<y_j<z_{k-1}).$