Nonparametric estimation of quantile versions of the Lorenz curve.

(1)

Agnieszka Magdalena Siedlaczek (Opole)

Nonparametric estimation of quantile versions of the Lorenz curve.

Abstract Estimators of quantile versions of the Lorenz curve are proposed. The pointwise consistency and asymptotic normality of the estimators is proved. The efficiency of the estimators is also studied in simulations.

2010 Mathematics Subject Classification: Primary: 62G05; Secondary: 62P10.

Key words and phrases: Lorenz curve, quantile version of the Lorenz curve, nonparametric estimation.

1. Introduction The Lorenz curve was first introduced by Lorenz in [7].

It measures the degree of inequality in the distribution of some features in a population. It is applied in many fields such as economics, biology, medical sciences, industry, etc.

Definition 1.1 Let X be a random variable with cumulative distribution function (CDF) F and µ = EX < ∞. Lorenz curve of X can be represented by the function

L(p) = 1 µ

Z p 0

F⁻¹(t)dt, where F⁻¹(t) = inf{x : F (x) ≥ t}, and p ∈ [0, 1].

Prendergast and Staudte in [9] redefined the basic concept of the Lorenz curve in terms of quantiles instead of moments. They introduced three quantile versions of the Lorenz curve. These versions are defined for the class of all CDFs F with F (0) = 0, not only for these with the finite expected value.

Denote F⁻¹(p) = xp.

Definition 1.2 Let F be the class of all CDFs F with F (0) = 0. For F ∈ F the quantile versions L₁, L₂, L₃ of the Lorenz curve are defined by functions

L₁(p) = px_p/2 x0.5

, L₂(p) = p x_p/2

x_1−p/2,

(2)

L3(p) = 2p x_p/2

x_p/2+ x_1−p/2 = 2 1/p + 1/L₂(p) for p ∈ [0, 1), and L_i(1) = 1, i = 1, 2, 3.

The article is organized as follows. In Section2the estimators of the quantile versions of the Lorenz curve are proposed. The pointwise consistency and asymptotic normality of the estimators is proved also in Section 2. In Sec- tion3 the estimators proposed are examined through a computer simulation study. Some real data are examined in Section4. Finally, in Section5, a brief discussion of the plug-in estimators of quantile versions of the Lorenz curves is provided.

2. Estimation of quantile versions of the Lorenz curve The plug-in estimators of each of the quantile versions of the Lorenz curve one can obtain by the substitution of the unknown quantiles by quantile estimators. In this section we recall the known quantile estimators based on the distribution function estimators and then we apply them to the estimation of the quantile versions of the Lorenz curve.

Let X₁, . . . , X_nbe i.i.d. non-negative random variables with a CDF F and a quantile function Q = F⁻¹. A traditional nonparametric estimator of the distribution function is the empirical distribution function (EDF) given by formula

Fˆ_n^E(t) := 1 n

n

X

i=1

I_(−∞,t](Xi),

while I_A(x) = 1 if x ∈ A and 0 otherwise. Accordingly, a nonparametric estimator of x_p is the empirical quantile

ˆ

x^E_p,n= X_([np]+1):n, (1)

where [x] denotes the greatest integer not greater than x.

In a widely cited article [3], the authors analysed nine different sample quantile definitions, which are commonly used in statistical packages. Most of them are based on the quantile function estimators ˆQ_nconstructed by linear interpola- tion between the so called plotting positions, i.e., the points p_k, k = 1, . . . , n, for which ˆQn(p_k) = X_k:n, where X1:n, . . . , Xn:n denote the order statistics of the sample X_n. From the nine mentioned estimators, the following three are recommended in literature:

• ˆx^H_p,n proposed by Hazen (1914) in [2], based on the plotting positions p^H_k = k − 1/2

n , (2)

• ˆx^HF_p,n proposed by Johnson and Kotz (1970) in [4] and recommended by Hyndman and Fan in [3], based on the plotting positions

p^HF_k = k − 1/3

n + 1/3, (3)

(3)

• ˆx^{M P}_p,n proposed by Weibull (1939) in [11] and Gumbel(1939) in [1] and recommended by Makkonen and Pajari (2014) in [8], based on the plotting positions

p^{M P}_k = k

n + 1. (4)

