Quantile Estimation in a Generalized Asymmetric Distributional Setting

Abstract

Allowing for symmetry in distributions is often a necessity in statistical modelling. This paper studies a broad family of asymmetric densities, which in a regression setting shares basic philosophy with generalized (non)linear models. The main focus, however, for the family of densities studied here is quantile estimation instead of mean estimation. In a similar fashion, a broad family of conditional densities is considered in the regression setting. We discuss estimation of the parameters in the unconditional case, and establish an asymptotic normality result, with explicit expression for the asymptotic variance–covariance matrix. In the regression setting, we allow for flexible modelling and estimate nonparametrically the location and scale functions, leading to semiparametric estimation of conditional quantiles, again in the unifying framework of the considered broad family. The practical use of the proposed methods is illustrated in a real data application on locomotor performance in small and large terrestrial mammals.

Appendix

1.1 A.1 Proof of Theorem 1

If Z is a random variable with asymmetric density $f_{\alpha }(\cdot ; \mu ,\phi )$ in (1), then the cumulative distribution function of Z is given by

$$\begin{aligned} F_{\alpha }(z;\mu ,\phi )=\left\{ \begin{array}{lll} 2\alpha F\big ((1-\alpha ) (\frac{z-\mu }{\phi })\big ) &{} \quad \text {if} &{} \, z \le \mu \\ 2\alpha -1+2(1-\alpha ) F\big (\alpha (\frac{z-\mu }{\phi })\big ) &{} \quad \text {if} &{} \, z > \mu ,\\ \end{array} \right. \end{aligned}$$

(29)

and for any $\beta \in (0,1)$, the $\beta$th-quantile of Z is

$$ F_{\alpha }^{-1}(\beta )=\left\{ \begin{array}{lll} \mu +\frac{\phi }{1-\alpha }F^{-1}\big (\frac{\beta }{2\alpha }\big ) &{} \quad \text {if} &{} \, \beta \le \alpha \\ \mu +\frac{\phi }{\alpha }F^{-1}\Big (\frac{1+\beta -2\alpha }{2(1-\alpha )}\Big )&{} \quad \text {if} &{}\, \beta > \alpha , \end{array} \right. $$

with $F_{\alpha }^{-1}(\alpha )=\mu$. These results are given in Corollary 2.1 of Gijbels et al. [10]. Using Expression (29), we find

$$\begin{aligned} F^g_{\alpha }(y;\eta ,\phi )= & {} \Pr (Y\le y)=\Pr \left( g^{-1}(Z)\le y\right) =\Pr \left( Z\le g(y)\right) \nonumber \\= & {} \left\{ \begin{array}{lll} 2\alpha F\big ((1-\alpha ) (\frac{g(y)-g(\eta )}{\phi })\big ) &{} \quad \text {if} &{} \, y \le \eta \\ 2\alpha -1+2(1-\alpha ) F\big (\alpha (\frac{g(y)-g(\eta )}{\phi })\big ) &{} \quad \text {if} &{} \, y > \eta .\\ \end{array} \right. \end{aligned}$$

From this we then easily obtain (5).

1.2 A.2 Proof of Theorem 2

Theorem 3.4 of Gijbels et al. [10] states that under the assumptions (B1)–(B4), the MLE $\widehat{\varvec{\theta }}^{(\text{ MLE })}_n$ centred with $\varvec{\theta }$ is asymptotically normally distributed with mean vector $\varvec{0}$ and variance–covariance matrix $[\mathcal {I}(\varvec{\theta })]^{-1}$:

$$ \sqrt{n}(\widehat{\varvec{\theta }}^{\text{ MLE }}_n-\varvec{\theta })\xrightarrow {d} {\mathcal {N}_3}(\varvec{0}, \mathcal {I}(\varvec{\theta })^{-1})\qquad \text {as}\quad n \rightarrow \infty , $$

where $\mathcal {I}(\varvec{\theta })$ is the Fisher information matrix given in Proposition 3.2 of Gijbels et al. [10], with inverse

$$\begin{aligned} \mathcal {I}(\varvec{\theta })^{-1}&=\begin{bmatrix} \frac{\gamma _3 \phi ^2}{2\alpha (1-\alpha )(\gamma _1\gamma _3-\gamma _2^2)} &{} \frac{(1-2\alpha )\gamma _2\phi ^2}{2\alpha (1-\alpha )(\gamma _1\gamma _3-\gamma _2^2)} &{} \frac{\gamma _2\phi }{2(\gamma _1\gamma _3-\gamma _2^2)} \\ \frac{(1-2\alpha )\gamma _2\phi ^2}{2\alpha (1-\alpha )(\gamma _1\gamma _3-\gamma _2^2)} &{}[\mathcal {I}(\varvec{\theta })^{-1}]_{22} &{} \frac{(1-2\alpha )\gamma _1\phi }{2(\gamma _1\gamma _3-\gamma _2^2)} \\ \frac{\gamma _2\phi }{2(\gamma _1\gamma _3-\gamma _2^2)} &{} \frac{(1-2\alpha )\gamma _1\phi }{2(\gamma _1\gamma _3-\gamma _2^2)} &{} \frac{\alpha (1-\alpha )\gamma _1}{2(\gamma _1\gamma _3-\gamma _2^2)} \end{bmatrix}, \end{aligned}$$

with $[\mathcal {I}(\varvec{\theta })^{-1}]_{22}= [\mathcal {I}(\varvec{\theta }^g)^{-1}]_{22}$ where the latter quantity is stated in Theorem 2.

We want to find an asymptotic distribution for $\widehat{\varvec{\theta }^g_n}^{\text{ MLE }} =(\widehat{\eta }^{\text{ MLE }}_n,\widehat{\phi }^{\text{ MLE }}_n,\widehat{\alpha }^{\text{ MLE }}_n)^T$, where $\widehat{\eta }^{\text{ MLE }}_n=g^{-1}(\widehat{\mu }_n^{\text{ MLE }})$, which is a function of $\widehat{\varvec{\theta }}^{\text{ MLE }}_n$. Using the multivariate delta method, we obtain

$$ {\displaystyle {{\sqrt{n}}[\widehat{\varvec{\theta }^g_n}^{\text{ MLE }}- \varvec{\theta }^g]\,{\xrightarrow {d}}\,{\mathcal {N}_3}\left( \varvec{0}, \mathcal {I}(\varvec{\theta }^g)^{-1}\right) }}, $$

with $\mathcal {I}(\varvec{\theta }^g)^{-1}$ as given in the statement of Theorem 2. Similarly, the results in (2) can be obtained if $\alpha$ is known.