# Estimating parameter $$\sigma$$ for the Rayleigh distribution

Let $$X, Y$$ be uncorrelated, jointly normally distributed cartesian coordinates with means $$\mu_{X}, \mu_{Y}$$, and equal variances $$\sigma_{X}^{2} = \sigma_{Y}^{2}$$. The radius of an $$(X,Y)$$-coordinate pair around the true mean is$R := \sqrt{(X-\mu_{X})^{2} + (Y-\mu_{Y})^{2}}$

With $$N$$ observations of $$(x,y)$$-coordinates, $$\text{SSR}$$ is the sum of squared radii$\begin{array}{rcl} \text{SSR} &:=& \sum\limits_{i=1}^{N} r_{i}^{2} = \sum\limits_{i=1}^{N} ((x_{i}-\mu_{X})^{2} + (y_{i}-\mu_{Y})^{2})\\ &=& \sum\limits_{i=1}^{N} (x_{i}-\mu_{X})^{2} + \sum\limits_{i=1}^{N} (y_{i}-\mu_{Y})^{2} \end{array}$

The Maximum-Likelihood-estimate of the variance of $$r$$ is the total variance of all $$2N$$ separate $$x$$- and $$y$$-coordinates$\widehat{\sigma^{2}_{ML}} = \frac{1}{2N} \cdot \text{SSR}$

## Unknown true center: Deriving Singh's (1992) $$C_{2}$$ estimator

When the $$(\mu_{X}, \mu_{Y})$$-center is estimated by $$(\bar{x}, \bar{y})$$, the Bessel correction is required to make the ML-estimate of the variance unbiased$\widehat{\sigma^{2}_{ub}} = \frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR} = \frac{1}{2(N-1)} \cdot \text{SSR}$

However, taking the square root of the unbiased variance estimate makes it biased (Jensen's inequality). Specifically, since the square root is concave, the bias is negative and $$\sqrt{\widehat{\sigma^{2}_{ub}}}$$ underestimates $$\sigma$$. When $$Q$$ is a $$\chi$$-distributed variable with $$k$$ degrees of freedom, its mean is$E(Q) = \sqrt{2} \cdot \frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}$

When $$X$$ is a normally distributed random variable with $$n$$ observations, and $$s^{2} := \frac{1}{n-1} \sum\limits_{i=1}^{n}(x-\bar{x})^{2}$$ is the Bessel-corrected variance estimate, $$E(s^{2}) = \sigma^{2}$$. Then $$c_{4}(n)$$ is the correction factor such that $$E(s) = c_{4}(n) \cdot \sigma$$. With $$Q$$ as given above, $$c_{4}(n) = \frac{1}{\sqrt{k}} \cdot E(Q)$$ with $$k := n-1$$$c_{4}(n) = \sqrt{\frac{2}{n-1}} \cdot \frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

We lose 2 degrees of freedom estimating the center. Therefore we really only have 2N - 2 degrees of freedom, so we set $$n := 2N-1$$ so that $$c_{4}(n)$$ is the scaled mean of a $$\chi$$-distributed variable with $$k = n-1 = 2N-1-1 = 2N-2$$ degrees of freedom. This gives$\begin{array}{rcl} \widehat{\sigma_{ub}} &=& \frac{1}{c_{4}(2N-1)} \cdot \sqrt{\widehat{\sigma^{2}_{ub}}} \\ &=& \frac{1}{\sqrt{\frac{2}{2N-1-1}} \cdot \frac{\Gamma\left(\frac{2N-1}{2}\right)}{\Gamma\left(\frac{2N-1-1}{2}\right)}} \cdot \sqrt{\frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{\frac{2(N-1)}{2}} \cdot \frac{\Gamma\left(\frac{2(N-1)}{2}\right)}{\Gamma\left(\frac{2N-1}{2}\right)} \cdot \sqrt{\frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{N} \cdot \frac{\Gamma(N-1)}{\Gamma\left(\frac{2N-1}{2}\right)} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \frac{\Gamma(N-1)}{\Gamma\left(N - \frac{1}{2}\right)} \cdot \sqrt{\frac{1}{2} \cdot \text{SSR}} \end{array}$

The penultimate expression is Singh's 1992 definition 3.3 for estimator $$C_{2}$$, also given by Moranda (1959).

## Known true center: Deriving Singh's (1992) $$C_{1}$$ estimator

When the $$(\mu_{X}, \mu_{Y})$$-center is known, the Bessel correction is not required to make the Maximum-Likelihood-estimate of the variance unbiased. In this case we simply set $$n := 2N+1$$ for the $$c_{4}(n)$$ correction factor so that $$c_{4}(n)$$ is the scaled mean of a $$\chi$$-distributed variable with $$k = n-1 = 2N+1-1 = 2N$$ degrees of freedom. This gives$\begin{array}{rcl} \widehat{\sigma_{ub}} &=& \frac{1}{c_{4}(2N+1)} \cdot \sqrt{\widehat{\sigma^{2}_{ub}}} \\ &=& \frac{1}{\sqrt{\frac{2}{2N+1-1}} \cdot \frac{\Gamma\left(\frac{2N+1}{2}\right)}{\Gamma\left(\frac{2N+1-1}{2}\right)}} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{N} \cdot \frac{\Gamma(N)}{\Gamma\left(\frac{2N+1}{2}\right)} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \frac{\Gamma(N)}{\Gamma\left(N + \frac{1}{2}\right)} \cdot \sqrt{\frac{1}{2} \cdot \text{SSR}} \end{array}$

The penultimate expression is Singh's 1992 definition 2.2 for estimator $$C_{1}$$.