A short note on quantile and expectile estimation in unequal probability samples 4. From expectiles to the distribution functionA short note on quantile and expectile estimation in unequal probability samples 4. From expectiles to the distribution function

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Both, the quantile function $Q\left(\alpha \right)$  and the expectile function $M\left(\alpha \right)$  uniquely define a distribution function $F\left(\mathrm{.}\right).$  While $Q\left(\alpha \right)$  is just the inversion of $F\left(\mathrm{.}\right)$  the relation between $M\left(\alpha \right)$  and $F\left(\mathrm{.}\right)$  is more complicated. Following Schnabel and Eilers (2009) and Yao and Tong (1996), we have the relation

$$M\left(\alpha \right)\mathrm{<mo>=</mo>}\frac{\left(1-\alpha \right)G\left(M\left(\alpha \right)\right)+\alpha \left\{M\left(0.5\right)-G\left(M\left(\alpha \right)\right)\right\}}{\left(1-\alpha \right)F\left(M\left(\alpha \right)\right)+\alpha \left\{1-F\left(M\left(\alpha \right)\right)\right\}}\mathrm{,}(4.1)$$

where $G\left(m\right)$ is the moment function defined through $G\left(m\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}{Y}_{i}1\left\{Y_i\le m\right\}/N.$ Expression (4.1) gives the unique relation of function $M\left(\alpha \right)$ to the distribution function $F\left(\mathrm{.}\right).$ The idea is now to solve (4.1) for $F\left(\mathrm{.}\right),$ that is to express the distribution $F\left(\mathrm{.}\right)$ in terms of the expectile function $M\left(\mathrm{.}\right).$ Apparently, this is not possible in analytic form but it may be calculated numerically. To do so, we evaluate the fitted function $\widehat{M}\left(\alpha \right)$ at a dense set of values $\mathrm{0\; <mo><</mo>}{\alpha }_1\mathrm{<mo><</mo>}{\alpha }_2\dots \mathrm{<mo><</mo>}{\alpha }_L\mathrm{<mo><</mo>\; 1}$ and denote the fitted values as ${\widehat{m}}_l\mathrm{<mo>=</mo>}\widehat{M}\left({\alpha }_l\right).$ We also define left and right bounds through ${\widehat{m}}_o\mathrm{<mo>=</mo>}{\widehat{m}}_1-c_0$ and ${\widehat{m}}_{L+1}\mathrm{<mo>=</mo>}{\widehat{m}}_L+c_{L+1},$ where $c_0$ and $c_L$ are some constants to be defined by the user. For instance, one may set $c_0\mathrm{<mo>=</mo>}{\widehat{m}}_2-{\widehat{m}}_1$ and $c_{L+1}\mathrm{<mo>=</mo>}{\widehat{m}}_L-{\widehat{m}}_{L-1}.$ By doing so we derive fitted values for the cumulative distribution function $F\left(\mathrm{.}\right)$ at ${\widehat{m}}_l$ which we write as ${\widehat{F}}_l\mathrm{<mo>:=</mo>}\widehat{F}\left({\widehat{m}}_l\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{\delta }}_j$ for non-negative steps ${\widehat{\delta }}_j\ge \mathrm{0,}j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}L$ with ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^L}{\widehat{\delta }}_j\le 1.$ We define ${\widehat{\delta }}_{L+1}\mathrm{<mo>=</mo>\; 1}-{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^L}{\widehat{\delta }}_l$ to make $\widehat{F}\left(\mathrm{.}\right)$ a distribution function. Assuming a uniform distribution between the dense supporting points ${\widehat{m}}_l$ we may express the moment function $G\left(\mathrm{.}\right)$ by simple stepwise integration as

$${\widehat{G}}_l\mathrm{<mo>:=</mo>}\widehat{G}\left({\widehat{m}}_l\right)\mathrm{<mo>=</mo>}{\displaystyle {\int }_{-\infty }^{m_l}}x\mathrm{<mi>\; </mi>}d\widehat{F}\left(x\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{d}}_j{\widehat{\delta }}_l\mathrm{,}$$

where ${\widehat{d}}_j\mathrm{<mo>=</mo>}\left({\widehat{m}}_j-{\widehat{m}}_{j-1}\right)/2$ with the constraint that ${\widehat{G}}_{L+1}\mathrm{<mo>=</mo>}\widehat{M}\left(0.5\right)$ and $\widehat{M}\left(0.5\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^n}\left(y_j/{\pi }_j\right)/{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^n}\left(1/{\pi }_j\right).$ With the steps ${\widehat{\delta }}_l\mathrm{,}l\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}L$ we can now re-express (4.1) as

$${\widehat{m}}_l\mathrm{<mo> = </mo>}\frac{\left(1-\alpha \right){\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{d}}_j{\widehat{\delta }}_j+\alpha \left(\widehat{M}\left(0.5\right)-{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{d}}_j{\widehat{\delta }}_j\right)}{\left(1-\alpha \right){\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{\delta }}_j+\alpha \left(1-{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l}{\widehat{\delta }}_j\right)}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\mathrm{ l}\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}L\mathrm{,}$$

which is then be solved for ${\widehat{\delta }}_1\mathrm{,}\dots \mathrm{,}{\widehat{\delta }}_L.$ This is a numerical exercise which is conceptually relatively straightforward. Details can be found in Schulze Waltrup et al. (2014). Once we have calculated ${\widehat{\delta }}_1\mathrm{,}\dots \mathrm{,}{\widehat{\delta }}_L$ we have an estimate for the cumulative distribution function which is denoted as ${\widehat{F}}_N^M\left(y\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{l\mathrm{<mo>:</mo>}{\widehat{m}}_l\mathrm{<mo><</mo>}y}}{\widehat{\delta }}_l.$ We may also invert ${\widehat{F}}_N^M\left(\mathrm{.}\right)$ which leads to a fitted quantile function which we denote with ${\widehat{Q}}_N^M\left(\alpha \right).$

As Kuk (1988) shows, both theoretically and empirically, ${\widehat{F}}_R\left(\mathrm{.}\right)$ is more efficient than ${\widehat{F}}_N(.).$ We make use of this relationship and apply it to ${\widehat{F}}_N^M(.),$ which yields the estimator

$${\widehat{F}}_R^M\mathrm{<mo>:=</mo>\; 1}-\frac{1}{N}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^n}1/{\pi }_j+\frac{{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^n}1/{\pi }_j}{N}{\widehat{F}}_N^M\mathrm{.}$$

In the next section we compare the quantiles calculated from the expectile based estimator ${\widehat{F}}_R^M$ with quantiles calculated from ${\widehat{F}}_R.$ Note that neither ${\widehat{F}}_R^M$ nor ${\widehat{F}}_R$ are proper distribution functions since they are not normed to take values between 0 and 1.

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