Inference for the Sharpe Ratio Using a Likelihood-Based Approach

Table 8

Results for Simulation Study.

CI	𝑛,𝑚	Method	Lower error	Upper error	Central coverage
		MLE	0.0616	0.0704	0.8680
		Likelihood ratio	0.0619	0.0712	0.8669
	𝑛=𝑚=12	Proposed	0.0484	0.0539	0.8977
		Nominal	0.0500	0.0500	0.9000
		Standard error	0.0022	0.0022	0.0030
		MLE	0.0645	0.0624	0.8731
		Likelihood ratio	0.0651	0.0628	0.8721
90%	𝑛=12,𝑚=20	Proposed	0.0501	0.0498	0.9001
		Nominal	0.0500	0.0500	0.9000
		Standard error	0.0022	0.0022	0.0030
		MLE	0.0550	0.0668	0.8782
		Likelihood ratio	0.0550	0.0681	0.8769
	𝑛=20,𝑚=12	Proposed	0.0508	0.0510	0.8982
		Nominal	0.0500	0.0500	0.9000
		Standard error	0.0022	0.0022	0.0030
		MLE	0.0328	0.0381	0.9291
		Likelihood ratio	0.0332	0.0393	0.9275
	𝑛=𝑚=12	Proposed	0.0235	0.0272	0.9493
		Nominal	0.0250	0.0250	0.9500
		Standard error	0.0016	0.0016	0.0022
		MLE	0.0331	0.0340	0.9329
		Likelihood ratio	0.0339	0.0345	0.9316
95%	𝑛=12,𝑚=20	Proposed	0.0254	0.0263	0.9483
		Nominal	0.0250	0.0250	0.9500
		Standard error	0.0016	0.0016	0.0022
		MLE	0.0283	0.0360	0.9357
		Likelihood ratio	0.0284	0.0381	0.9335
	𝑛=20,𝑚=12	Proposed	0.0250	0.0259	0.9491
		Nominal	0.0250	0.0250	0.9500
		Standard error	0.0016	0.0016	0.0022
		MLE	0.0085	0.0093	0.9822
		Likelihood ratio	0.0094	0.0099	0.9807
	𝑛=𝑚=12	Proposed	0.0043	0.0061	0.9896
		Nominal	0.0050	0.0050	0.9900
		Standard error	0.0007	0.0007	0.0010
		MLE	0.0070	0.0083	0.9847
		Likelihood ratio	0.0081	0.0085	0.9834
99%	𝑛=12,𝑚=20	Proposed	0.0052	0.0046	0.9902
		Nominal	0.0050	0.0050	0.9900
		Standard error	0.0007	0.0007	0.0010
		MLE	0.0066	0.0078	0.9856
		Likelihood ratio	0.0066	0.0089	0.9845
	𝑛=20,𝑚=12	Proposed	0.0053	0.0057	0.9890
		Nominal	0.0050	0.0050	0.9900
		Standard error	0.0007	0.0007	0.0010