A Bimodal Model Based on Truncation Positive Normal with Application to Height Data
Abstract
:1. Introduction
2. A Bimodal Truncation Positive Normal Distribution
2.1. Stochastic Representation, pdf and cdf
2.2. Moments and Moment-Generating Function
2.3. Mode and Unimodality and Bimodality Regions
2.4. Particular Cases
- btpn tpn;
- btpn N, i.e., the normal distribution with mean 0 and variance ;
- btpn esn, i.e., the epsilon skew-normal distribution (Mudholkar and Hutson [10]).
3. Inference
3.1. Maximum Likelihood Function
3.2. Computational Aspects
- est.btpn(y)
4. Simulation Study
5. Application
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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True Value | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
par. | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | ||
−0.75 | −0.5 | 0.894 | 3.271 | 8.403 | 0.814 | 0.170 | 1.089 | 3.165 | 0.856 | 0.035 | 0.514 | 0.584 | 0.898 | |
−0.831 | 3.405 | 8.561 | 0.862 | −0.132 | 1.237 | 3.050 | 0.891 | −0.028 | 0.638 | 0.691 | 0.917 | |||
−0.018 | 0.112 | 0.121 | 0.940 | −0.008 | 0.078 | 0.081 | 0.945 | −0.003 | 0.055 | 0.057 | 0.947 | |||
0.75 | 0.736 | 2.490 | 7.381 | 0.802 | 0.233 | 1.218 | 3.497 | 0.862 | 0.041 | 0.538 | 0.630 | 0.901 | ||
−0.663 | 2.663 | 7.325 | 0.855 | −0.205 | 1.368 | 3.453 | 0.897 | −0.029 | 0.660 | 0.728 | 0.924 | |||
0.030 | 0.142 | 0.167 | 0.936 | 0.014 | 0.099 | 0.106 | 0.945 | 0.009 | 0.070 | 0.073 | 0.947 | |||
1 | −0.5 | −0.057 | 0.350 | 0.360 | 0.887 | −0.026 | 0.248 | 0.250 | 0.914 | −0.012 | 0.175 | 0.176 | 0.936 | |
0.073 | 0.406 | 0.416 | 0.938 | 0.035 | 0.285 | 0.289 | 0.946 | 0.014 | 0.201 | 0.201 | 0.949 | |||
−0.014 | 0.086 | 0.095 | 0.932 | −0.008 | 0.061 | 0.064 | 0.940 | −0.003 | 0.043 | 0.045 | 0.944 | |||
0.75 | −0.055 | 0.350 | 0.355 | 0.884 | −0.024 | 0.248 | 0.251 | 0.919 | −0.015 | 0.174 | 0.176 | 0.931 | ||
0.072 | 0.406 | 0.412 | 0.935 | 0.035 | 0.285 | 0.288 | 0.943 | 0.020 | 0.200 | 0.203 | 0.942 | |||
0.028 | 0.109 | 0.127 | 0.940 | 0.012 | 0.076 | 0.082 | 0.942 | 0.006 | 0.054 | 0.055 | 0.943 | |||
3 | −0.5 | −0.049 | 0.203 | 0.214 | 0.919 | −0.022 | 0.145 | 0.152 | 0.930 | −0.013 | 0.103 | 0.106 | 0.936 | |
0.103 | 0.356 | 0.383 | 0.948 | 0.047 | 0.248 | 0.262 | 0.947 | 0.028 | 0.174 | 0.180 | 0.948 | |||
−0.007 | 0.051 | 0.054 | 0.937 | −0.003 | 0.036 | 0.036 | 0.949 | −0.001 | 0.025 | 0.026 | 0.950 | |||
0.75 | −0.048 | 0.203 | 0.214 | 0.917 | −0.023 | 0.145 | 0.148 | 0.932 | −0.012 | 0.103 | 0.103 | 0.946 | ||
0.105 | 0.356 | 0.385 | 0.946 | 0.046 | 0.248 | 0.253 | 0.952 | 0.024 | 0.174 | 0.175 | 0.953 | |||
0.011 | 0.064 | 0.072 | 0.933 | 0.005 | 0.045 | 0.048 | 0.939 | 0.002 | 0.032 | 0.033 | 0.942 |
True Value | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
par. | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | ||
−0.75 | −0.5 | 2.279 | 10.597 | 19.571 | 0.802 | 0.961 | 5.254 | 11.695 | 0.864 | 0.221 | 2.538 | 3.411 | 0.900 | |
