Inference for quantile measures of skewness

Abstract

Given a location-scale family generated by a distribution with smooth positive density, the aim is to provide distribution-free tests and confidence intervals for a skewness coefficient determined by three quantiles. It is the Bowley–Hinkley ratio $S_r/R_r$, where $S_r=x_r+x_{1-r}-2x_{0.5}$ is the sum of two symmetric quantiles minus twice the median, and $R_r=x_{1-r}-x_r$ is the $r$th interquantile range. Here, $0<r< 0.5$ is to be chosen and fixed. The sample version of this ratio depends only on three order statistics and is the basis for tests and confidence intervals. It is shown that the variance stabilized version of this statistic leads to more powerful tests than the Studentized version of the sample version of $S_r$. Sample sizes required to obtain accurate coverage of confidence intervals with a prespecified width are provided.

Appendices

Appendices

1.1 A1: Frequency tables of visits to doctors data, by gender

http://www.unige.ch/ses/dsec/staff/faculty/Cantoni-Eva/Books/RobustBiostat.html

The data are found at the above website. After extracting Chapter5.rdata, loading it into R and employing the following R commands, one obtains the table of data listed below.

There were, for example, 141 females who visited the doctor twice during the observation period. The number of females is 2,079 and the number of males is 789.

1.2 A2: Asymptotic widths of confidence intervals for $\gamma _r$

We now give a somewhat heuristic derivation, corroborated by simulation studies, for the expression (6), (7) for $W=U-L$ and also $W_\text {DF}=U_\text {DF}-L_\text {DF}$ of (9). We assume $W_\text {DF} = W_{\text {DF},n,\alpha }(\hat{\gamma };\hat{a}_0,\hat{a}_1,\hat{a}_2)=W_{n,\alpha }(\hat{\gamma }; ,a_0,a_1,a_2) +o_p(n^{-1/2})$. Let $b=\sinh ^{-1}\{l(\hat{\gamma })/D\}$ and $c=z_{1-\alpha /2}\sqrt{a_2/n}\,$. Then, using (2), (4), and (5) one can write with the help of hyperbolic function relations:

$$\begin{aligned} W&= k^{-1}(De^{b+c})-k^{-1}(De^{b-c}) \nonumber \\&= \frac{D}{2a_2}\left\{ \sinh (b+c)-\sinh (b-c)\right\} \nonumber \\&= \frac{D}{a_2}\left\{ \cosh (b)\sinh (c)\right\} \nonumber \\&= \frac{D}{a_2}\left\{ c\;\cosh (b)+o(c)\right\} \text {\qquad as } c\rightarrow 0 \nonumber \\&= \frac{D}{a_2}\left[ c\;\cosh \left\{ \sinh ^{-1}\left( \frac{l(\hat{\gamma }) }{D}\right) \right\} \right] +o(c) \nonumber \\&= \frac{\sqrt{l^2(\hat{\gamma })+D^2}\,}{\sqrt{a_2}\,}\;\frac{z_{1-\alpha /2}}{\sqrt{n}\,}+o(n^{-1/2}) \text {\qquad as } n\rightarrow \infty \nonumber \\&= \frac{w_\text {asym}(\hat{\gamma })\;z_{1-\alpha /2}}{\sqrt{n}\,}+o(n^{-1/2}). \end{aligned}$$

(13)

Elementary calculations show that $w_{asym}(\gamma )=2\sqrt{q(t)}=2\sqrt{a_0+a_1\gamma +a_2\gamma ^2}\,$ is convex with minimum at $\gamma _\text {min}=-a_1/(2a_2)=\hbox {Cov}[\hat{R}_r,\hat{S}_r]/\hbox {Var}[\hat{R}_r].$ Further,

$$\begin{aligned} w_\text {asym}(\gamma _\text {min})=\frac{2\{(n\hbox {Var}[\hat{S}_r])\, (1-\hbox {Corr}^2[\hat{R}_r,\hat{S}_r])\}^{1/2}\,}{R_r}. \end{aligned}$$

In general, $\gamma _\text {min}\ne S_r/R_r=\gamma _r=\lim \hat{\gamma }_r$, although they are both equal to 0 for symmetric $F$.

The maximum of $w_{asym}(\gamma )$ over $[-1,1]$ is the larger of $2\sqrt{a_0-a_1+a_2}\,$ and $2\sqrt{a_0+a_1+a_2}\,$. A glance at Table 3 shows that these values can be much larger than $w_\text {asym}(\gamma _r)$, but it is the latter quantity that is of interest because of the consistency of $\hat{\gamma }_r$ for $\gamma _r$. It follows from (13) that to obtain a large sample 100($1-\alpha )$ % confidence interval for $\gamma _{r}$ of desired width $W_0$, one requires

$$\begin{aligned} n\ge n_0=n_0(r,\alpha ,W_0)= \biggl \{\frac{\max _F w_\text {asym}(\gamma _r)\;z_{1-\alpha /2}}{W_0}\biggr \}^2. \end{aligned}$$

(14)

Table 6 shows that for the 15 families considered here, and each $r$ ranging from $0.05$ to $0.25$, the $\max _F w_\text {asym}(\gamma _{r})$ occurs for $F$ equal to the Cauchy distribution. Thus, we want the value of $n_0(r )$ for this worst case. Letting $\theta _r=\pi (r-0.5)$, one finds $x_r=\tan (\theta _r)$ and $g_r=\pi /\cos ^{2}(\theta _r),$ so that the constants (8) are

$$\begin{aligned} a_0(r)&= \frac{\pi ^2}{4}\left\{ \frac{1}{\tan ^{2}(\theta _r)}+ \frac{8r}{\sin ^{2}(2\theta _r)}-\frac{4r}{\sin ^{2}(\theta _r)}\right\} \\ a_2(r)&= \frac{2\pi ^2r(1-2r)}{\sin ^{2}(2\theta _r)}.\nonumber \end{aligned}$$

(15)

A plot of $a_0(r)$ against $0<r<0.5$ shows the graph is U-shaped and symmetric about $r=0.25$, with a minimum of $a_0(0.25)=\pi ^2/4.$ The graph of $a_2(r)$ is also U-shaped, symmetric and nearly identical to that of $a_0(r)$ for $r$ near 0.25, but diverges from it as $r$ approaches the boundaries of $[0,0.5].$

Similarly, the graph of $w_\text {asym}(\gamma _r)=2\sqrt{a_0(r)}\,$ is also U-shaped and symmetric about $r=0.25$, but with minimum $\pi .$ Thus, Bowley’s coefficient $\hat{\gamma }_{0.25}$ minimizes over $r$ the maximum over $F$ of the asymptotic interval widths.