Journal of Probability and StatisticsVolume 2012 (2012), Article ID 138450, 18 pagesdoi:10.1155/2012/138450

Research Article

New Bandwidth Selection for Kernel Quantile Estimators

Ali Al-Kenani and Keming Yu

Department of Mathematical Sciences, Brunel University, Uxbridge UBB 3PH, UK

Received 8 August 2011; Revised 26 September 2011; Accepted 10 October 2011

Academic Editor: Junbin B. Gao

Copyright © 2012 Ali Al-Kenani and Keming Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We propose a cross-validation method suitable for smoothing of kernel quantile estimators. In particular, our proposed method selects the bandwidth parameter, which is known to play a crucial role in kernel smoothing, based on unbiased estimation of a mean integrated squared error curve of which the minimising value determines an optimal bandwidth. This method is shown to lead to asymptotically optimal bandwidth choice and we also provide some general theory on the performance of optimal, data-based methods of bandwidth choice. The numerical performances of the proposed methods are compared in simulations, and the new bandwidth selection is demonstrated to work very well.

1. Introduction

The estimation of population quantiles is of great interest when one is not prepared to assume a parametric form for the underlying distribution. In addition, due to their robust nature, quantiles often arise as natural quantities to estimate when the underlying distribution is skewed [1]. Similarly, quantiles often arise in statistical inference as the limits of confidence interval of an unknown quantity.

Let be independent and identically distributed random sample drawn from an absolutely continuous distribution function with density . Further, let denote the corresponding order statistics. For a quantile function is defined as follows: If denotes th sample quantile, then where denotes the integral part of. Because of the variability of individual order statistics, the sample quantiles suffer from lack of efficiency. In order to reduce this variability, different approaches of estimating sample quantiles through weighted order statistics have been proposed. A popular class of these estimators is called kernel quantile estimators. Parzen [2] proposed a version of the kernel quantile estimator as below: From (1.2) one can readily observe that puts most weight on the order statistics , for which is close to. In practice, the following approximation to is often used: Yang [3] proved that and are asymptotically equivalent in terms of mean square errors. Similarly, Falk [4] demonstrates that, from a relative deficiency perspective, the asymptotic performance of is better than that of the empirical sample quantile.

In this paper, we propose a cross-validation method suitable for smoothing of kernel quantile estimators. In particular, our proposed method selects the bandwidth parameter, which is known to play a crucial role in kernel smoothing, based on unbiased estimation of a mean integrated squared error curve of which the minimising value determines an optimal bandwidth. This method is shown to lead to asymptotically optimal bandwidth choice and we also provide some general theory on the performance of optimal, data-based methods of bandwidth choice. The numerical performances of the proposed methods are compared in simulations, and the new bandwidth selection is demonstrated to work very well.

2. Data-Based Selection of the Bandwidth

Bandwidth plays a critical role in the implementation of practical estimation. Specifically, the choice of the smoothing parameter determines the tradeoff between the amount of smoothness obtained and closeness of the estimation to the true distribution [5].

Several data-based methods can be made to find the asymptotically optimal bandwidth in kernel quantile estimators for given by (1.3). One of these methods use derivatives of the quantile density for .

Building on Falk [4], Sheather and Marron [1] gave the MSE of as follows. If is not symmetric or is symmetric but , where , and is the antiderivative of .

If then where , .

There is no single optimal bandwidth minimizing the when is symmetric and. Also, If , we need higher terms and the can be shown to bewhere see Cheng and Sun [6].

In order to obtain we need to estimate and . It follows from (1.3) that the estimator of can be constructed as follows: Jones [7] derived that the as By minimizing (2.5), we obtain the asymptotically optimal bandwidth for : To estimate in (2.2), we employ the known result and it readily follows that which represents the asymptotically optimal bandwidth for . By substituting in (2.4) and in (2.7) we can compute .

3. Cross-Valdation Bandwidth Selection

When measuring the closeness of an estimated and true function the mean integrated squared (MISE) defined as is commonly used as a global measure of performance.

The value which minimises is the optimal smoothing parameter, and it is unknown in practice. The following is the discrete form of error criterion approximating:

The unknown is replaced by and a function of cross-validatory procedure is created as:where denotes the kernel estimator evaluated at observation , but constructed from the data with observation omitted.

The general approach of crossvalidation is to compare each observation with a value predicted by the model based on the remainder of the data. A method for density estimation was proposed by Rudemo [8] and Bowman [9]. This method can be viewed as representing each observation by a Dirac delta function , whose expectation is , and contrasting this with a density estimate based on the remainder of the data. In the context of distribution functions, a natural characterisation of each observation is by the indicator function whose expectation is . This implies that the kernel method for density estimation can be expressed as

when .

