Properties and Estimations of a Multivariate Folded Normal Distribution
Abstract
:1. Introduction
2. Properties of a Multivariate Folded Normal Distribution
- (1)
- and
- (2)
- , where
3. Parameter Estimation
3.1. Simulation Studies
3.1.1. An Example of Estimating Parameters
3.1.2. Accuracy of Estimated Parameters
3.2. Real Data Application
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 2.5) | (1, 2.5) | (1.5, 2.5) | (2, 2.5) | (2.5, 2.5) |
20 | (0.72, 0.94) | (0.90, 0.94) | (0.94, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
30 | (0.73, 0.95) | (0.88, 0.95) | (0.93, 0.95) | (0.94, 0.95) | (0.93, 0.95) |
40 | (0.79, 0.94) | (0.93, 0.94) | (0.95, 0.94) | (0.93, 0.94) | (0.93, 0.94) |
50 | (0.78, 0.94) | (0.91, 0.93) | (0.96, 0.93) | (0.94, 0.93) | (0.94, 0.93) |
60 | (0.77, 0.94) | (0.90, 0.94) | (0.95, 0.94) | (0.95, 0.94) | (0.95, 0.94) |
70 | (0.77, 0.94) | (0.90, 0.94) | (0.95, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
80 | (0.78, 0.94) | (0.91, 0.94) | (0.96, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
90 | (0.78, 0.96) | (0.93, 0.96) | (0.96, 0.96) | (0.96, 0.96) | (0.95, 0.96) |
100 | (0.78, 0.95) | (0.93, 0.95) | (0.95, 0.95) | (0.94, 0.95) | (0.94, 0.95) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 2.5) | (1, 2.5) | (1.5, 2.5) | (2, 2.5) | (2.5, 2.5) |
20 | (0.74, 0.84, 0.87) | (0.83, 0.91, 0.87) | (0.86, 0.94, 0.87) | (0.87, 0.94, 0.87) | (0.88, 0.94, 0.87) |
30 | (0.76, 0.86, 0.88) | (0.83, 0.90, 0.88) | (0.87, 0.94, 0.88) | (0.89, 0.96, 0.88) | (0.90, 0.96, 0.88) |
40 | (0.80, 0.87, 0.91) | (0.84, 0.90, 0.91) | (0.90, 0.93, 0.91) | (0.91, 0.94, 0.91) | (0.91, 0.94, 0.91) |
50 | (0.81, 0.87, 0.92) | (0.85, 0.93, 0.92) | (0.91, 0.96, 0.92) | (0.93, 0.96, 0.92) | (0.93, 0.96, 0.92) |
60 | (0.81, 0.86, 0.92) | (0.84, 0.92, 0.92) | (0.89, 0.94, 0.92) | (0.91, 0.95, 0.92) | (0.91, 0.95, 0.92) |
70 | (0.81, 0.85, 0.91) | (0.84, 0.91, 0.91) | (0.91, 0.95, 0.91) | (0.92, 0.95, 0.91) | (0.92, 0.96, 0.91) |
80 | (0.82, 0.87, 0.92) | (0.86, 0.92, 0.92) | (0.93, 0.94, 0.92) | (0.94, 0.94, 0.92) | (0.92, 0.94, 0.92) |
90 | (0.82, 0.86, 0.93) | (0.87, 0.92, 0.93) | (0.93, 0.96, 0.93) | (0.94, 0.95, 0.93) | (0.94, 0.95, 0.93) |
100 | (0.82, 0.87, 0.93) | (0.86, 0.93, 0.93) | (0.93, 0.96, 0.93) | (0.94, 0.96, 0.93) | (0.93, 0.96, 0.93) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.66, 0.70) | (0.89, 0.90) | (0.94, 0.94) | (0.94, 0.95) | (0.94, 0.95) |
30 | (0.66, 0.66) | (0.87, 0.89) | (0.94, 0.95) | (0.93, 0.94) | (0.93, 0.94) |
40 | (0.71, 0.72) | (0.91, 0.90) | (0.96, 0.95) | (0.94, 0.94) | (0.93, 0.94) |
50 | (0.71, 0.69) | (0.89, 0.90) | (0.95, 0.95) | (0.93, 0.94) | (0.93, 0.94) |
