On the Sample Coefficient of Nominal Variation

Abstract

Categorical dispersion is commonly measured in terms of the index of qualitative variation (Gini index), but a transformed version of it, the coefficient of nominal variation (CNV), is recommended as being better interpretable. We consider the sample version of the CNV and derive its asymptotic distribution both for independent and time series data. The finite-sample performance of this approximation is analyzed in a simulation study. The CNV is also applied to a real-data example.

Appendices

Appendix 1

The NDARMA model has been proposed by [4], and its definition might be given as follows [11]:

Let $(X_t)_{\mathbb {Z}}$ and $(\epsilon _t)_{\mathbb {Z}}$ be nominal processes with state space $\mathcal S$, where $(\epsilon _t)_{\mathbb {Z}}$ is i.i.d. with marginal distribution ${\varvec{p}}$, and where $\epsilon _t$ is independent of $(X_s)_{s<t}$. Let

$$ (\alpha _{t,1},\ldots ,\alpha _{t,{\text {p}}},\beta _{t,0},\ldots ,\beta _{t,{\text {q}}}) \quad \sim \ {\text {MULT}}(1;\ \phi _1, \ldots ,\phi _{{\text {p}}},\varphi _0,\ldots ,\varphi _{{\text {q}}}) $$

be i.i.d. multinomial random vectors, which are independent of $(\epsilon _t)_{\mathbb {Z}}$ and of $(X_s)_{s<t}$. Then $(X_t)_{\mathbb {Z}}$ is said to be an NDARMA(p, q) process (and the cases ${\text {q}}=0$ and ${\text {p}}=0$ are referred to as a DAR(p) process and DMA (q) process, respectively) if it follows the recursion

$$\begin{aligned} X_t\ =\ \alpha _{t,1}\cdot X_{t-1}+\cdots +\alpha _{t,{\text {p}}}\cdot X_{t-{\text {p}}}\ +\ \beta _{t,0}\cdot \epsilon _{t}+\cdots +\beta _{t,{\text {q}}}\cdot \epsilon _{t-{\text {q}}}. \end{aligned}$$

(7)

(Here, if the state space $\mathcal S$ is not numerically coded, we assume $0\cdot s = 0$, $1\cdot s=s$ and $s+0=s$ for each $s\in \mathcal S$.)

NDARMA processes have several attractive properties, e.g., $X_t$ and $\epsilon _{t}$ have the same stationary marginal distribution: $P(X_t=s_i)\,=\,p_i\,=\,P(\epsilon _t=s_i)$ for all $i\in \mathcal S$. Their serial dependence structure is characterized by a set of Yule-Walker-type equations for the serial dependence measure Cohen’s $\kappa$ from (3) [11]:

$$\begin{aligned} \textstyle \kappa (h)\ =\ \sum _{j=1}^{{\text {p}}} \phi _j\, \kappa (|h-j|)\ +\ \sum _{i=0}^{{\text {q}}-h} \varphi _{i+h}\, r(i)\qquad \text {for } h\ge 1, \end{aligned}$$

(8)

where the r(i) satisfy $r(i)\, =\, \sum _{j=\max {\{0,i-{\text {p}}\}}}^{i-1} \phi _{i-j}\cdot r(j)\, +\, \varphi _i\,\mathbbm {1}(0\le i\le {\text {q}})$. The bivariate distributions at lag h, in turn, are $p_{i|j}(h)\, =\, p_i\, +\, \kappa (h)\,(\delta _{i,j}-p_i)$. Consequently, see (4), we always have $\vartheta (h)=\kappa (h)$ for NDARMA processes.

Finally, [9] showed that an NDARMA process is $\phi$-mixing with exponentially decreasing weights such that the CLT on p. 200 in [2] is applicable.

Appendix 2

From Sect. 2, we know that $\widehat{\text {IQV}}$ is asymptotically normally distributed. Furthermore, with $f(y)=1-\sqrt{1-y}$, it holds that $f(\text {IQV})=\text {CNV}$. Since the derivative of f equals $f'(y)=1/\big (2\,\sqrt{1-y}\big )=1/\big (2\,(1-f(y))\big )$, the Delta method immediately implies that the asymptotic variance of $\widehat{\text {CNV}}$ equals $\sigma _{\text {CNV}}^2/n$ with

$$\begin{aligned} \sigma _{\text {CNV}}^2\ =\ \frac{\sigma _{\text {IQV}}^2}{4\,(1-\text {IQV})} \ \overset{(6)}{=}\ c_\vartheta \, \left( \tfrac{m+1}{m}\right) ^2\,\frac{s_3({\varvec{p}}) - s_2^2({\varvec{p}})}{(1-\text {CNV})^2}. \end{aligned}$$

(9)

Note that (9) also implies the relation $\sigma _{\text {IQV}}^2\,=\,4\,(1-\text {CNV})^2\,\sigma _{\text {CNV}}^2$.

