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which is a normal distribution's density function. Thus, the generalized error density function is a generalization of the normal distribution's density function. There are many ways to generalize a normal distribution's density function. Another example, but with the normal distribution's density function only as a limiting value, and not with mid-range substitution values like the generalized error density function, is the Student'st 's density function. Using the Student's-t density function, we would have a rather more restricted selection of kurtosis, and $\textit{df}\geq2$ is the shape parameter because the second moment does not exist for $\textit{df}<2$. Moreover, df is not actually limited to positive integer values, it is in general real $\geq1$. The Student's-t only becomes normal in the limit as $\textit{df}\rightarrow\infty$, which is why I did not choose it as an example. It is neither a good example nor is it a counter example, and in this I disagree with @Xi'an and @whuber.

Thus, using the example of the Student's-t PDF, the choices are to either consider it to be a generalization of a normal PDF, in which case a normal PDF has a permissible form for a Student's-t's PDF, or not, in which case the Student's-t's PDF is of a different form from the normal PDF and thus is irrelevant to the question posed.

We can argue this many ways. My opinion is that a normal PDF is a sub-selected form of a Student's-t 's PDF, but that a normal PDF is not a sub-selection of a gamma PDF even though a limiting value of a gamma PDF can be shown to be a normal PDF, and, my reason for this is that in the normal/Student's-t case, the support is the same, but in the normal/gamma case the support is infinite versus semi-infinite, which is the required incompatibility.

which is a normal distribution's density function. Thus, the generalized error density function is a generalization of the normal distribution's density function. There are many ways to generalize a normal distribution's density function. Another example, but with the normal distribution's density function only as a limiting value, and not with mid-range substitution values like the generalized error density function, is the Student's$-t$ 's density function. Using the Student's$-t$ density function, we would have a rather more restricted selection of kurtosis, and $\textit{df}\geq2$ is the shape parameter because the second moment does not exist for $\textit{df}<2$. Moreover, df is not actually limited to positive integer values, it is in general real $\geq1$. The Student's$-t$ only becomes normal in the limit as $\textit{df}\rightarrow\infty$, which is why I did not choose it as an example. It is neither a good example nor is it a counter example, and in this I disagree with @Xi'an and @whuber.

Thus, using the example of the Student's$-t$ PDF, the choices are to either consider it to be a generalization of a normal PDF, in which case a normal PDF has a permissible form for a Student's$-t$'s PDF, or not, in which case the Student's$-t$ 's PDF is of a different form from the normal PDF and thus is irrelevant to the question posed.

We can argue this many ways. My opinion is that a normal PDF is a sub-selected form of a Student's$-t$ 's PDF, but that a normal PDF is not a sub-selection of a gamma PDF even though a limiting value of a gamma PDF can be shown to be a normal PDF, and, my reason for this is that in the normal/Student'$-t$ case, the support is the same, but in the normal/gamma case the support is infinite versus semi-infinite, which is the required incompatibility.

which is a normal distribution's density function. Thus, the generalized error density function is a generalization of the normal distribution's density function. There are many ways to generalize a normal distribution's density function. Another example, but with the normal distribution's density function only as a limiting value, and not with mid-range substitution values like the generalized error density function, is the Student's-t's density function. Using the Student's-t density function, we would have a rather more restricted selection of kurtosis, and $df\geq2$ is the shape parameter because the second moment does not exist for $df<2$. Moreover, $df$ is not actually limited to positive integer values, it is in general real $\geq1$. The Student's-t only becomes normal in the limit as $df\rightarrow\infty$, which is why I did not choose it as an example. It is neither a good example nor is it a counter example, and in this I disagree with @Xi'an and @whuber.

Thus, using the example of the Student's-t PDF, the choices are to either consider it to be a generalization of a normal PDF, in which case a normal PDF has a permissible form for a Student's-t's PDF, or not, in which case the Student's-t's PDF is of a different form from the normal PDF and thus is irrelevant to the question posed.

We can argue this many ways. My opinion is that a normal PDF is a sub-selected form of a Student's-t's PDF, but that a normal PDF is not a sub-selection of a gamma PDF even though a limiting value of a gamma PDF can be shown to be a normal PDF, and, my reason for this is that in the normal/Student's-t case, the support is the same, but in the normal/gamma case the support is infinite versus semi-infinite, which is the required incompatibility.

which is a normal distribution's density function. Thus, the generalized error density function is a generalization of the normal distribution's density function. There are many ways to generalize a normal distribution's density function. Another example, but with the normal distribution's density function only as a limiting value, and not with mid-range substitution values like the generalized error density function, is the Student'st 's density function. Using the Student's-t density function, we would have a rather more restricted selection of kurtosis, and $\textit{df}\geq2$ is the shape parameter because the second moment does not exist for $\textit{df}<2$. Moreover, df is not actually limited to positive integer values, it is in general real $\geq1$. The Student's-t only becomes normal in the limit as $\textit{df}\rightarrow\infty$, which is why I did not choose it as an example. It is neither a good example nor is it a counter example, and in this I disagree with @Xi'an and @whuber.

Thus, using the example of the Student's-t PDF, the choices are to either consider it to be a generalization of a normal PDF, in which case a normal PDF has a permissible form for a Student's-t's PDF, or not, in which case the Student's-t's PDF is of a different form from the normal PDF and thus is irrelevant to the question posed.

We can argue this many ways. My opinion is that a normal PDF is a sub-selected form of a Student's-t 's PDF, but that a normal PDF is not a sub-selection of a gamma PDF even though a limiting value of a gamma PDF can be shown to be a normal PDF, and, my reason for this is that in the normal/Student's-t case, the support is the same, but in the normal/gamma case the support is infinite versus semi-infinite, which is the required incompatibility.

Comment: There are now 4 downvotes and 12 upvotes on this answer, it started off negative, but was edited. Another similarly good answer, and one without any downvotes is given by @kjetilbhalvorsen below.

There is apparently some confusion as to what a family of distributions is and how to count free parameters versus free plus fixed (assigned) parameters. Those questions are an aside that is unrelated to the intent of the OP, and of this answer. I do not use the word family herein because it is confusing. For example, a family according to one source is the result of varying the shape parameter. @whuber states that A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family. I will use the word form which covers both the intended usage of the word family and parameter identification and counting. For example the formula $x^2-2x+4$ has the form of a quadratic formula, i.e., $a_2x^2+a_1x+a_0$ and if $a_1=0$ the formula is still of quadratic form. However, when $a_2=0$ the formula is linear and the form is no longer complete enough to contain a quadratic shape term. Those who wish to use the word family in a proper statistical context are encouraged to contribute to that separate question.

There is apparently some confusion as to what a family of distributions is and how to count free parameters versus free plus fixed (assigned) parameters. Those questions are an aside that is unrelated to the intent of the OP, and of this answer. I do not use the word family herein because it is confusing. For example, a family according to one source is the result of varying the shape parameter. @whuber states that A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family. I will use the word form which covers both the intended usage of the word family and parameter identification and counting. For example the formula $x^2-2x+4$ has the form of a quadratic formula, i.e., $a_2x^2+a_1x+a_0$ and if $a_1=0$ the formula is still of quadratic form. However, when $a_2=0$ the formula is linear and the form is no longer complete enough to contain a quadratic shape term. Those who wish to use the word family in a proper statistical context are encouraged to contribute to that separate question.