Zieliński in [12] has proposed a continuous estimator of a distribution function based on a kernel estimator with random bandwidth. The estimator is constant on some intervals of the real line R, hence the quantile function estimator based on this function is not continuous. Therefore Jokiel-Rokita and Pulit in [5] proposed a continuous and easily invertible estimator ˆF_n^{J P} of the distribution function. The quantile function estimator ˆQ^{J P}_n , based on the estimator ˆF_n^{J P} is of the form

Qˆ^{J P}_n (p) = npX_(k+1):n− X_(k−1):n

2 − (k − 1)X_(k+1):n− X_(k−1):n 2

+X_(k−1):n+ X_k:n

2 ,

if ^k−1_n < p ≤ _n^k for k = 1, . . . , n, and ˆ

x^{J P}_p,n= ˆQ^{J P}_n (p). (5) The ˆx^{J P}_p,n estimator does not satisfy a symmetry property, which is desirable in quantile estimation [3]. For that reason Jokiel-Rokita and Siedlaczek in [6]

proposed a modification ˆF_n^M of the ˆF_n^{J P} estimator which leads to the following estimator

Qˆ^M_n (p) =











X0:n+ D1

Fˆ_n^M(X1:n)p, p ∈ [0, ˆF_n^M(X1:n)], Xk:n+ Dk+1

2[_n^k− ˆF_n^M(X_k:n)][p − ˆF_n^M(Xk:n)], p ∈ [ ˆF_n^M(Xk:n),^k_n], X_k:n+ X_(k+1):n

2 + Dk+1(p −_n^k)

2[ ˆF_n^M(X_(k+1):n) −^k_n], p ∈ [_n^k, ˆF_n^M(X_(k+1):n)], Xn:n+ D_n+1

1 − ˆF_n^M(X_n:n)[p − ˆF_n^M(Xn:n)], p ∈ [ ˆF_n^M(Xn:n), 1], for k = 1, . . . , n − 1, of the quantile function, where D_i = Xi:n− X_(i−1):n, i = 1, . . . , n + 1, X_0:n= 0, X_(n+1):n = X_n:n+^X^n:n^−X₂^(n−1):n = ³₂X_n:n−¹₂X_(n−1):n, and

ˆ

x^M_p,n= ˆQ^M_n (p). (6)

We will denote ˆL^E_i (p), ˆL^H_i (p), ˆL^HF_i (p), ˆL^{M P}_i (p), ˆL^{J P}_i (p) and ˆL^M_i (p), i = 1, 2, 3, the plug-in estimators obtained by the substitution of the unknown

(4)

quantiles by the quantile estimators given by equations (1), (2), (3), (4), (5) and (6), respectively.

Theorem 2.1 (consistency) The proposed estimators ˆL^j_i(p) are consistent for L_i(p) for each p ∈ (0, 1), where i=1,2,3 and j = E, H, HF, M P, J P, M . Proof Consistency of ˆx^E_p is for example proved in [10] (Theorem 2.3.1).

Consistency of other quantile estimators can be concluded from Theorem 3 at [6]. By Slutsky’s theorem we have consistency of L^j_i(p), for each p ∈ (0, 1).

Theorem 2.2 (asymptotic normality) The proposed estimators ˆL^j_i(p) are asymptotically normal for L_i(p) for each p ∈ (0, 1), where i=1,2,3 and j = E, H, HF, M P, J P, M .

Proof Asymptotic normality of ˆx^E_p is for example proved in [10] (Theorem 2.3.3.A). Asymptotic normality of other quantile estimators can be concluded from Theorem 4 at [6]. By Slutsky’s theorem we have asymptotic normality

of L^j_i(p), for each p ∈ (0, 1).

3. Simulation study Simulations are made for samples of size n = 30 from the generalized Pareto distribution with following CDF

F_ξ,σ(t) =











1 −

1 +^ξt_σ

−1/ξ

I_[0,∞)(t) for ξ > 0,

1 −

1 +^ξt_σ−1/ξ

I_[0,−σ/ξ](t) for ξ < 0,

1 − exp −_σ^t I_[0,∞)(t) for ξ = 0,

where σ > 0. We will take into account the scale parameter σ = 1 and the shape parameter ξ ∈ {−1/4, 1/4, 2/3}. The quantile function is

Q_ξ,σ(p) =







σ

ξ (1 − p)^−ξ − 1

for ξ 6= 0,

−σ ln(1 − p) for ξ = 0.

