On the Folded Normal Distribution
<p>The black line is the density of the <math display="inline"> <mrow> <mi>N</mi> <mfenced separators="" open="(" close=")"> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math> and the red line of the <math display="inline"> <mrow> <mi>F</mi> <mi>N</mi> <mfenced separators="" open="(" close=")"> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math>. The parameters in the left figure (<b>a</b>) are <math display="inline"> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math> and <math display="inline"> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>3</mn> </mrow> </math> and in the right figure (<b>b</b>) <math display="inline"> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math> and <math display="inline"> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>4</mn> </mrow> </math>.</p> "> Figure 2
<p>Entropy values for a range of values of <math display="inline"> <mrow> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>μ</mi> <mi>σ</mi> </mfrac> </mrow> </math> with <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> (<b>a</b>) and <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> (<b>b</b>).</p> "> Figure 3
<p>Kullback–Leibler divergence from the normal for a range of values of <math display="inline"> <mrow> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>μ</mi> <mi>σ</mi> </mfrac> </mrow> </math> with <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> (<b>a</b>) and <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> (<b>b</b>).</p> "> Figure 4
<p>Kullback–Leibler divergence from the half normal for a range of values of <math display="inline"> <mrow> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>μ</mi> <mi>σ</mi> </mfrac> </mrow> </math> with <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> (<b>a</b>) and <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> (<b>b</b>).</p> "> Figure 5
<p>The left graph (<b>a</b>) shows the three solutions of the log-likelihood. The right three-dimensional figure (<b>b</b>) shows the values of the log-likelihood for a range of mean and variance values.</p> "> Figure 6
<p>The histogram on the left shows the body mass indices of 700 New Zealand adults. The green line is the fitted folded normal and the blue line is the kernel density. The perspective plot on the right shows the log-likelihood of the body mass index data as a function of the mean and the variance.</p> ">
Abstract
:1. Introduction
2. The Folded Normal
2.1. Relations to Other Distributions
2.2. Mode of the Folded Normal Distribution
2.3. Characteristic Function and Other Related Functions of the Folded Normal Distribution
- The moment generating function of Equation (2) exists and is equal to:
- The cumulant generating function is simply the logarithm of the moment generating function:
- The Laplace transformation can easily be derived from the moment generating function and is equal to:
- The Fourier transformation is:
- The mean residual life is given by:
3. Entropy and Kullback–Leibler Divergence
3.1. Entropy
3.2. Kullback–Leibler Divergence from the Normal Distribution
3.3. Kullback–Leibler Divergence from the Half Normal Distribution
4. Parameter Estimation
4.1. An Example with Simulated Data
4.2. Simulation Studies
Values | of | θ | ||||||
---|---|---|---|---|---|---|---|---|
Sample size | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
20 | 0.689 | 0.930 | 0.955 | 0.931 | 0.926 | 0.940 | 0.930 | 0.948 |
30 | 0.679 | 0.921 | 0.949 | 0.943 | 0.925 | 0.926 | 0.941 | 0.915 |
40 | 0.690 | 0.916 | 0.936 | 0.933 | 0.941 | 0.948 | 0.944 | 0.928 |
50 | 0.718 | 0.944 | 0.955 | 0.938 | 0.933 | 0.948 | 0.946 | 0.946 |
60 | 0.699 | 0.950 | 0.968 | 0.948 | 0.949 | 0.941 | 0.942 | 0.946 |
70 | 0.721 | 0.931 | 0.956 | 0.939 | 0.939 | 0.939 | 0.949 | 0.945 |
80 | 0.691 | 0.930 | 0.950 | 0.940 | 0.946 | 0.936 | 0.945 | 0.939 |
90 | 0.720 | 0.932 | 0.960 | 0.949 | 0.949 | 0.939 | 0.954 | 0.944 |
