Timc(n, mu2n, mu4n,icoef)$$\mu_2(\lambda)=\frac{B(3\lambda_1^{-1},\lambda_2+1)}{B(\lambda_1^{-1},\lambda_2+1)} \qquad \mu_4(\lambda)=\frac{B(5\lambda_1^{-1},\lambda_2+1)}{B(\lambda_1^{-1},\lambda_2+1)}$$
where $B(x,y)$ denotes the beta function between the positive real-valued numbers $x$ and $y$. The second and fourth moments of the null distributions of the rank correlations are known polynomials in $n$.
$$\mu_2(r_1)=\frac{1}{n-1}$$ $$\mu_2(r_2)=\frac{2(2n+5)}{9n(n-1)}$$ $$\mu_2(r_3)=\left[\frac{2}{3(n-1)}\right]\left[\frac{n^2+2+k_n}{n^2-k_n}\right]$$ $$\mu_4(r_1)=\frac{3(25n^3-38n^2-35n+72)} {25n(n+1)(n-1)^3}$$ $$\mu_4(r_2)=\frac{100n^4+328n^3-127n^2-997n-372} {1350\left[0.5n(n-1)\right]^3}$$ $$\mu_4(r_3)=\frac{4[35n^7-(111-35k_n)n^6+(153+29k_n)n^5-(366-59k_n)n^4+304+11k_n)n^3]} {n^{k_n}(105-2k_n)(n+k_n)^3(\!n-k_n)^4(n-3+k_n)}+$$ $$\frac{-[(456-114k_n)n^2-(912-492k_n)n+(1248-933k_n)]} {n^{k_n}(105-2k_n)(n+k_n)^3(\!n-k_n)^4(n-3+k_n)}$$
The approximation to the null distribution of $r_4$ is based on an estimation of the second moment obtained through a regression strategy. In particular,
$$\mu_2 (r_4)\approx \frac{1.00762}{(n-1)}$$ $$\mu_4(r_4)\approx \frac{1.0949159471}{\sqrt{n-1}}+\frac{38.7820781157}{(n-1)^2}-\frac{208.8267798530}{(n-1)^3}+\frac{396.3338168921}{(n-1)^4}$$
See Tarsitano and Amerise (2016). The second and fourth moments of $r_5$ are given in Fieller and Pearson (1961):
$$\mu_2(r_5)=\frac{1}{(n-1)}$$ $$\mu_4(r_5)=\frac{1}{(n-1)^2}\Big[\frac{3(n-1)}{n+1}+\frac{(n-2)(n-3)}{n(n^2-1)}\Big(\frac{k_4}{k_2^2}\Big)^2 \Big]$$
with
$$k_2=\frac{\sum_{i=1}^n\xi(x_i|n)^2}{n-1}$$ $$k_4=\frac{n\left{(n+1)\sum_{i=1}^n\xi(x_i|n)^4-\frac{3(n-1)}{n}\Big[\sum_{i=1}^n\xi(x_i|n)^2\Big]^2\right}}{(n-1)(n-2)(n-3)}$$
where $\xi(x_i|n)$ is the expected values of the i-th largest standardized deviate in a sample of size $n$ from a Gaussian population. With regard to $r_6$, the GGFR approximation is provisionally implemented with the same structure as $r_5$ using the medians in place of the means of the Gaussian order statistics. See Amerise & Tarsitano (2016).
The estimate of the parameter vector $\lambda=(\lambda_1,\lambda_2)$ is obtained by solving
$$C(\lambda)=\min[\max[g_2(\lambda),g_4(\lambda)]]$$ with $g_2(\lambda)=\mu_2(\lambda)-\mu_{2,n}$ and $g_4(\lambda)=\mu_4(\lambda)-\mu_{4,n}$ where $\mu_{2,n}$ and $\mu_{4,n}$ are the second and fourth moments of the given rank correlation.
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Tarsitano, A. and Amerise, I. L. (2016). "Modelling of the null distribution of rank correlations". Submitted.
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