When ξ ≥ 1 the mean is not defined, neither is the Lorenz curve. When ξ < 1 the Lorenz curve is

L_ξ(p) =







1−(1−p)^1−ξ−(1−ξ)p

ξ for ξ 6= 0, ξ < 1, ln(1 − p)(1 − p) + p for ξ = 0.

(5)

The quantile versions of the Lorenz curve are given by

L^GP₁ (p) =





 p⁽¹⁻

p 2)^−ξ−1

(¹₂)^−ξ−1 dla ξ 6= 0, p^{− ln(1−}

p 2)

ln 2 dla ξ = 0,

L^GP₂ (p) =









 p⁽¹⁻

p 2)^−ξ−1

(^p₂)^−ξ−1 dla ξ 6= 0, p^ln(1−

p 2)

ln^p₂ dla ξ = 0,

L^GP₃ (p) =











2p ⁽¹⁻

p 2)^−ξ−1

(1−^p₂)^−ξ−1+(^p₂)^−ξ−1 dla ξ 6= 0, 2p ^ln(1−

p 2)

ln(^p₂−^p2₄ ) dla ξ = 0.

As a measure of error of the estimators ˆLi, i = 1, 2, 3, we take into account the integrated mean squared error IMSE, which is defined as

IM SE = E

Z 1 0

(Li(p) − ˆLi(p))²dp

.

Table 1 presents the estimated IMSEs of the proposed estimators, multiplied by 100 for a clarity of the results, based on 1000 replications.

Table 1: Simulation results for the Pareto distribution

q_E q_H q_HF q_{M P} q_{J P} q_M

ξ = −1/4

L1 0.4046 0.3074 0.3057 0.3046 0.2834 0.2939 L₂ 0.365 0.3072 0.3035 0.2986 0.2795 0.2829 L3 0.4599 0.4256 0.4095 0.38 0.4084 0.4108 ξ = 1/4

L1 0.4189 0.3082 0.306 0.3038 0.2815 0.2924 L2 0.3622 0.2925 0.2884 0.2826 0.2595 0.2632 L₃ 0.4582 0.4154 0.399 0.3693 0.3937 0.397 ξ = 2/3

L₁ 0.51 0.3688 0.3665 0.3644 0.3371 0.3505 L₂ 0.4056 0.3223 0.3188 0.3144 0.2843 0.2879 L3 0.4704 0.4181 0.4031 0.3761 0.3919 0.3958 In the considered cases of the generalized Pareto distribution, while esti- mating ˆL₁(p) and ˆL₂(p), the ˆL^{J P} estimator has the least estimated IMSEs.

In the case of ˆL₃(p), the ˆL^{M P} estimator has the least estimated IMSEs. The worst estimator, in the sense of IMSE, is the ˆL^E.

Let us consider also the Weibull We(ξ, σ) distribution with the following CDF

Fξ,σ(x) = 1 − exp

−x σ

ξ ,

(6)

where x ≥ 0, σ > 0, ξ > 0. In the simulations we take into account the scale parameter σ = 1 and the shape parameter ξ ∈ {1/2, 3/2, 3}. The quantile function of the Weibull We(ξ, σ) distribution is

Q_ξ,σ(p) = σ(ln(1 − p))¹^ξ. The corresponding Lorenz curve is

Lξ(p) = γ(1 + ¹_ξ, − ln(1 − p)) Γ(1 +¹_ξ) ,

where Γ(a) is the gamma function, and γ(a, x) is the incomplete gamma function.

Quantile versions of the Lorenz curve of the Weibull We(ξ, σ) distribution are given by

L1(p) = p^ξ

sln(1 −^p₂) ln(¹₂) , L₂(p) = p^ξ

sln(1 −^p₂) ln(^p₂) ,

L₃(p) = 2p

ξ

q

ln(1 −^p₂)

ξ

q

ln(1 −^p₂) + ^ξ q

ln(^p₂) .