60 | (0.67, 0.73) | (0.90, 0.92) | (0.96, 0.95) | (0.95, 0.94) | (0.94, 0.94) |
70 | (0.70, 0.72) | (0.89, 0.91) | (0.95, 0.95) | (0.95, 0.95) | (0.95, 0.94) |
80 | (0.71, 0.74) | (0.91, 0.91) | (0.95, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
90 | (0.72, 0.73) | (0.92, 0.90) | (0.97, 0.96) | (0.96, 0.96) | (0.96, 0.96) |
100 | (0.71, 0.72) | (0.90, 0.92) | (0.94, 0.96) | (0.94, 0.96) | (0.93, 0.95) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.70, 0.78, 0.70) | (0.81, 0.87, 0.81) | (0.86, 0.93, 0.85) | (0.87, 0.95, 0.87) | (0.87, 0.95, 0.86) |
30 | (0.72, 0.77, 0.70) | (0.82, 0.86, 0.82) | (0.88, 0.94, 0.87) | (0.89, 0.96, 0.88) | (0.89, 0.95, 0.89) |
40 | (0.77, 0.80, 0.76) | (0.82, 0.89, 0.84) | (0.88, 0.95, 0.89) | (0.90, 0.95, 0.90) | (0.91, 0.96, 0.90) |
50 | (0.78, 0.77, 0.76) | (0.84, 0.91, 0.85) | (0.90, 0.96, 0.91) | (0.92, 0.96, 0.92) | (0.92, 0.95, 0.92) |
60 | (0.75, 0.77, 0.78) | (0.85, 0.90, 0.87) | (0.90, 0.94, 0.91) | (0.92, 0.95, 0.92) | (0.92, 0.95, 0.92) |
70 | (0.78, 0.79, 0.79) | (0.84, 0.88, 0.84) | (0.92, 0.94, 0.90) | (0.93, 0.94, 0.91) | (0.94, 0.94, 0.91) |
80 | (0.80, 0.79, 0.80) | (0.87, 0.91, 0.86) | (0.93, 0.94, 0.91) | (0.94, 0.95, 0.90) | (0.94, 0.95, 0.91) |
90 | (0.79, 0.79, 0.78) | (0.86, 0.90, 0.87) | (0.93, 0.95, 0.93) | (0.93, 0.95, 0.93) | (0.92, 0.94, 0.94) |
100 | (0.78, 0.78, 0.78) | (0.86, 0.92, 0.87) | (0.93, 0.97, 0.93) | (0.93, 0.97, 0.93) | (0.93, 0.97, 0.93) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.73, 0.74) | (0.91, 0.91) | (0.94, 0.94) | (0.95, 0.95) | (0.95, 0.94) |
30 | (0.71, 0.71) | (0.91, 0.91) | (0.94, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
40 | (0.74, 0.73) | (0.92, 0.92) | (0.94, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
50 | (0.71, 0.72) | (0.92, 0.92) | (0.94, 0.94) | (0.93, 0.94) | (0.93, 0.93) |
60 | (0.76, 0.76) | (0.91, 0.91) | (0.95, 0.95) | (0.94, 0.94) | (0.94, 0.94) |
70 | (0.73, 0.73) | (0.92, 0.92) | (0.94, 0.94) | (0.94, 0.94) | (0.94, 0.94) |
80 | (0.76, 0.76) | (0.93, 0.92) | (0.95, 0.95) | (0.95, 0.95) | (0.95, 0.95) |
90 | (0.75, 0.76) | (0.92, 0.92) | (0.96, 0.96) | (0.95, 0.96) | (0.96, 0.96) |
100 | (0.71, 0.71) | (0.92, 0.92) | (0.95, 0.95) | (0.94, 0.94) | (0.95, 0.94) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.58, 0.58, 0.59) | (0.74, 0.75, 0.74) | (0.82, 0.82, 0.82) | (0.84, 0.84, 0.84) | (0.85, 0.84, 0.84) |
30 | (0.62, 0.61, 0.61) | (0.78, 0.79, 0.78) | (0.85, 0.85, 0.85) | (0.88, 0.87, 0.87) | (0.87, 0.88, 0.88) |
40 | (0.69, 0.69, 0.69) | (0.84, 0.83, 0.83) | (0.88, 0.88, 0.88) | (0.89, 0.89, 0.89) | (0.89, 0.89, 0.89) |