−0.400 | 2.336 | 4.023 | 0.857 | −0.161 | 1.209 | 2.328 | 0.896 | −0.034 | 0.632 | 0.780 | 0.924 | |||
−0.020 | 0.112 | 0.126 | 0.938 | −0.009 | 0.079 | 0.083 | 0.949 | −0.004 | 0.055 | 0.056 | 0.950 | |||
0.75 | 1.748 | 9.147 | 17.694 | 0.797 | 0.608 | 5.063 | 7.312 | 0.859 | 0.157 | 2.496 | 3.225 | 0.891 | ||
−0.272 | 2.046 | 3.498 | 0.856 | −0.093 | 1.172 | 1.548 | 0.895 | −0.021 | 0.623 | 0.734 | 0.920 | |||
0.038 | 0.143 | 0.177 | 0.933 | 0.013 | 0.099 | 0.103 | 0.951 | 0.007 | 0.069 | 0.072 | 0.945 | |||
1 | −0.5 | −0.225 | 1.772 | 1.829 | 0.887 | −0.126 | 1.237 | 1.260 | 0.916 | −0.057 | 0.876 | 0.891 | 0.932 | |
0.065 | 0.408 | 0.415 | 0.938 | 0.037 | 0.285 | 0.289 | 0.940 | 0.015 | 0.201 | 0.205 | 0.943 | |||
−0.014 | 0.086 | 0.095 | 0.936 | −0.005 | 0.060 | 0.063 | 0.943 | −0.002 | 0.043 | 0.043 | 0.952 | |||
0.75 | −0.284 | 1.744 | 1.756 | 0.886 | −0.136 | 1.236 | 1.255 | 0.913 | −0.059 | 0.875 | 0.869 | 0.933 | ||
0.073 | 0.405 | 0.405 | 0.941 | 0.037 | 0.285 | 0.288 | 0.943 | 0.016 | 0.201 | 0.201 | 0.946 | |||
0.028 | 0.109 | 0.131 | 0.926 | 0.012 | 0.076 | 0.081 | 0.945 | 0.008 | 0.054 | 0.056 | 0.946 | |||
3 | −0.5 | 1.985 | 2.784 | 30.165 | 0.913 | 1.699 | 1.824 | 26.982 | 0.928 | 1.113 | 0.799 | 22.887 | 0.932 | |
0.023 | 0.420 | 1.320 | 0.941 | −0.032 | 0.287 | 1.178 | 0.947 | −0.025 | 0.184 | 0.955 | 0.941 | |||
−0.004 | 0.051 | 0.054 | 0.934 | −0.002 | 0.036 | 0.037 | 0.944 | −0.001 | 0.026 | 0.026 | 0.948 | |||
0.75 | 0.386 | 4.618 | 13.399 | 0.918 | 0.596 | 1.816 | 14.121 | 0.924 | 0.355 | 0.929 | 10.561 | 0.943 | ||
0.071 | 0.474 | 0.692 | 0.954 | 0.016 | 0.285 | 0.680 | 0.941 | 0.003 | 0.188 | 0.503 | 0.952 | |||
0.019 | 0.065 | 0.076 | 0.925 | 0.015 | 0.046 | 0.051 | 0.935 | 0.008 | 0.033 | 0.034 | 0.941 |
Data Set | n | ||||
---|---|---|---|---|---|
weight measured | 126 | 0 | 1 |
Estimated | btpn | esig | asn |
---|---|---|---|
0.813 (0.113) | 1.304 (0.148) | 0.996 (0.063) | |
0.496 (0.316) | 0.527 (0.073) | 0.014 (3.422) | |
−0.002 (0.048) | 0.095 (0.059) | 0.014 (3.409) | |
AIC | 360.76 | 415.79 | 362.67 |
BIC | 369.27 | 424.30 | 375.91 |
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Gómez, H.J.; Caimanque, W.E.; Gómez, Y.M.; Magalhães, T.M.; Concha, M.; Gallardo, D.I. A Bimodal Model Based on Truncation Positive Normal with Application to Height Data. Symmetry 2022, 14, 665. https://doi.org/10.3390/sym14040665
Gómez HJ, Caimanque WE, Gómez YM, Magalhães TM, Concha M, Gallardo DI. A Bimodal Model Based on Truncation Positive Normal with Application to Height Data. Symmetry. 2022; 14(4):665. https://doi.org/10.3390/sym14040665
Chicago/Turabian StyleGómez, Héctor J., Wilson E. Caimanque, Yolanda M. Gómez, Tiago M. Magalhães, Miguel Concha, and Diego I. Gallardo. 2022. "A Bimodal Model Based on Truncation Positive Normal with Application to Height Data" Symmetry 14, no. 4: 665. https://doi.org/10.3390/sym14040665