Table2presents the estimated IMSEs for the Weibull distribution, multiplied by 100 for a clarity of the results, based on 1000 replications.

Table 2: Simulation results for the Weibull distribution

q_E q_H q_HF q_{M P} q_{J P} q_M

ξ = 1/2

L₁ 0.7515 0.5357 0.5298 0.5209 0.4784 0.4994 L2 0.5516 0.4156 0.4093 0.399 0.3546 0.3603 L₃ 0.494 0.4043 0.3996 0.3931 0.3607 0.3658 ξ = 3/2

L1 0.291 0.2215 0.2213 0.2226 0.2063 0.2132 L₂ 0.2603 0.222 0.2207 0.2204 0.2034 0.2059 L3 0.1728 0.153 0.1534 0.1564 0.1429 0.1446 ξ = 3

L1 0.1117 0.0855 0.086 0.0882 0.0802 0.0827 L2 0.1218 0.1069 0.1069 0.1083 0.0994 0.1005 L₃ 0.0594 0.0531 0.0536 0.0558 0.0499 0.0504 The ˆL^{J P} estimator has the least estimated IMSE for all considered cases of the Weibull distribution. The worst estimator, in the sense of IMSE, is again the ˆL^E.

4. Real data analysis We applied our results to a medical problem of investigating the effectiveness of treatment. We used data from Jokiel-

(7)

Rokita and Pulit [5]. The data concerns the effectiveness of the combined treatment of interferon alpha and metronomic cyclophosphamide in patients with metastatic kidney cancer. There were 31 patients, while we consider hemoglobin level for those, who had clinical response observed at 24-th week of treatment (17 patients), i.e. 7.2, 8.6, 9.1, 9.5, 10.9, 10.9, 11.1, 11.5, 11.7, 11.9, 11.9, 12.7, 12.9, 13.9, 14.1, 14.5, 14.7. The estimated quantile versions of the Lorenz curve for these data are illustrated in Figure1. We present only curves for the best estimator ˆL^{J P}, in the context of IMSE, and in the most cases considered, and the worst empirical estimator ˆL^E, for comparison.

0.0 0.2 0.4 0.6 0.8 1.0

0.00.20.40.60.81.0

p LE

LJP

(a) L1

0.0 0.2 0.4 0.6 0.8 1.0

0.00.20.40.60.81.0

p LE

LJP

(b) L2

0.0 0.2 0.4 0.6 0.8 1.0

0.00.20.40.60.81.0

p LE

LJP

(c) L3

Figure 1: Estimators of the quantile versions of the Lorenz curve of the real data

5. Conclusion and further remarks New estimators of the quantile versions of the Lorenz curve have been proposed. The consistency and asymptotic normality of the estimators were proved. In the simulation, the accuracy, measured by the integrated mean squared error, of the plug-in estimators of the quantile version of the Lorenz curve was investigated when the small samples were generated from the Weibull and generalized Pareto distributions.

The ˆL^{J P} and ˆL^{M P} estimators have the lowest estimated IMSEs. The real data analysis showed the great difference between the empirical estimator and estimator ˆL^{J P}.

The considered estimators of the quantile versions of the Lorenz curve can be applied to the estimation of measures of inequalities in distributions such as the quantile versions of the Gini index.

References

[1] E. Gumbel. La probabilité des hypothèses. C. R. Acad. Sci., Paris, 209:

645–647, 1939. ISSN 0001-4036. Zbl 0022.37203. Cited on p.151.

[2] A. Hazen. Storage to be provided in impounding reservoirs for municipal water supply. Transactions of the American Society of Civil Engineers, 77:1539–1669, 1914. URL https://catalog.hathitrust.org/Record/

000506421. Cited on p.150.

(8)

[3] R. J. Hyndman and Y. Fan. Sample quantiles in statistical packages. The American Statistician, 50(4):361–365, 1996. ISSN 00031305.

doi: 10.2307/2684934. URL http://www.jstor.org/stable/2684934.

Cited on pp. 150 and 151.

