**6. Concluding Remarks**

This paper first develops the relationship between the first- and second-order SD with the Omega ratio dominance. We then illustrate the applicability of the theory developed in this paper to examine the relationship between property size and property investment in the Hong Kong real estate market and draw the conclusion that the Hong Kong real estate market is inefficient, and there are expected arbitrage opportunities and anomalies in the Hong Kong real estate market. Our findings are useful for real estate investors and policy makers in real estate for their policy making to make the real estate market become efficient.

We note that the stochastic dominance tests have been well developed by now. For example, one could apply the SD tests developed by Bai et al. (2015) to examine whether there is any FSD, SSD or SRSD relationship between any two prospects. Then, one could apply the theory developed in this paper to draw inference on the preference of the corresponding Omega ratios under different conditions and for different types of investors, including risk averters, risk seekers and investors with increasing utility functions. We note that recently, Hoang et al. (2015) hypothesized that the preference of the Omega ratios implies the preference of the corresponding assets for risk averters or risk seekers. We note that this is not so straight-forward, and this is another good direction of further study in this area (Guo and Wong (2017)). Another direction of related research is to extend Niu et al. (2016, 2017) and others to develop risk measures with a different order of stochastic dominance. This could further be used to examine whether the market is efficient and whether there is arbitrage opportunity in the market.

**Acknowledgments:** The authors are grateful to the Editor and two anonymous referees for constructive comments and suggestions that led to a significant improvement of an early manuscript. The third author would like to thank Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. Xu Guo's work is partially supported by the China Postdoctoral Science Foundation (2017M610058), the National Natural Science Foundation of China (No. 11601227 and No. 11626130) and the Natural Science Foundation of Jiangsu Province, China (No. BK20150732). Xuejun Jiang's work is partially supported by the National Natural Science Foundation of China (No. 11101432) and the Natural Science Foundation of Guangdong Province, China (No. 2016A030313856). Wing-Keung Wong's work is partially supported by grants from Asia University, Hang Seng Management College, Lingnan University, Ministry of Science and Technology (MOST), Taiwan, and the Research Grants Council (RGC) of Hong Kong.

**Author Contributions:** Xu Guo presents the basic ideas and obtains the main results in Section 3; Xuejun Jiang conducts the Illustration section; Wing-keung Wong writes the Section 4 and also the whole paper.

**Conflicts of Interest:** The authors declare no conflicts of interests.