To derive an approximate bias correction for $\widehat{\text {CNV}}$, we have to go back to the asymptotic result (5) for $\sqrt{n}\,\big (\hat{{\varvec{p}}}-{\varvec{p}}\big )$. Define $g({\varvec{x}})=\frac{m+1}{m}\,(1-\sum _{i=0}^{m} x_i^2)$ such that $g({\varvec{p}}) = \text {IQV}$ and $f\big (g({\varvec{p}})\big ) = \text {CNV}$. Our approach is to consider the second-order Taylor expansion of $f\big (g({\varvec{x}})\big )$, where $f\big (g({\varvec{x}})\big )$ satisfies

$$ \begin{array}{rl} \frac{\partial }{\partial x_k}\,f\big (g({\varvec{x}})\big )\ =&{} f'\big (g({\varvec{x}})\big )\,\frac{\partial }{\partial x_k}\,g({\varvec{x}}),\\ \frac{\partial ^2}{\partial x_k\,\partial x_l}\,f\big (g({\varvec{x}})\big )\ =&{} f''\big (g({\varvec{x}})\big )\,\big (\frac{\partial }{\partial x_k}\,g({\varvec{x}})\big )\,\big (\frac{\partial }{\partial x_l}\,g({\varvec{x}})\big ) \ +\ f'\big (g({\varvec{x}})\big )\,\frac{\partial ^2}{\partial x_k\,\partial x_l}\,g({\varvec{x}}). \end{array} $$

Here,

$$ \textstyle f''(y)=\frac{1/4}{(1-f(y))^3},\quad \tfrac{\partial }{\partial x_k}\,g({\varvec{x}})=-2\,\frac{m+1}{m}\,x_k,\ \tfrac{\partial ^2}{\partial x_k\,\partial x_l}\,g({\varvec{x}}) = -2\,\frac{m+1}{m}\,\delta _{k,l}. $$

Hence, the matrix $\mathbf{H }$, defined as the Hessian of $f\big (g({\varvec{x}})\big )$ evaluated in ${\varvec{x}}={\varvec{p}}$, has the entries

$$ h_{k,l} \ =\ \frac{(\frac{m+1}{m})^2\,p_k\,p_l}{(1-\text {CNV})^3} \ -\ \frac{\frac{m+1}{m}\,\delta _{k,l}}{1-\text {CNV}}. $$

Since $E[\hat{{\varvec{p}}}-{\varvec{p}}]={\varvec{0}}$, the second-order Taylor expansion implies that $n\,\Big (E\big [f\big (g(\hat{{\varvec{p}}})\big )\big ]-f\big (g({\varvec{p}})\big )\Big )\,\approx \, \frac{1}{2}\,E\big [\sqrt{n}(\hat{{\varvec{p}}}-{\varvec{p}})^\top \,\mathbf{H }\,\sqrt{n}(\hat{{\varvec{p}}}-{\varvec{p}})\big ]$. Thus, using (5), a bias correction for $n\,\widehat{\text {CNV}}$ is computed as

$$ \textstyle \tfrac{1}{2}\sum \limits _{k=0}^{m} h_{k,k}\,\sigma _{k,k} \ +\ \sum \limits _{k=0}^{m-1} \sum \limits _{l=k+1}^{m} h_{k,l}\,\sigma _{k,l} \ =\ \tfrac{1}{2}\sum \limits _{k,l=0}^m\frac{(\frac{m+1}{m})^2\,p_k\,p_l}{(1-\text {CNV})^3}\,\sigma _{k,l} \ -\ \tfrac{1}{2}\sum \limits _{k=0}^{m} \frac{\frac{m+1}{m}}{1-\text {CNV}}\,\sigma _{k,k}. $$

Using (6) and that $\sum _{k=0}^{m} \sigma _{k,k} = c_\kappa \left( 1-s_2({\varvec{p}})\right)$ according to (5), this expression becomes

$$ \tfrac{1}{8}\,\frac{\sigma _{\text {IQV}}^2}{(1-\text {CNV})^3} \ -\ \tfrac{1}{2}\,\frac{c_\kappa \,\text {IQV}}{1-\text {CNV}} \ \overset{(9)}{=}\ \tfrac{1}{2}\,\frac{\sigma _{\text {CNV}}^2}{1-\text {CNV}} \ -\ \tfrac{1}{2}\,\frac{c_\kappa \,\text {CNV}\,(2-\text {CNV})}{1-\text {CNV}}. $$

So the proof of Theorem 1 is complete.