50 | (0.69, 0.69, 0.70) | (0.84, 0.84, 0.84) | (0.90, 0.90, 0.90) | (0.92, 0.92, 0.92) | (0.93, 0.92, 0.92) |
60 | (0.72, 0.73, 0.73) | (0.86, 0.86, 0.85) | (0.89, 0.90, 0.90) | (0.91, 0.91, 0.91) | (0.92, 0.92, 0.92) |
70 | (0.72, 0.72, 0.72) | (0.85, 0.86, 0.85) | (0.88, 0.88, 0.88) | (0.90, 0.90, 0.90) | (0.90, 0.91, 0.91) |
80 | (0.72, 0.72, 0.72) | (0.86, 0.86, 0.86) | (0.89, 0.90, 0.89) | (0.90, 0.90, 0.90) | (0.90, 0.90, 0.90) |
90 | (0.73, 0.73, 0.73) | (0.85, 0.86, 0.86) | (0.91, 0.91, 0.91) | (0.92, 0.92, 0.93) | (0.94, 0.94, 0.94) |
100 | (0.74, 0.74, 0.73) | (0.87, 0.87, 0.86) | (0.92, 0.92, 0.93) | (0.94, 0.94, 0.94) | (0.95, 0.94, 0.94) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 2.5) | (1, 1, 1, 2.5) | (1.5, 1.5, 1.5, 2.5) | (2, 2, 2, 2.5) | (2.5, 2.5, 2.5, 2.5) |
20 | (0.70, 0.71, 0.70, 0.93) | (0.82, 0.84, 0.84, 0.94) | (0.90, 0.93, 0.91, 0.94) | (0.92, 0.94, 0.93, 0.94) | (0.93, 0.94, 0.93, 0.93) |
30 | (0.65, 0.68, 0.66, 0.94) | (0.82, 0.84, 0.84, 0.94) | (0.93, 0.92, 0.93, 0.94) | (0.94, 0.93, 0.93, 0.94) | (0.94, 0.93, 0.93, 0.94) |
40 | (0.67, 0.67, 0.70, 0.94) | (0.85, 0.82, 0.84, 0.94) | (0.93, 0.93, 0.93, 0.94) | (0.94, 0.93, 0.93, 0.94) | (0.94, 0.94, 0.92, 0.94) |
50 | (0.67, 0.68, 0.70, 0.94) | (0.85, 0.84, 0.85, 0.94) | (0.94, 0.94, 0.94, 0.94) | (0.95, 0.94, 0.94, 0.94) | (0.95, 0.94, 0.94, 0.94) |
60 | (0.68, 0.69, 0.67, 0.95) | (0.86, 0.86, 0.86, 0.95) | (0.94, 0.94, 0.94, 0.95) | (0.94, 0.94, 0.94, 0.95) | (0.94, 0.94, 0.94, 0.95) |
70 | (0.70, 0.68, 0.69, 0.95) | (0.88, 0.87, 0.85, 0.95) | (0.94, 0.96, 0.94, 0.95) | (0.94, 0.96, 0.93, 0.95) | (0.93, 0.96, 0.94, 0.95) |
80 | (0.72, 0.68, 0.69, 0.95) | (0.88, 0.86, 0.89, 0.95) | (0.95, 0.95, 0.95, 0.95) | (0.95, 0.94, 0.94, 0.95) | (0.94, 0.95, 0.94, 0.95) |
90 | (0.70, 0.70, 0.72, 0.95) | (0.89, 0.87, 0.88, 0.95) | (0.95, 0.95, 0.96, 0.95) | (0.95, 0.95, 0.96, 0.95) | (0.95, 0.95, 0.95, 0.95) |
100 | (0.70, 0.70, 0.70, 0.96) | (0.90, 0.90, 0.88, 0.96) | (0.95, 0.96, 0.93, 0.96) | (0.95, 0.95, 0.94, 0.96) | (0.95, 0.95, 0.94, 0.96) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 2.5) | (1, 1, 1, 2.5) | (1.5, 1.5, 1.5, 2.5) | (2, 2, 2, 2.5) | (2.5, 2.5, 2.5, 2.5) |
20 | |||||
50 | |||||
100 |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | (0.68, 0.68, 0.68, 0.71) | (0.82, 0.82, 0.83, 0.83) | (0.91, 0.91, 0.90, 0.92) | (0.93, 0.93, 0.93, 0.93) | (0.93, 0.94, 0.93, 0.93) |
30 | (0.64, 0.60, 0.64, 0.67) | (0.84, 0.81, 0.82, 0.83) | (0.92, 0.91, 0.93, 0.92) | (0.94, 0.94, 0.94, 0.94) | (0.93, 0.93, 0.93, 0.94) |
40 | (0.62, 0.61, 0.63, 0.64) | (0.84, 0.82, 0.83, 0.82) | (0.94, 0.93, 0.94, 0.93) | (0.95, 0.93, 0.95, 0.93) | (0.95, 0.93, 0.95, 0.93) |
