On Homogeneous Multivariate Distributions in Random Occupancy Models and Their Applications

Abstract

In this article, we consider random occupancy models and the related problems based on the methods of generating functions. The waiting time distributions associated with sequential random occupancy models are investigated through the probability generating functions. We provide the effective computational tools for the evaluation of the probability functions by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of occupancy models. Finally, we treat several examples in order to demonstrate how our theoretical results are employed for the investigation of the random occupancy models along with numerical results.

Appendices

Appendix A: Bell Polynomials

We provide the necessary facts of the Bell polynomials briefly. For more details on these polynomials and their generalizations, the interested reader is referred to Charalambides (2002).

The polynomial B_n,k ≡ B_n,k(x₁,...,x_n) in the variables x₁,...,x_n of degree k defined by the sum

$$ B_{n,k}(x_{1},...,x_{n}) = \underset{\underset{k_{1} + 2k_{2} + {\cdots} + n k_{n}=n}{\scriptstyle k_{1} + {\cdots} + k_{n}=k}}{\sum} \frac{n!}{k_{1}! {\cdots} k_{n}!} \prod\limits_{i=1}^{n} \left( \frac{x_{i}}{i!} \right)^{k_{i}} $$

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is called (exponential) partial Bell partition polynomial, where k₁, k₂,...,k_n are nonnegative integers. The Eq. 1 readily yields

$$ B_{n,k}(abx_{1},a^{2}bx_{2},...,a^{n} b x_{n}) = a^{n} b^{k} B_{n,k}(x_{1},...,x_{n}). $$

The generating function of the partial Bell polynomials B_n,k(x₁,...,x_n), k = 0, 1,...n, n = 0, 1,..., is given by

$$ \begin{array}{@{}rcl@{}} B(t,u) &=& \sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{n} B_{n,k}(x_{1},...,x_{n}) u^{k} \frac{t^{n}}{n!} = 1 + \sum\limits_{n=1}^{\infty} \left\{ \sum\limits_{k=1}^{n} u^{k} B_{n,k}(x_{1},...,x_{n}) \right\} \frac{t^{n}}{n!} \\ & & = \exp \left( u \sum\limits_{m=1}^{\infty} x_{m} \frac{t^{m}}{m!}\right). \end{array} $$

The partial Bell polynomials B_n,k(x₁,...,x_n), k = 0, 1,...,n, n = 0, 1,..., with B_0,0 ≡ 1 can be evaluated through the well-known recurrence relations

$$ \begin{array}{@{}rcl@{}} & & B_{n+1,k+1}(x_{1},...,x_{n+1}) = \sum\limits_{i=0}^{n-k} {n \choose i} x_{i+1} B_{n-i,k}(x_{1},...,x_{n-i}), \\ & & B_{n+1,k+1}(x_{1},...,x_{n+1}) = \frac{1}{k+1} \sum\limits_{i=0}^{n-k} {n+1 \choose i+1} x_{i+1} B_{n-i,k}(x_{1},...,x_{n-i}), \end{array} $$

for k = 0, 1,...,n, n = 0, 1,....

Appendix B: Faa di Bruno Formula

We show that the derivative of order n of a composite function is expressed as a partial Bell polynomial.

Theorem 1

Let f(u) and g(z) be two functions of real variables for which all the derivatives,

$$ g_{j}=\left. \frac{d^{j} g(z)}{d z^{j}} \right|_{z = a}, j=1,2,..., \ \ f_{k}= \left. \frac{d^{k} f(u)}{d u^{k}} \right|_{u = g(a)}, k=0,1,..., $$

exist. Then the derivatives of the composite function $h(z)=f \left (g \left (z \right ) \right )$ are expressed as the partial Bell polynomials

$$ \begin{array}{@{}rcl@{}} & & \frac{ d^{n} } {d z^{n}} h(z) = \sum\limits_{k=0}^{n} f_{k} B_{n,k}(g_{1},...,g_{n}). \end{array} $$

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Appendix C: Proofs

We provide the proofs of the theorems and propositions presented in Sections 3 and 4.

1.1 C.1 Proof of Theorem 1

Observing that

$$ \begin{array}{@{}rcl@{}} \phi_{n}(\mathbf{z}) & = & E \left[ z_{1}^{{N_{n}^{1}}} z_{2}^{{N_{n}^{2}}} {\cdots} z_{m}^{{N_{n}^{m}}} \right] = E \left[ E[z_{1}^{{N_{n}^{1}}} z_{2}^{{N_{n}^{2}}} {\cdots} z_{m}^{{N_{n}^{m}}} ] \left| \right. X_{1},...,X_{n} \right] \\ & = & E \left[ (p_{1} z_{1} + {\cdots} + p_{m} z_{m})^{X_{1} + {\cdots} + X_{n}} \right] = \left[ g(p_{1} z_{1} + {\cdots} + p_{m} z_{m}) \right]^{n}, \\ {\varPhi}(\textbf{z},t) & = & \sum\limits_{n=0}^{\infty} \phi(\mathbf{z}) t^{n} = \frac{1}{1- t g \left( p_{1} z_{1} + {\cdots} + p_{m} z_{m} \right)}, \end{array} $$

the proof is completed.

