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QUESTION

What is a citation of a book whose scope includes the uniform distribution [1] that is generalized to an $n$-ball [2]?

Among other things, I'd like to read a book that include such information as the distribution's parameters, its support, its probability density function, and its moment generating function.

Answers to this question may be useful to other users as well (cf., [3]).

Bibliography

[1] https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

[2] https://en.wikipedia.org/wiki/Ball_(mathematics)

[3] https://math.stackexchange.com/questions/3002527/generalised-general-uniform-distribution-continuous

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    $\begingroup$ Please explain what generalizations you have in mind: would it be to balls, spheres, tori, general products of such, or what? What kind of information should this book be providing about these distributions? Why do you mention the vM-F distribution, which is not uniform--does that indicate you mean something besides having a constant density? $\endgroup$
    – whuber ♦
    Commented Aug 19, 2019 at 21:13
  • $\begingroup$ Re the edit: thank you for making your question more precise. There's little to say about the density, support, or parameters that isn't trivial (the post on the math site indicates as much), so at best you can hope to find some mention of these in passing within a larger context; but the questions of its moment generating function, characteristic function, and cumulant generating function require some calculation: why not just ask for that information directly? $\endgroup$
    – whuber ♦
    Commented Aug 20, 2019 at 10:55

1 Answer 1

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The book "Directional Statistics", Mardia and Jupp (1999), has information on distributions on spheres in arbitrary dimensions. If you want a distribution on closed balls instead, you could take a product distribution of a spherical distribution with a distribution on the interval [0,R], where R>0 is the radius of the ball. The key notion of the uniform distribution on an interval of the real line is that it is invariant under shifts on the axis. Invariance properties are underpinning the definitions of distributions on spheres, and I would think that in the case of balls this should be similar, hence the above approach is natural.

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    $\begingroup$ I mean ball with an inside. $\endgroup$
    – Michael Levy
    Commented Aug 23, 2019 at 13:49
  • $\begingroup$ Then you should revise the title. Spherical distribution is usually taken to mean distribution in the surface of a ball, that is, a sphere. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 24, 2019 at 20:51

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