Abstract
A collection of specific probability density functional forms that have found substantial use in statistical applications are examined in this chapter. The selection includes a number of the more commonly used densities, but the collection is by no means an exhaustive account of the vast array of probability densities that are available and that have been applied in the literature.
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- 1.
Johnson, Kotz, Balakrishnan and Kemp provide a set of volumes that provide an extensive survey of a large array of density functions that have been used in statistical applications. These include:
Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition, by Samuel Kotz, N. Balakrishnan and Normal L. Johnson, 2000;
Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson, 1994;
Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson, 1995;
Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson, 1997;
Univariate Discrete Distributions, 3rd Edition by Normal L. Johnson, Adrienne Kemp, and Samuel Kotz, 2008; all published by John Wiley and Sons, New York.
- 2.
Limiting densities will be discussed further in Chapter 5.
- 3.
o(Δt) is a generic notation applied to any function of Δt, whose values approach zero at a rate faster than Δt, so that \( {{\lim}_{{\Delta t\to 0}}}\left( {o\left( {\Delta t} \right)} \right)/\left( {\Delta t} \right) = 0 \). The “o(Δt)” stands for “of smaller order of magnitude than Δt.” For example, h(Δt) = (Δt)2 is a function to which we could affix the label o(Δt), while h(Δt) = (Δt)1/2 is not. More will be said about orders of magnitude in Chapter 5.
- 4.
Ultimately, this assumption could be tested using a nonparametric test of hypothesis. We will examine tests of independence in our discussion of hypothesis testing procedures in Chapter 10.
- 5.
The exponential density is the only density for continuous nonnegative-valued random variables that has the memoryless property. See V.K. Rohatgi (1976) An Introduction to Probability Theory and Mathematical Statistics. New York: John Wiley, p. 209.
- 6.
K. Pearson, (1956), Tables of the Incomplete Beta Function, New York: Cambridge Univ. Press.
- 7.
There are substantive conceptual considerations underlying binary choice situations that naturally lead to the use of the logistic distribution in this context. See Train, K. (1986). Qualitative Choice Analysis: Theory, Econometrics, and an Application to Automobile Demand, MIT Press, Chapter 2.
- 8.
Note that this “normalization” of the logistic distribution in the current model context can be done without loss of generality since the probabilities of the decision events are unaffected thereby. This has to do with the concept of “parameter identification” in probability models, which we will discuss in later chapters when we examine statistical inference issues.
- 9.
Recall that the characteristic roots and vectors of a square matrix A are the scalars, λ, and associated vectors, p, that satisfy the equation [A − λ I] p = 0. There will be as many roots and associated vectors as there are rows (or columns) in the square matrix A.
- 10.
These arguments extend in a natural way to higher dimensions, in which case we are examining n axes of the ellipsoid. See B. Bolch, and C. Huang (1974), Multivariate Statistical Methods for Business and Economics, Englewood Cliffs, NJ: Prentice Hall, p. 19–23, for the matrix theory underlying the derivation of the results on axis length and orientation discussed here.
- 11.
The symmetric matrix square root Σ 1/2 of Σ could be chosen for A. Alternatively, there exists a lower triangular matrix, called the Cholesky decomposition of Σ, which satisfies A′A = Σ. Either choice of A can be calculated straightforwardly on the computer, for example, the chol(.) command in GAUSS.
- 12.
We warn the reader that some authors refer to this collection of densities as the exponential family rather than the exponential class. The collection of densities referred to in this section is a broader concept than that of a parametric family, and nests a large number of probability density families within it, as noted in the text. We (and others) use the term class to distinguish this broader density collection from that of a parametric family, and to also avoid confusion with the exponential density family discussed in Section 4.2.
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Mittelhammer, R.C. (2013). Parametric Families of Density Functions. In: Mathematical Statistics for Economics and Business. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5022-1_4
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DOI: https://doi.org/10.1007/978-1-4614-5022-1_4
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