1 Correction to: Metrika (2019) 82:385–408 https://doi.org/10.1007/s00184-018-0687-7

The statement of Theorem 3 (p. 397) in Nikolov and Stoimenova (2019) should be corrected to:

Theorem 3

Let \(M(\theta ,N)=\left\{ m_{ij}(\theta ,N)\right\} _{i,j=1}^{N}\) be the Marginals matrix based on Lee distance. Then

$$\begin{aligned} m_{ij}(\theta _N,N) \frac{N}{ \exp \left( \theta _N\mu + \displaystyle \frac{\theta _N^{2}\nu ^{2}}{2}\right) } \xrightarrow [N \rightarrow \infty ] \displaystyle 1, \quad for \;\; i,j=1,2,\ldots ,N, \end{aligned}$$

where

$$\begin{aligned} \mu= & {} \frac{N c_{N}(i,j)}{N-1}-\frac{1}{N-1}\left[ \frac{N+1}{2}\right] \left[ \frac{N}{2}\right] ,\\ \nu ^{2}= & {} {\left\{ \begin{array}{ll} \displaystyle \frac{2 N^{2} \left( c_{N}\left( i,j\right) \right) ^{2}- N^{3}c_{N}\left( i,j\right) }{2\left( N-2\right) \left( N-1\right) ^{2}}-\frac{N^{2}\left( N^{2}-2N+4\right) }{48(N-1)^{2}}, &{} \text{ for } N\hbox { even} \\ \displaystyle \frac{2 N^{2} \left( c_{N}\left( i,j\right) \right) ^{2}- N\left( N^{2}-1\right) c_{N}\left( i,j\right) }{2\left( N-2\right) \left( N-1\right) ^{2}}-\frac{N\left( N+1\right) \left( N-3\right) }{48(N-2)}, &{} \text{ for } N\hbox { odd.} \end{array}\right. } \end{aligned}$$

and \(\left\{ \theta _N\right\} _{N=1}^{\infty }\) is a sequence of real numbers such that \(\lim \limits _{N\rightarrow \infty }\displaystyle \theta _{N}N^{\frac{3}{2}}=a,\) for some real constant a.

The quantity \(m_{ij}\left( \theta ,N\right) \) for fixed \(\theta \) in the proof (p. 407) of the original statement of Theorem 3 (p. 397) can be written as

$$\begin{aligned} m_{ij}\left( \theta ,N\right) =\frac{1}{N}\exp \left( \theta \left( \tilde{\mu }(N)-\mu (N)\right) +\frac{\theta ^2}{2}\left( \tilde{\sigma }^{2}(N)-\sigma ^{2}(N)\right) \right) \frac{1+\tilde{f}\left( \theta \tilde{\sigma }(N),N\right) }{1+f\left( \theta \sigma (N),N\right) }, \end{aligned}$$

where \(\mu (N)=\mathbf {E}\left( D_{L}\right) \), \(\sigma ^2(N)=\mathbf {Var}\left( D_{L}\right) \), \(\tilde{\mu }(N)=\mathbf {E}\left( \tilde{D}_{N-1}\right) \), \(\tilde{\sigma }^2(N)=\mathbf {Var}\left( \tilde{D}_{N-1}\right) \) and \(f(\theta ,N)\) and \(\tilde{f}(\theta ,N)\) are functions such that \(\lim \nolimits _{N\rightarrow \infty }f(\theta ,N)=0\) and \(\lim \nolimits _{N\rightarrow \infty }\tilde{f}(\theta ,N)=0\). Thus, the original statement is equivalent to

$$\begin{aligned} \lim \limits _{N\rightarrow \infty } \displaystyle \frac{1+\tilde{f}\left( \theta \tilde{\sigma }(N),N\right) }{1+f\left( \theta \sigma (N),N\right) } =1, \end{aligned}$$

which is not trivial to be proved and even might not hold. However, if instead of fixed \(\theta \) we consider a sequence of real numbers \(\left\{ \theta _N\right\} _{N=1}^{\infty }\) such that \(\lim \nolimits _{N\rightarrow \infty }\displaystyle \theta _{N}N^{\frac{3}{2}}=a,\) for some real constant a, then

$$\begin{aligned} \lim \limits _{N\rightarrow \infty }\displaystyle \theta _{N}\sigma (N)=c, \quad \lim \limits _{N\rightarrow \infty }\displaystyle \theta _{N}\tilde{\sigma }(N)=\tilde{c}, \end{aligned}$$

where c and \(\tilde{c}\) are some real constants, and \(\lim \nolimits _{N\rightarrow \infty }\displaystyle \frac{1+\tilde{f}\left( \theta _{N}\tilde{\sigma }(N),N\right) }{1+f\left( \theta _{N}\sigma (N),N\right) }=1\). This completes the proof of the corrected formulation.

Furthermore, since the theorem holds for a sequence of real numbers \(\left\{ \theta _N\right\} _{N=1}^{\infty }\), it could be used to approximate the Marginals matrix \(M(\theta ,N)\) only when the value of \(\theta \) is close to 0. However, our experience from simulation studies indicates that the original statement of Theorem 3 (p. 397) might be true and the approximation could be applied even for large values of \(\theta \). As an illustration, see the example in Section 4.2 (pp. 400–402).