# Adjusted Empirical Likelihood Method in the Presence of Nuisance Parameters with Application to the Sharpe Ratio

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Empirical Likelihood and the Adjusted Empirical Likelihood Methods

## 3. The Adjusted Empirical Likelihood Method in the Presence of Nuisance Parameters

**Lemma**

**1.**

**Result**

**1.**

**Result**

**2.**

**Theorem**

**1.**

## 4. Simulation Study

## 5. Real Data Analysis

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## References

- Sharpe, W.F. Mutual fund performance. J. Bus.
**1966**, 39, 119–138. [Google Scholar] [CrossRef] - Jobson, J.D.; Korkie, B.M. Performance hypothesis testing with the sharpe and treynor measures. J. Financ.
**1981**, 36, 889–908. [Google Scholar] [CrossRef] - Fama, E. The behavior of stock-market prices. J. Bus.
**1965**, 38, 34–105. [Google Scholar] [CrossRef] - Leland, H.E. Beyond mean-variance: Risk and performance measurement in a nonsymmetric world. Financ. Anal. J.
**1999**, 1, 27–36. [Google Scholar] [CrossRef] - Agarwal, V.; Naik, N.Y. Risk and portfolio decisions involving hedge funds. Rev. Financ. Stud.
**2004**, 17, 63–98. [Google Scholar] [CrossRef] - Ingersoll, J.; Spiegel, M.; Goetzmann, W. Portfolio performance manipulation and manipulation-proof performance measures. Rev. Financ. Stud.
**2007**, 20, 1503–1546. [Google Scholar] - Samuelson, P. The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments. Rev. Econ. Stud.
**1970**, 37, 537–542. [Google Scholar] [CrossRef] - Scott, R.; Horvath, P. On the direction of preference for moments of higher order than variance. J. Financ.
**1980**, 35, 915–919. [Google Scholar] [CrossRef] - Zakamouline, V.; Koekebakker, S. Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance. J. Bank. Financ.
**2009**, 33, 1242–1254. [Google Scholar] [CrossRef] - Pierro, M.D.; Mosevich, J. Effects of skewness and kurtosis on portfolio rankings. Quant. Financ.
**2011**, 11, 1449–1453. [Google Scholar] [CrossRef] - Thomas, D.R.; Grunkemeier, G.L. Confidence interval estimation of survival probabilities for censored data. J. Am. Stat. Assoc.
**1975**, 70, 865–871. [Google Scholar] [CrossRef] - Owen, A. Empirical likelihood ratio confidence intervals for a single functional. Biometrika
**1988**, 75, 237–249. [Google Scholar] [CrossRef] - Owen, A. Empirical likelihood ratio confidence regions. Ann. Stat.
**1990**, 18, 90–120. [Google Scholar] [CrossRef] - Hall, P.; La Scala, B. Methodology and algorithms of empirical likelihood. Inter. Stat. Rev.
**1990**, 58, 109–127. [Google Scholar] [CrossRef] - Qin, J.; Lawless, J. Empirical likelihood and general estimating equations. Ann. Stat.
**1994**, 22, 300–325. [Google Scholar] [CrossRef] - Chen, J.; Variyath, A.M.; Abraham, B. Adjusted empirical likelihood and its properties. J. Comput. Graph. Stat.
**2008**, 17, 426–443. [Google Scholar] [CrossRef] - Rao, C.R. Linear Statistical Inference and Its Applications; Wiley: New York, NY, USA, 1973. [Google Scholar]
- Wilks, S.S. The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Ann. Math. Stat.
**1938**, 9, 60–62. [Google Scholar] [CrossRef] - Wang, H.J.; Zhu, Z. Empirical likelihood for quantile regression models with longitudinal data. J. Stat. Plan. Inference
**2011**, 141, 1603–1615. [Google Scholar] [CrossRef] - Chen, J.; Huang, Y. Finite-sample properties of the adjusted empirical likelihood. J. Nonparametric Stat.
**2013**, 25, 147–159. [Google Scholar] [CrossRef] - Mertens, E. Comments on the Correct Variance of Estimated Sharpe Ratios in Lo (2002, FAJ) When Returns Are IID. Research Note. Available online: http://www.elmarmertens.com/research/discussion/soprano01.pdf (accessed on 30 January 2018).
- Gallier, J. Geometric Methods and Applications for Computer Science and Engineering; Springer: New York, NY, USA, 2011. [Google Scholar]

$1-\mathit{\alpha}$ | Method | $\mathit{n}=20$ | $\mathit{n}=50$ | $\mathit{n}=200$ | $\mathit{n}=500$ |
---|---|---|---|---|---|

