Natural Exponential Family - Natural Exponential Families With Quadratic Variance Functions (NEF-QVF) - The Six NEF-QVFs

The Six NEF-QVFs

The six NEF-QVF are written here in increasing complexity of the relationship between variance and mean.

1. The normal distribution with fixed variance is NEF-QVF because the variance is constant. The variance can be written, so variance is a degree 0 function of the mean.

2. The Poisson distribution is NEF-QVF because all Poisson distributions have variance equal to the mean, so variance is a linear function of the mean.

3. The Gamma distribution is NEF-QVF because the mean of the Gamma distribution is and the variance of the Gamma distribution is, so the variance is a quadratic function of the mean.

4. The binomial distribution is NEF-QVF because the mean is and the variance is which can be written in terms of the mean as

5. The negative binomial distribution is NEF-QVF because the mean is and the variance is

6. The (not very famous) distribution generated by the generalized hyperbolic secant distribution (NEF-GHS) has and

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