50 | (0.60, 0.63, 0.65, 0.64) | (0.82, 0.82, 0.83, 0.84) | (0.93, 0.94, 0.94, 0.94) | (0.93, 0.94, 0.95, 0.94) | (0.93, 0.94, 0.95, 0.94) |
60 | (0.62, 0.65, 0.62, 0.63) | (0.85, 0.85, 0.82, 0.86) | (0.94, 0.94, 0.93, 0.95) | (0.95, 0.94, 0.94, 0.95) | (0.94, 0.94, 0.93, 0.94) |
70 | (0.60, 0.59, 0.59, 0.63) | (0.84, 0.83, 0.85, 0.85) | (0.95, 0.94, 0.94, 0.95) | (0.96, 0.94, 0.93, 0.96) | (0.96, 0.94, 0.93, 0.96) |
80 | (0.62, 0.64, 0.60, 0.64) | (0.84, 0.86, 0.86, 0.86) | (0.95, 0.95, 0.96, 0.96) | (0.95, 0.94, 0.95, 0.94) | (0.95, 0.94, 0.94, 0.94) |
90 | (0.63, 0.59, 0.59, 0.65) | (0.88, 0.85, 0.88, 0.87) | (0.95, 0.95, 0.95, 0.95) | (0.95, 0.95, 0.95, 0.95) | (0.95, 0.94, 0.95, 0.95) |
100 | (0.63, 0.65, 0.63, 0.64) | (0.87, 0.88, 0.89, 0.87) | (0.95, 0.95, 0.95, 0.95) | (0.95, 0.94, 0.95, 0.94) | (0.95, 0.94, 0.95, 0.94) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | |||||
50 | |||||
100 |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | (0.84, 0.84, 0.83, 0.84) | (0.88, 0.87, 0.88, 0.87) | (0.90, 0.90, 0.90, 0.90) | (0.92, 0.92, 0.92, 0.92) | (0.93, 0.93, 0.93, 0.92) |
30 | (0.79, 0.79, 0.78, 0.79) | (0.89, 0.89, 0.89, 0.89) | (0.92, 0.92, 0.92, 0.92) | (0.93, 0.93, 0.93, 0.93) | (0.93, 0.93, 0.93, 0.93) |
40 | (0.78, 0.77, 0.78, 0.78) | (0.87, 0.86, 0.86, 0.87) | (0.93, 0.92, 0.93, 0.92) | (0.94, 0.93, 0.93, 0.93) | (0.94, 0.93, 0.94, 0.93) |
50 | (0.74, 0.72, 0.74, 0.73) | (0.88, 0.88, 0.88, 0.88) | (0.93, 0.93, 0.93, 0.93) | (0.93, 0.93, 0.94, 0.94) | (0.93, 0.93, 0.94, 0.94) |
60 | (0.73, 0.72, 0.73, 0.72) | (0.90, 0.89, 0.89, 0.90) | (0.93, 0.93, 0.93, 0.93) | (0.94, 0.94, 0.94, 0.94) | (0.94, 0.94, 0.94, 0.94) |
70 | (0.72, 0.72, 0.72, 0.72) | (0.89, 0.89, 0.89, 0.89) | (0.95, 0.94, 0.95, 0.95) | (0.95, 0.95, 0.95, 0.95) | (0.96, 0.96, 0.95, 0.96) |
80 | (0.73, 0.73, 0.74, 0.73) | (0.89, 0.89, 0.88, 0.89) | (0.94, 0.94, 0.94, 0.94) | (0.94, 0.94, 0.94, 0.94) | (0.94, 0.95, 0.94, 0.94) |
90 | (0.74, 0.74, 0.74, 0.74) | (0.91, 0.90, 0.90, 0.91) | (0.94, 0.94, 0.94, 0.94) | (0.94, 0.95, 0.94, 0.94) | (0.95, 0.95, 0.95, 0.95) |
100 | (0.70, 0.70, 0.69, 0.70) | (0.88, 0.88, 0.88, 0.88) | (0.93, 0.93, 0.93, 0.93) | (0.95, 0.94, 0.94, 0.95) | (0.95, 0.95, 0.95, 0.95) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | |||||
50 | |||||
100 |
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Scenario | Values of | Value of | Values of | Table Index of Accuracy of Estimated Mean and Variance Parameters | Noted Changes (Comparing with Baseline) |
---|---|---|---|---|---|
1 | , , | , | Table 2 and Table 3 | Baseline | |
2 | Table A1 and Table A2 | ||||
3 | Table A3 and Table A4 | ||||