$N(1,0.25)$ | |||||

0.9 | JK | 0.8956 | 0.9022 | 0.8968 | 0.9060 |

Mertens | 0.8258 | 0.8694 | 0.8894 | 0.9004 | |

EL | 0.8210 | 0.8760 | 0.8906 | 0.9040 | |

AEL | 0.8486 | 0.8874 | 0.8942 | 0.9058 | |

Delta | 0.8428 | 0.8840 | 0.8896 | 0.8976 | |

0.95 | JK | 0.9460 | 0.9488 | 0.9488 | 0.9522 |

Mertens | 0.8926 | 0.9270 | 0.9414 | 0.9494 | |

EL | 0.8762 | 0.9214 | 0.9440 | 0.9514 | |

AEL | 0.8980 | 0.9312 | 0.9466 | 0.9534 | |

Delta | 0.9054 | 0.9334 | 0.9408 | 0.9476 | |

${t}_{3}$ | |||||

0.9 | JK | 0.8960 | 0.9004 | 0.9030 | 0.9040 |

Mertens | 0.8390 | 0.8646 | 0.8782 | 0.8890 | |

EL | 0.8428 | 0.8738 | 0.8884 | 0.8946 | |

AEL | 0.8794 | 0.8896 | 0.8944 | 0.8976 | |

Delta | 0.8240 | 0.8586 | 0.8766 | 0.8858 | |

0.95 | JK | 0.9494 | 0.9538 | 0.9516 | 0.9550 |

Mertens | 0.9028 | 0.9144 | 0.9326 | 0.9442 | |

EL | 0.9042 | 0.9268 | 0.9438 | 0.9508 | |

AEL | 0.9340 | 0.9402 | 0.9466 | 0.9514 | |

Delta | 0.8910 | 0.9092 | 0.9318 | 0.9372 | |

${t}_{6}$ | |||||

0.9 | JK | 0.8982 | 0.8984 | 0.8934 | 0.8976 |

Mertens | 0.8738 | 0.8840 | 0.8900 | 0.8954 | |

EL | 0.8700 | 0.8860 | 0.8928 | 0.8962 | |

AEL | 0.8986 | 0.9018 | 0.8966 | 0.8978 | |

Delta | 0.8634 | 0.8828 | 0.8936 | 0.9008 | |

0.95 | JK | 0.9504 | 0.9482 | 0.9466 | 0.9476 |

Mertens | 0.9246 | 0.9364 | 0.9426 | 0.9458 | |

EL | 0.9240 | 0.9394 | 0.9438 | 0.9466 | |

AEL | 0.9466 | 0.9470 | 0.9480 | 0.9481 | |

Delta | 0.9214 | 0.9330 | 0.9460 | 0.9494 | |

${\chi}_{4}^{2}$ | |||||

0.9 | JK | 0.9640 | 0.9532 | 0.9476 | 0.9474 |

Mertens | 0.8048 | 0.8474 | 0.8676 | 0.8938 | |

EL | 0.7800 | 0.8352 | 0.8626 | 0.8942 | |

AEL | 0.8216 | 0.8536 | 0.8664 | 0.8952 | |

Delta | 0.8072 | 0.8354 | 0.8660 | 0.8914 | |

0.95 | JK | 0.9872 | 0.9808 | 0.9808 | 0.9774 |

Mertens | 0.8780 | 0.9072 | 0.9278 | 0.9422 | |

EL | 0.8562 | 0.8972 | 0.9194 | 0.9418 | |

AEL | 0.8924 | 0.9126 | 0.9228 | 0.9430 | |

Delta | 0.8872 | 0.9046 | 0.9252 | 0.9388 | |

${\chi}_{6}^{2}$ | |||||

0.9 | JK | 0.9466 | 0.9476 | 0.9392 | 0.9414 |

Mertens | 0.8048 | 0.8460 | 0.8760 | 0.8916 | |

EL | 0.7996 | 0.8392 | 0.8728 | 0.8904 | |

AEL | 0.8346 | 0.8568 | 0.8780 | 0.8926 | |

Delta | 0.8236 | 0.8524 | 0.8766 | 0.8846 | |

0.95 | JK | 0.9796 | 0.9776 | 0.9758 | 0.9754 |

Mertens | 0.8780 | 0.9168 | 0.9352 | 0.9412 | |

EL | 0.8626 | 0.9032 | 0.9284 | 0.9402 | |

AEL | 0.8894 | 0.9156 | 0.9326 | 0.9410 | |

Delta | 0.8916 | 0.9170 | 0.9336 | 0.9432 |

$1-\mathit{\alpha}$ | Method | Estimate | Lower Bound | Upper Bound |
---|---|---|---|---|

0.9 | JK | 0.1907 | −0.0441 | 0.4254 |

Mertens | 0.1907 | −0.0350 | 0.4163 | |

Delta | 0.1926 | −0.0329 | 0.4181 | |

EL | 0.1926 | −0.0376 | 0.4140 | |

AEL | 0.1926 | −0.0479 | 0.4241 | |

0.95 | JK | 0.1907 | −0.0890 | 0.4703 |

Mertens | 0.1907 | −0.0783 | 0.4596 | |

Delta | 0.1926 | −0.0761 | 0.4613 | |

EL | 0.1926 | −0.0827 | 0.4558 | |

AEL | 0.1926 | −0.0949 | 0.4683 |

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**MDPI and ACS Style**

Fu, Y.; Wang, H.; Wong, A. Adjusted Empirical Likelihood Method in the Presence of Nuisance Parameters with Application to the Sharpe Ratio. *Entropy* **2018**, *20*, 316.
https://doi.org/10.3390/e20050316

**AMA Style**

Fu Y, Wang H, Wong A. Adjusted Empirical Likelihood Method in the Presence of Nuisance Parameters with Application to the Sharpe Ratio. *Entropy*. 2018; 20(5):316.
https://doi.org/10.3390/e20050316

**Chicago/Turabian Style**

Fu, Yuejiao, Hangjing Wang, and Augustine Wong. 2018. "Adjusted Empirical Likelihood Method in the Presence of Nuisance Parameters with Application to the Sharpe Ratio" *Entropy* 20, no. 5: 316.
https://doi.org/10.3390/e20050316