Estimators of the Parameters of Beta Distribution

Abstract

The iteration algorithm of computation of effective estimators of the shape parameters of beta distributions using the unbiased estimators of the end point parameters of the random variable were obtained and investigated. For the cases when more accurate estimations of the parameters are required, one more step of computation, realized optimization of the obtained estimations, is necessary. The computation results, realized on the basis of the simulation of the appropriate random samples, demonstrate the correctness of the obtained theoretical outcomes.

Appendices

Appendix 1. Formulae for Determination of Direct and Inverse Incomplete Beta-Functions

Let us consider a standard form of the incomplete beta distribution, i.e.,

$$B_{x} (p,q) ={{\int}_{0}^{x}} t^{p-1} \,(1 -t)^{q -1} \,d t \;, $$

B(p,q) = B₁(p,q) is a complete beta function and

$$B(p,q) =\frac{{\Gamma}(p)\cdot {\Gamma}(q)}{{\Gamma}(p +q)} \;. $$

Let us introduce denotations:

$$\begin{array}{@{}rcl@{}} &&F_{m} (\lambda) =\prod\limits_{k = 0}^{m-1} (\lambda +k) \;, \quad \chi_{n} (\lambda,\mu,z) =\sum\limits_{k = 0}^{n-1} \frac{F_{k} (\lambda)}{F_{k} (\mu)}\cdot z^{k} \;,\\ &&I_{z} (p,q) =B_{z} (p,q) /B(p,q) \;. \end{array} $$

Then we can represent the beta function in the form of the series

$$B_{z} (\alpha,\beta) =\alpha^{-1} \,z^{\alpha} \cdot {~}_{2} F_{1} (\alpha,\, 1-\beta;\, \alpha + 1;\, z) \;, $$

or

$$B_{z} (\alpha,\beta) =\alpha^{-1} \,\left( \frac{z}{1-z}\right)^{\alpha} \cdot {~}_{2} F_{1} \left( \alpha,\, \alpha +\beta;\, \alpha + 1;\, \frac{z}{1 -z}\right) \;, $$

where

$${~}_{2} F_{1} (\alpha_{1},\, \alpha_{2};\, \gamma;\, z) =\sum\limits_{k = 0}^{\infty} \frac{F_{k} (\alpha_{1}) \,F_{k} (\alpha_{2})}{F_{k} (\gamma)}\cdot \frac{z^{k}}{k!} \qquad (|z|<1) \;. $$

At integer values of the parameters, the following relations are correct

$$\begin{array}{@{}rcl@{}} &&B(n, \alpha) =\frac{(n-1)!}{F_{n} (\alpha)} =\frac{1}{\alpha} \,\prod\limits_{k = 1}^{n-1} \frac{k}{\alpha +k} \;, \\ &&B_{z} (\alpha, n) =B(\alpha,n)\cdot z^{\alpha} \,\chi_{n} (\alpha, 1, 1-z) \;, \\ &&B_{z} (n,\alpha) =B(n,\alpha) -B(n,\alpha)\cdot (1-z)^{\alpha} \,\chi_{n} (\alpha, 1, z) \;. \end{array} $$

In particular,

$$\begin{array}{@{}rcl@{}} &&B(1,\alpha) = 1/\alpha \;, \\ &&I_{z} (\alpha,1) =z^{\alpha} \;,\qquad I_{z} (1,\alpha) = 1 -(1 -z)^{\alpha} \;. \end{array} $$

At half-integer values of the parameters, we have

$$\begin{array}{@{}rcl@{}} &&B(m + 1/2,\, n + 1/2) =\pi \,\frac{F_{m} (1/2) \,F_{n} (1/2)}{(m+n)!} \;, \\ &&I_{z} (m + 1/2,\, n + 1/2) =(2/\pi)\cdot \arcsin \sqrt{z} +(2/\pi) \,\sqrt{z \,(1 -z)}\cdot {\Psi}_{m,n} \;, \end{array} $$

where

$$\begin{array}{@{}rcl@{}} {\Psi}_{m,n} &=&\chi_{n} (1,\, 3/2,\, 1-z) -\frac{n!}{F_{n} (1/2)}\cdot (1-z)^{n} \,\chi_{m} (1 +n,\, 3/2,\, z) \\ &=&-\chi_{m} (1,\, 3/2,\, z) -\frac{m!}{F_{m} (1/2)}\cdot z^{m} \,\chi_{n} (1 +m,\, 3/2,\, 1-z) \;. \end{array} $$

In particular,

$$\begin{array}{@{}rcl@{}} I_{z} (1/2,\, 1/2) &=&(2/\pi)\cdot \arcsin \sqrt{z} =(2/\pi)\cdot \arctan\sqrt{z/(1-z)} \;, \\ B_{z} (1/2,\, 1/2) &=&2\cdot \arcsin \sqrt{z} = 2\cdot \arctan\sqrt{z/(1-z)} \;. \end{array} $$

For computation of the inverse incomplete beta function, let us introduce Q(α,β,ξ), which is the inverse function of I_z(α,β) for given α and β. It satisfies the relation

$$Q(\alpha,\beta,\xi) = 1 -Q(\beta,\alpha,\, 1-\xi) \;. $$

At ξ → 0

$$Q(\alpha,\beta,\xi) \approx \eta +\frac{\beta -1}{\alpha + 1} \,\eta^{2} +... \;, $$

where

$$\eta =\left( \xi \cdot \left( \alpha /B(\alpha,\beta)\right)\right)^{1/\alpha} \;. $$

At ξ → 1

$$Q(\alpha,\beta,\xi) \approx 1 -\eta -\frac{\alpha -1}{\beta + 1} \,\eta^{2} +... \;, $$

where

$$\eta =\left( (1 -\xi)\cdot \left( \beta /B(\alpha,\beta)\right)\right)^{1/\beta} \;. $$

For determination of the inverse incomplete beta function, we used the following iteration method.

Let us consider the equation

$$ I_{x} (p,q) =y \;, $$

(1)

at 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and (1 − p)(1 − q)≠ 0. Let us denote

$$x_{\mu} =(p-1)/(p +q -2) \;. $$

For the solution of (1), the method of Newton’s iteration was used. The initial approximation x₀ of the iterative process is to be defined as follows:

• if p > 1 and q > 1 then x₀ = x_μ;

• if p ≤ 1 and q ≥ 1 then x₀ = 0;

• if p ≥ 1 and q ≤ 1 then x₀ = 1;

• if α < 1 and β < 1 and $I_{x_{\mu }}(p,q) >y$ then x₀ = 0;

• if α < 1 and β < 1 and $I_{x_{\mu }}(p,q) <y$ then x₀ = 1.

It should be noted that the monotonous convergence of the iteratively obtained sequence of the solutions of (1) to the desired value is guaranteed.

Appendix 2. Examples

Examples of the graphs of the divergences between empirical and estimated distribution functions by the samples of different size n for case 11 at a = 12.0, b = 21.0, p = 7.21; q = 1.252 (see Table 6) are represented in Fig. 6.

Figure 6

Examples of the graphs of the divergences between empirical and estimated distribution functions