4 | Table A5 and Table A6 |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.68, 0.67) | (0.90, 0.88) | (0.94, 0.93) | (0.94, 0.94) | (0.94, 0.94) |
30 | (0.67, 0.67) | (0.87, 0.89) | (0.93, 0.94) | (0.94, 0.95) | (0.93, 0.94) |
40 | (0.71, 0.71) | (0.92, 0.89) | (0.95, 0.94) | (0.93, 0.94) | (0.94, 0.93) |
50 | (0.69, 0.71) | (0.91, 0.90) | (0.96, 0.94) | (0.94, 0.94) | (0.94, 0.93) |
60 | (0.70, 0.68) | (0.90, 0.92) | (0.94, 0.95) | (0.95, 0.94) | (0.95, 0.94) |
70 | (0.73, 0.72) | (0.90, 0.91) | (0.95, 0.95) | (0.94, 0.94) | (0.94, 0.94) |
80 | (0.74, 0.71) | (0.91, 0.90) | (0.96, 0.95) | (0.94, 0.94) | (0.94, 0.94) |
90 | (0.72, 0.73) | (0.92, 0.91) | (0.96, 0.96) | (0.96, 0.96) | (0.95, 0.96) |
100 | (0.70, 0.69) | (0.92, 0.90) | (0.96, 0.96) | (0.94, 0.95) | (0.94, 0.95) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5) | (1, 1) | (1.5, 1.5) | (2, 2) | (2.5, 2.5) |
20 | (0.71, 0.75, 0.69) | (0.81, 0.86, 0.80) | (0.86, 0.93, 0.86) | (0.87, 0.94, 0.88) | (0.88, 0.94, 0.87) |
30 | (0.72, 0.79, 0.70) | (0.83, 0.88, 0.82) | (0.87, 0.94, 0.87) | (0.89, 0.95, 0.89) | (0.91, 0.94, 0.91) |
40 | (0.76, 0.81, 0.76) | (0.84, 0.91, 0.84) | (0.90, 0.94, 0.89) | (0.91, 0.94, 0.91) | (0.93, 0.96, 0.92) |
50 | (0.77, 0.78, 0.78) | (0.85, 0.91, 0.85) | (0.90, 0.95, 0.91) | (0.93, 0.97, 0.92) | (0.94, 0.98, 0.97) |
60 | (0.77, 0.80, 0.77) | (0.83, 0.90, 0.84) | (0.89, 0.95, 0.91) | (0.91, 0.94, 0.92) | (0.91, 0.95, 0.92) |
70 | (0.78, 0.80, 0.77) | (0.84, 0.90, 0.85) | (0.92, 0.94, 0.91) | (0.92, 0.96, 0.91) | (0.92, 0.96, 0.91) |
80 | (0.77, 0.79, 0.76) | (0.86, 0.90, 0.84) | (0.93, 0.94, 0.91) | (0.94, 0.94, 0.92) | (0.92, 0.94, 0.92) |
90 | (0.79, 0.78, 0.78) | (0.87, 0.91, 0.86) | (0.93, 0.96, 0.93) | (0.94, 0.95, 0.93) | (0.94, 0.95, 0.93) |
100 | (0.79, 0.80, 0.74) | (0.86, 0.92, 0.85) | (0.93, 0.96, 0.93) | (0.94, 0.96, 0.93) | (0.93, 0.96, 0.93) |
Scenario | Values of | Value of | Values of | Table Index of Accuracy of Estimated Mean and Variance Parameters | Noted Settings (Compared with Baseline) | Accuracy of Estimated Mean and Variance Parameters (Compared with Baseline) |
---|---|---|---|---|---|---|
1 | , , | , | Table 2 and Table 3 | Baseline | increases for mean and variance if sample size increases or increases | |
2 | Table A1 and Table A2 | increases for and when | ||||
3 | Table A3 and Table A4 | almost equal to baseline | ||||
4 | Table A5 and Table A6 | almost equal to baseline for mean, decreases for variance |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | (0.66, 0.68, 0.66, 0.67) | (0.81, 0.82, 0.83, 0.83) | (0.90, 0.92, 0.90, 0.91) | (0.92, 0.94, 0.93, 0.93) | (0.93, 0.94, 0.93, 0.93) |
