A comment on "New non-parametric inferences for low-income proportions" by Shan Luo and Gengsheng Qin

TL;DR

This paper comments on Luo and Qin's non-parametric methods for low-income proportions, comparing their approach to Zieliński's 2009 method to evaluate differences and implications.

Contribution

It provides a critical comparison between Luo and Qin's approach and Zieliński's 2009 method for non-parametric inference in low-income proportion studies.

Findings

Luo and Qin's method differs from Zieliński's approach in key statistical assumptions.

The comparison highlights strengths and limitations of both methods.

Implications for future research in non-parametric inference are discussed.

Abstract

Shan Luo and Gengsheng Qin published the article "New non-parametric inferences for low-income proportions" Ann Inst Stat Math, 69, 599-626. In the note their approach is compared to Zieli\'nski 2009 approach.

Equations

$${\theta }_{\alpha \beta }=F(\alpha {\xi }_{\beta }).$$

$${\tilde{\theta }}_{\alpha \beta }=\frac{1}{n}\#\{X_i\le \alpha \cdot {\tilde{\xi }}_{\beta }\},$$

$${\tilde{\theta }}_{\alpha \beta }=\frac{1}{n}\max \left\{j:X_{j:n}\le \alpha X_{\lfloor \beta n\rfloor +1:n}\right\},$$

$${\hat{\theta }}_{\alpha \beta }=F_n(\alpha {\hat{\xi }}_{\beta }),$$

$$\left(q{B}^{-1}\left(\eta ,M-\eta +1;\frac{1-\gamma }{2}\right);q{B}^{-1}\left(\eta +1,M-\eta ;\frac{1-\gamma }{2}\right)\right),$$

$$(\ast )$$

Full text

A comment on

“New non-parametric inferences for low-income proportions”

by Shan Luo and Gengsheng Qin

Wojciech Zieli ski

Department of Econometrics and Statistics

Warsaw University of Life Sciences

Nowoursynowska 159, 02-776 Warszawa

e-mail: wojciech_zielinski@sggw.pl

http://wojtek.zielinski.statystyka.info

Abstract. Shan Luo and Gengsheng Qin published the article ”New non-parametric inferences for low-income proportions” Ann Inst Stat Math, 69, 599-626. In the note their approach is compared to Zieli ski 2009 approach.

Key words Low-Income-Proportion, At-Risk-of-Poverty-Rate, nonparametric inference

Low-income proportion (LIP) known also as At-Risk-of-Poverty-Rate (ARPR) is an important index in describing the inequality of an income distribution. It is often used to evaluate the social economic and poverty status of a population. A low-income proportion is defined as the proportion of the population income below a given fraction $\alpha$ ($0<\alpha<1$) of the $\beta$ ($0<\beta<1$) quantile of an income distribution. Let $X\in[0,\infty)$ be an income variable with cumulative distribution function $F(\cdot)$ and denote $\xi_{\beta}$ as the $\beta$ quantile $F^{-1}(\beta)$ of $F$. Then, the $\alpha$ fraction of the $\beta$ quantile $\alpha\xi_{\beta}$ is the income line, and the low-income proportion (At-Risk-of-Poverty-Rate) is

[TABLE]

Let $X_{1},\ldots,X_{n}$ be a sample of disposable incomes of randomly drawn $n$ persons and let $\hat{\xi}_{\beta}$ denote the estimator of $\xi_{\beta}$. The natural estimator $\tilde{\theta}_{\alpha\beta}$ is defined as

[TABLE]

where $\#S$ denotes the cardinality of the set $S$. This estimator may be equivalently written as

[TABLE]

where $X_{1:n}\leq\cdots\leq X_{n:n}$ is the ordered sample and $X_{\lfloor\beta n\rfloor+1:n}$ is the estimator of $\xi_{\beta}$, i.e. $\tilde{\xi}_{\beta}=X_{\lfloor\beta n\rfloor+1:n}$.

Luo and Qin proposed the following estimator of $\theta_{\alpha\beta}$:

[TABLE]

where $F_{n}(\cdot)$ is the empirical distribution function and $\hat{\xi}_{\beta}=F_{n}^{-1}(\beta)$. Note that if $F_{n}$ is the classical empirical distribution function than $(LQ)$ and $(W)$ are the same.

Luo and Qin made an asymptotic consideration. In their approach they firstly estimate the cumulative distribution function $F$ using a kernel estimator and then investigate the properties of the obtained estimator. By making use of the asymptotic normality of $\hat{\theta}_{\alpha\beta}$ the confidence interval for $\theta_{\alpha\beta}$ was constructed.

Zieli ski R. (2006) showed that $\tilde{\theta}_{\alpha\beta}$ is almost unbiased even for small sample sizes. He also calculated its variance. Zieli ski W. (2009) showed that the random variable $\eta=\#\{X_{i}\leq\alpha\cdot\tilde{\xi}_{\beta}\}$ is distributed as (almost) binomial with parameters $M=\lfloor\beta n\rfloor$ and $\theta_{\alpha\beta}/\beta$ (if $F$ is a power distribution function it is exactly binomial). The confidence interval for $\theta_{\alpha\beta}$ (at the confidence level $\gamma$) has the form (Clopper and Pearson 1934)

[TABLE]

where $B^{-1}(a,b;\delta)$ is the $\delta$ quantile of beta distribution with parameters $(a,b)$.

The coverage probability of $(*)$ (for $\alpha=0.5$, different $\beta$ and $\gamma=0.95$) calculated (not simulated) for chi-square $\chi^{2}_{3}$ and lognormal $\log N(0,1)$ distributions are given in Tables 1 and 2, respectively. Results are similar to those obtained in simulations by Luo and Qin.

[TABLE]

[TABLE]

For ARPR defined in Eurostat document (Doc. IPSE/65/04/EN page 11) i.e. for $\alpha=0.6$ and $\beta=0.5$ Zieli ski W. (2009) calculated coverage probabilities for different parent distributions and sample sizes $20$, $200$ and $2000$. Those results show that the coverage probability of $(*)$ is near the nominal confidence level.

The estimator $\tilde{\theta}_{\alpha\beta}$ may be recommended in applications because of its simplicity and good properties in small as well as in large samples. The estimator $\hat{\theta}_{\alpha\beta}$ is more complicated and is applicable only for large sample sizes.

References

Luo, S., Qin, G. (2017): “New non-parametric inferences for low-income proportions,” Ann Inst Stat Math, 69, 599-626, DOI 10.1007/s10463-016-0554-0.

Clopper, C. J., Pearson, E. S. (1934): “The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial,” Biometrika, 26, 404-413.

Zieli ski, R. (2006): “Exact distribution of the natural ARPR estimator in small samples from infinite populations,” Statistics in Transition new series, 7, 881-888.

Zieli ski, W. (2009): “A nonparametric confidence interval for At-Risk-of-Poverty-Rate,” Statistics in Transition new series, 10, 437-444.