[4] N. L. Johnson and S. Kotz. Distributions in statistics. Continuous uni- variate distributions. 2. Houghton Mifflin Co., Boston, Mass., 1970. MR 0270476. Cited on p.150.

[5] A. Jokiel-Rokita and M. Pulit. Nonparametric estimation of the ROC curve based on smoothed empirical distribution functions. Stat. Comput., 23(6):703–712, 2013. ISSN 0960-3174; 1573-1375/e. doi: 10.1007/s11222- 012-9340-x. Zbl 1322.62122. Cited on pp.151 and 155.

[6] A. Jokiel-Rokita and A. M. Siedlaczek. Quantile estimation via distribution fitting. Technical report, Politechnika Wrocławska, 2017. Cited on pp. 151 and 152.

[7] M. O. Lorenz. Methods of measuring the concentration of wealth. Pub- lications of the American Statistical Association, 9(70):209–219, 1905.

ISSN 15225437. doi: 10.2307/2276207. URL http://www.jstor.org/

stable/2276207. Cited on p.149.

[8] L. Makkonen and M. Pajari. Defining sample quantiles by the true rank probability. J. Probab. Stat., pages Art. ID 326579, 6, 2014. ISSN 1687- 952X. doi: 10.1155/2014/326579. URL https://doi.org/10.1155/

2014/326579. MR 3293006. Cited on p. 151.

[9] L. A. Prendergast and R. G. Staudte. Quantile versions of the Lorenz curve. Electron. J. Stat., 10(2):1896–1926, 2016. ISSN 1935-7524. doi:

10.1214/16-EJS1154. URL https://doi.org/10.1214/16-EJS1154.

Cited on p. 149.

[10] R. J. Serfling. Approximation theorems of mathematical statistics. John Wiley & Sons, Inc., New York, 1980. ISBN 0-471-02403-1. Wiley Series in Probability and Mathematical Statistics. Cited on p. 152.

[11] W. Weibull. The phenomenon of rupture in solids., volume 153 of Ingeniörsvetenskapsakademiens handlingar. Generalstabens litografiska anstalts förlag, Stokholm, 1939. URLhttp://www.barringer1.com/wa_

files/The-Phenomenon-Of-Rupture-In-Solids-Weibull-1939.pdf.

Cited on p. 151.

[12] R. Zieliński. Kernel estimators and the Dvoretzky-Kiefer-Wolfowitz inequality. Appl. Math., 34(4):401–404, 2007. ISSN 1233-7234; 1730- 6280/e. doi: 10.4064/am34-4-3. Zbl 1130.62031. Cited on p.151.

(9)

Nieparametryczna estymacja kwantylowych wersji krzywych Lorenza.

Agnieszka Magdalena Siedlaczek

Streszczenie Zaproponowano nieparametryczne estymatory kwantylowych wersji krzywej Lorenza. Udowodniono punktową zgodność i asymptotyczną normalność zaproponowanych estymatorów. Porównano średnie scałkowane błędy kwadratowe wybranych nieparametrycznych estymatorów kwantylowych wersji krzywej Lorenza przy użyciu symulacji komputerowych.

Klasyfikacja tematyczna AMS (2010): Primary: 62G05; Secondary: 62P10.

Słowa kluczowe: krzywa Lorenza; kwantylowe wersje krzywej Lorenza; estymacja nieparametryczna.

Agnieszka Magdalena Siedlaczek is a PhD student in Mathemat- ics at the University of Opole. She received MSc in Mathematics in the specialization of mathematical modeling and data analysis in 2016, a bachelor degree in Mathematics in 2014 in the specialization of financial mathematics and engineer in Computer Science in 2016. During studies she has been receiving a schol- arship for the best students. In 2017 she got the Fidelius prize for young mathematicians for the best paper presented at the XLVI National Conference on the Applications of Mathematics.

Her current scientific interests are estimation of the Lorenz curve, estimation of quantiles and statistics in medicine in general.

Agnieszka Magdalena Siedlaczek University of Opole

Institute of Mathematics and Computer Science ul. Oleska 48, 45-052 Opole

E-mail: asiedlaczek@uni.opole.pl Communicated by: Urszula Foryś

(Received: 22nd of February 2018; revised: 10th of March 2018)