Calls a fortran subroutine which performs a controlled random search to compute the parameters of the Vianelli Generalized Gaussian with finite range distribution (VGGFR).
Timc(n, mu2n, mu4n,icoef)number of ranks
variance of \(r_{h,n}\)
kurtosis of \(r_{h,n}\)
an integer indicating the coefficient to be interpolated: 1=Spearman, 2=Gini, 3=Kendall, 4=r4, 5=Fisher-Yates, 6=Filliben.
A numeric vector giving the estimates of the two parameters of the VGGFR
Final value of the loss function
The VGGFR is a flexible density to which we can resort in the case the number of ranks is larger than the threshold for which the exact null distribution of a rank correlation is known, but lower than the threshold for which the asymptotic Gaussian approximation becomes valid.
Although there are several methods for estimating the parameters of the GGFR, we follow the moment-matching method as applied by Karian and Dudewicz (2000) [Ch 2.6, p. 104] in similar estimations. The second and fourth centered moments of a GGFR distribution are:
$$\mu_2(\lambda)=\frac{B(3\lambda_1^{-1},\lambda_2+1)}{B(\lambda_1^{-1},\lambda_2+1)} \qquad \mu_4(\lambda)=\frac{B(5\lambda_1^{-1},\lambda_2+1)}{B(\lambda_1^{-1},\lambda_2+1)}$$
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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