30 | (0.64, 0.64, 0.63, 0.66) | (0.80, 0.84, 0.83, 0.82) | (0.94, 0.92, 0.92, 0.92) | (0.94, 0.93, 0.93, 0.93) | (0.94, 0.93, 0.93, 0.94) |
40 | (0.61, 0.63, 0.63, 0.64) | (0.82, 0.80, 0.82, 0.86) | (0.93, 0.93, 0.92, 0.94) | (0.94, 0.94, 0.93, 0.95) | (0.94, 0.94, 0.92, 0.94) |
50 | (0.64, 0.61, 0.62, 0.62) | (0.83, 0.84, 0.82, 0.83) | (0.94, 0.94, 0.93, 0.94) | (0.95, 0.94, 0.94, 0.95) | (0.95, 0.94, 0.94, 0.94) |
60 | (0.64, 0.63, 0.63, 0.62) | (0.85, 0.84, 0.83, 0.84) | (0.94, 0.95, 0.94, 0.95) | (0.94, 0.94, 0.94, 0.95) | (0.94, 0.94, 0.94, 0.95) |
70 | (0.62, 0.61, 0.60, 0.60) | (0.85, 0.84, 0.82, 0.85) | (0.94, 0.96, 0.94, 0.95) | (0.94, 0.96, 0.93, 0.95) | (0.93, 0.96, 0.94, 0.95) |
80 | (0.62, 0.63, 0.63, 0.59) | (0.86, 0.86, 0.86, 0.85) | (0.96, 0.95, 0.95, 0.95) | (0.95, 0.94, 0.94, 0.94) | (0.94, 0.95, 0.94, 0.95) |
90 | (0.64, 0.63, 0.60, 0.61) | (0.87, 0.86, 0.86, 0.89) | (0.96, 0.95, 0.96, 0.95) | (0.95, 0.95, 0.96, 0.95) | (0.95, 0.95, 0.95, 0.95) |
100 | (0.63, 0.62, 0.61, 0.60) | (0.89, 0.89, 0.87, 0.87) | (0.95, 0.96, 0.93, 0.96) | (0.95, 0.95, 0.94, 0.96) | (0.95, 0.95, 0.94, 0.96) |
Values | of | () | |||
---|---|---|---|---|---|
Sample Size | (0.5, 0.5, 0.5, 0.5) | (1, 1, 1, 1) | (1.5, 1.5, 1.5, 1.5) | (2, 2, 2, 2) | (2.5, 2.5, 2.5, 2.5) |
20 | |||||
50 | |||||
100 |
Scenario | Values of | Value of | Values of | Table Index of Accuracy of Estimated Mean and Variance Parameters | Noted Changes (Comparing with Baseline) | Accuracy of Estimated Mean and Variance Parameters (Comparing with Baseline) |
---|---|---|---|---|---|---|
5 | Table 5 and Table 6 | Baseline | increases for mean and variance if Sample size increases or increases | |||
6 | Table A7 and Table A8 | increases for and when | ||||
7 | Table A9 and Table A10 | almost equal to baseline | ||||
8 | Table A11 and Table A12 | almost equal to baseline for mean, decreases for variance |
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Liu, X.; Jin, Y.; Yang, Y.; Pan, X. Properties and Estimations of a Multivariate Folded Normal Distribution. Mathematics 2023, 11, 4860. https://doi.org/10.3390/math11234860
Liu X, Jin Y, Yang Y, Pan X. Properties and Estimations of a Multivariate Folded Normal Distribution. Mathematics. 2023; 11(23):4860. https://doi.org/10.3390/math11234860
Chicago/Turabian StyleLiu, Xi, Yiqiao Jin, Yifan Yang, and Xiaoqing Pan. 2023. "Properties and Estimations of a Multivariate Folded Normal Distribution" Mathematics 11, no. 23: 4860. https://doi.org/10.3390/math11234860
APA StyleLiu, X., Jin, Y., Yang, Y., & Pan, X. (2023). Properties and Estimations of a Multivariate Folded Normal Distribution. Mathematics, 11(23), 4860. https://doi.org/10.3390/math11234860