1.2 C.2 Proof of Theorem 2

It is easy to see that

$$ \frac{ \partial } {\partial z_{i} } h(\mathbf{z}) = p_{i} \left. \frac{ d } {d z } f(g(z)) \right|_{z = p_{1} z_{1} + {\cdots} + p_{m} z_{m}}. $$

Further,

$$ \left. \frac{ \partial^{i_{1} + {\cdots} + i_{m}} } {\partial z_{1}^{i_{1}} {\cdots} \partial z_{m}^{i_{m}}} h(\mathbf{z}) = p_{1}^{i_{1}} {\cdots} p_{m}^{i_{m}} \frac{ d^{i_{1} + {\cdots} + i_{m}} } {d z^{i_{1} + {\cdots} + i_{m}} } f(g(z)) \right|_{z = p_{1} z_{1} + {\cdots} + p_{m} z_{m}} $$

and from Faa di Bruno formula (see Theorem 4 in Appendix B), we have

$$ \left. \frac{ d^{i_{1} + {\cdots} + i_{m}} } {d z^{i_{1} + {\cdots} + i_{m}} } f(g(z)) \right|_{z = p_{1} z_{1} + {\cdots} + p_{m} z_{m}} = {\sum}_{k=0}^{i} f_{k} B_{i,k}(g_{1},...,g_{i}). $$

The proof is completed.

1.3 C.3 Proof of Proposition 1

We see that ϕ_n(z) = [g(p₁z₁ + ⋯ + p_mz_m)]ⁿ with [g(p₁ + ⋯ + p_m)]ⁿ = 1. Therefore, the multivariate random variable $\mathbf {N}_{n}=({N_{n}^{1}},{N_{n}^{2}},...,{N_{n}^{m}})$ is of the homogeneous type from the Definition 1. The proof is completed.

1.4 C.4 Proof of Proposition 3

Observing first that

$$ P \left( {N_{n}^{1}} \geq r_{1} \ \ \text{or} \ ,\ ...\ ,\ \text{or} \ \ {N_{n}^{m}} \geq r_{m} \right)=1-P \left( {N_{n}^{1}} < r_{1},..., {N_{n}^{m}} < r_{m} \right), $$

we get

$$ \begin{array}{@{}rcl@{}} A_{n}(\mathbf{z}) &=& \underset{ r_{1},...,r_{m} \geq 0}{\sum} P \left( {N_{n}^{1}} \geq r_{1} \ \ \text{or} \ ,\ ...\ ,\ \text{or} \ \ {N_{n}^{m}} \geq r_{m} \right) z_{1}^{r_{1}} {\cdots} z_{m}^{r_{m}} \\ & =& \underset{ r_{1},...,r_{m} \geq 0}{\sum} \left[ 1 - P \left( {N_{n}^{1}} < r_{1},...,{N_{n}^{m}} < r_{m} \right) \right] z_{1}^{r_{1}} {\cdots} z_{m}^{r_{m}} \\ & =& \frac{1}{{\prod}_{i=1}^{m} (1-z_{i})} \left[1 - {\sum}_{i_{1},...,i_{m} \geq 0} P \left( {N_{n}^{1}} =i_{1},...,{N_{n}^{m}} = i_{m} \right) \right. \\ && \Biggl. \times z_{1}^{r_{1}+1} {\cdots} z_{m}^{r_{m}+1} \Biggr ]. \end{array} $$

It is easy to see that

$$ \begin{array}{@{}rcl@{}} & & B_{n}(\mathbf{z}) = {\sum}_{r_{1},...,r_{m} \geq 0} \ P \left( {N_{n}^{1}} \geq r_{1} ,..., {N_{n}^{m}} \geq r_{m} \right) z_{1}^{r_{1}} {\cdots} z_{m}^{r_{m}} \\ & & = {\sum}_{r_{1},...,r_{m} \geq 0} \underset{\underset{\scriptstyle j=1,2,...,m}{\scriptstyle i_{j} \geq r_{j}}}{\sum} P \left( {N_{n}^{1}} = i_{1} ,..., {N_{n}^{m}} = i_{m} \right) z_{1}^{r_{1}} {\cdots} z_{m}^{r_{m}} \\ & & = {\sum}_{i_{1},...,i_{m} \geq 0 } P\left( {N_{n}^{1}} = i_{1} ,..., {N_{n}^{m}} = i_{m} \right) \ {\prod}_{j=1}^{m} \frac{ 1-z_{j}^{i_{j}+1} }{1 - z_{j}} \\ & & = \frac{1}{{\prod}_{i=1}^{m}(1-z_{i}) } \left( 1 + {\sum}_{j=1}^{m} (-1)^{j} {\sum}_{1 \leq i_{1} < {\cdots} < i_{j} \leq m} {\sum}_{i_{1},...,i_{j} \geq 0} \right. \\ & & \Biggl. P \left( N_{n}^{i_{1}} = i_{1} ,..., N_{n}^{i_{j}} = i_{j} \right) z_{i_{1}}^{i_{1}+1} {\cdots} z_{i_{j}}^{i_{j}+1} \Biggr). \end{array} $$

The proof is completed.

1.5 C.5 Proof of Theorem 3

Inoue and Aki (2009) elucidated the relation between the distributions of T_r(x), x = 1,m and $({N_{n}^{1}},...,{N_{n}^{m}})$ in terms of the double generating functions:

$$ \begin{array}{@{}rcl@{}} & & H(t,\textbf{z};x) = \frac{1}{{\prod}_{i=1}^{m}(1-z_{i}) } \left( 1 + \sum\limits_{j=m-x+1}^{m} (-1)^{j-m+x} \underset{1 \leq i_{1} < {\cdots} < i_{j} \leq m}{\sum} \ \prod\limits_{u=1}^{j} z_{i_{u}} \right. \\ & & \left. {\kern115pt}\times (1-t) {\varPhi}(z_{i_{1}},...,z_{i_{j}},t)\vphantom{\sum\limits_{j=m-x+1}^{m}} \right). \end{array} $$

By virtue of Eq. 1, the proof is completed.