100 | 0.738 | 0.945 | 0.949 | 0.938 | 0.943 | 0.926 | 0.946 | 0.952 |
Values | of | θ | ||||||
---|---|---|---|---|---|---|---|---|
Sample size | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
20 | 0.890 | 0.925 | 0.939 | 0.921 | 0.918 | 0.940 | 0.929 | 0.942 |
30 | 0.894 | 0.931 | 0.933 | 0.943 | 0.926 | 0.922 | 0.942 | 0.910 |
40 | 0.910 | 0.925 | 0.927 | 0.933 | 0.941 | 0.947 | 0.946 | 0.928 |
50 | 0.914 | 0.943 | 0.942 | 0.934 | 0.934 | 0.945 | 0.946 | 0.943 |
60 | 0.904 | 0.949 | 0.953 | 0.950 | 0.941 | 0.938 | 0.943 | 0.944 |
70 | 0.893 | 0.934 | 0.943 | 0.936 | 0.937 | 0.938 | 0.949 | 0.939 |
80 | 0.918 | 0.940 | 0.939 | 0.939 | 0.944 | 0.935 | 0.946 | 0.938 |
90 | 0.920 | 0.934 | 0.952 | 0.948 | 0.946 | 0.939 | 0.951 | 0.947 |
100 | 0.918 | 0.940 | 0.936 | 0.932 | 0.946 | 0.925 | 0.945 | 0.949 |
Values | of | θ | ||||||
---|---|---|---|---|---|---|---|---|
Sample size | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
20 | 0.649 | 0.765 | 0.854 | 0.853 | 0.876 | 0.870 | 0.862 | 0.885 |
30 | 0.697 | 0.794 | 0.870 | 0.898 | 0.892 | 0.898 | 0.894 | 0.896 |
40 | 0.723 | 0.849 | 0.893 | 0.914 | 0.919 | 0.913 | 0.909 | 0.902 |
50 | 0.751 | 0.867 | 0.916 | 0.907 | 0.911 | 0.924 | 0.899 | 0.912 |
60 | 0.745 | 0.865 | 0.911 | 0.913 | 0.916 | 0.906 | 0.920 | 0.933 |
70 | 0.769 | 0.874 | 0.928 | 0.928 | 0.912 | 0.930 | 0.926 | 0.935 |
80 | 0.776 | 0.883 | 0.927 | 0.919 | 0.934 | 0.936 | 0.916 | 0.924 |
90 | 0.795 | 0.901 | 0.931 | 0.932 | 0.925 | 0.930 | 0.940 | 0.941 |
100 | 0.824 | 0.904 | 0.927 | 0.933 | 0.925 | 0.936 | 0.932 | 0.942 |
Values | of | θ | ||||||
---|---|---|---|---|---|---|---|---|
Sample size | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
20 | 0.657 | 0.814 | 0.862 | 0.842 | 0.840 | 0.832 | 0.818 | 0.824 |
30 | 0.701 | 0.850 | 0.885 | 0.891 | 0.882 | 0.867 | 0.869 | 0.866 |
40 | 0.743 | 0.881 | 0.896 | 0.913 | 0.912 | 0.886 | 0.881 | 0.878 |
50 | 0.772 | 0.895 | 0.921 | 0.916 | 0.897 | 0.901 | 0.885 | 0.892 |
60 | 0.797 | 0.907 | 0.912 | 0.910 | 0.906 | 0.897 | 0.907 | 0.916 |
70 | 0.807 | 0.904 | 0.925 | 0.915 | 0.909 | 0.918 | 0.908 | 0.924 |
80 | 0.822 | 0.895 | 0.925 | 0.914 | 0.925 | 0.917 | 0.909 | 0.909 |
90 | 0.869 | 0.916 | 0.932 | 0.922 | 0.919 | 0.915 | 0.934 | 0.929 |
100 | 0.873 | 0.915 | 0.918 | 0.925 | 0.906 | 0.931 | 0.920 | 0.939 |
Values | of | θ | ||||||
---|---|---|---|---|---|---|---|---|
Sample size | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
20 | −0.600 | −0.495 | −0.272 | −0.086 | −0.025 | −0.006 | −0.001 | 0.000 |
30 | −0.638 | −0.537 | −0.262 | −0.089 | −0.022 | −0.005 | −0.001 | 0.000 |
40 | −0.695 | −0.548 | −0.251 | −0.081 | −0.021 | −0.005 | −0.001 | 0.000 |
50 | −0.723 | −0.580 | −0.259 | −0.076 | −0.020 | −0.005 | −0.001 | 0.000 |
60 | −0.750 | −0.597 | −0.251 | −0.075 | −0.019 | −0.004 | −0.001 | 0.000 |
70 | −0.771 | −0.588 | −0.256 | −0.073 | −0.019 | −0.004 | −0.001 | 0.000 |
80 | −0.774 | −0.604 | −0.253 | −0.074 | −0.019 | −0.004 | −0.001 | 0.000 |
90 | −0.796 | −0.599 | −0.245 | −0.073 | −0.018 | −0.004 | −0.001 | 0.000 |
100 | −0.804 | −0.611 | −0.252 | −0.072 | −0.019 | −0.004 | −0.001 | 0.000 |
Values | of | θ | |||||
---|---|---|---|---|---|---|---|
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
0.309 | 0.159 | 0.067 | 0.023 | 0.006 | 0.001 | 0.000 | 0.000 |
5. Application to Body Mass Index Data
6. Discussion
Conflicts of Interest
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Tsagris, M.; Beneki, C.; Hassani, H. On the Folded Normal Distribution. Mathematics 2014, 2, 12-28. https://doi.org/10.3390/math2010012
Tsagris M, Beneki C, Hassani H. On the Folded Normal Distribution. Mathematics. 2014; 2(1):12-28. https://doi.org/10.3390/math2010012
Chicago/Turabian StyleTsagris, Michail, Christina Beneki, and Hossein Hassani. 2014. "On the Folded Normal Distribution" Mathematics 2, no. 1: 12-28. https://doi.org/10.3390/math2010012