The kernel method for distribution function

where is a distribution function, is the bandwidth controls the degree of smoothing. When

where is the indicator function

Now, from (1.3) when and thus a cross-validation function can be written as The smoothing parameter is then chosen to minimise this function. By subtracting a term that characterise the performance of the true we have which does not involve . By expanding the braces and taking expectation, we obtain When the th order statistic is asymptotically normally distributed where the notation with positive subscript denotes a kernel estimator based on a sample size of . The proceeding arguments demonstrate that provides an asymptotic unbiased estimator of the true curve for a sample size . The identity at (3.12) strongly suggests that crossvalidation should perform well.

4. Theoretical Properties

From (3.1), we can write .

Sheather and Marron [1] have shown that while Falk [4, page 263] proved that

On combining the expressions for bias and variance we can express the mean integrated square error as

and for and the MISE can be expressed as Therefore, the asymptotically optimal bandwidth is , where.

We can see from (3.12) that may be a good approximation to or at least to that function evaluated for a sample of size rather than . Additionally, this is true if we adjusted by adding the quantity This quantity is demean and does not depend on which makes it attractive for obtaining a particularly good approximation to .

Theorem 4.1. Suppose that is bounded on and right continuous at the point 0, and that is a compactly supported density and symmetric about 0. Then, for each , with probability, uniformly in , as .

(An outline proof of the above theorem is in the appendix).

From the above theorem, we can conclude that minimisation of produces a bandwidth that is asymptotically equivalent to the bandwidth that minimises .

Corollary 4.2. Suppose that the conditions of previous theorem hold. If denotes the bandwidth that minimises in the range , for any and any , then with probability 1 as .

5. A Simulation Study

A numerical study was conducted to compare the performances of the two bandwidth selection methods. Namely, the method presented by Sheather and Marron [1] and our proposed method.

In order to account for different shapes for our simulation study we consider a standard normal, Exp, Log-normal(0,1) and double exponential distributions and we calculate 18 quantiles ranging from to . Through the numerical study the Gaussian kernel was used as the kernel function. Sample sizes of 100, 200 and 500 were used, with 100 simulations in each case. The performance of the methods was assessed through the mean squared errors criterion (MSE). . And the relative efficiency (R.E)

Further, for comparison purposes we refer to our proposed method and that of Sheather and Marron [1] as method 1 and method 2 respectively.

(a) Standard normal distribution (see Table 1 and Figure 1).

Table 1: Mean squared errors results for bandwidth selection methods for different sample sizes and for data from a normal distribution.

0.05
method 1
0.34841956
0.321073870
0.298936771

method 2
0.29636758
0.164364738
0.090598082

0.10
method 1
0.07645956
0.065440575
0.054697205

method 2
0.04947745
0.022846355
0.015566907

0.15
method 1
0.02291501
0.013920668
0.007384189

method 2
0.02939708
0.013386234
0.005005849

0.20
method 1
0.01891919
0.009273746
0.003152866

method 2
0.02228828
0.010094172
0.003812209

0.25
method 1
0.01596948
0.008581398
0.003000777

method 2
0.01835912
0.008880639
0.003568772

0.30
method 1
0.01614981
0.008035667
0.003208531

method 2
0.01639148
0.008299838
0.003375445

0.35
method 1
0.01461880
0.007677567
0.003534028

method 2
0.01544790
0.007763629
0.003012045

0.40
method 1
0.01279474
0.007375428
0.002899081

method 2
0.01494506
0.007248497
0.002661230

0.45
method 1
0.01224268
0.006128817
0.002183302

method 2
0.01444153
0.006790490
0.002295830

0.55
method 1
0.01414050
0.006348893
0.001922013

method 2
0.01373258
0.006702430
0.002099446

0.60
method 1
0.01375373
0.006392721
0.002007274

method 2
0.01341763
0.006762798
0.002254869

0.65
method 1
0.01344773
0.006063502
0.002589679

method 2
0.01290569
0.006801901
0.002507202

0.70
method 1
0.01320832
0.006394102
0.002456085

method 2
0.01233948
0.007001064
0.002691678

0.75
method 1
0.01503264
0.007011867
0.002789939

method 2
0.01219829
0.007216326
0.002679609

0.80
method 1
0.01604847
0.007246605
0.002715445

method 2
0.01327836
0.007602346
0.002791240

0.85
method 1
0.01757171
0.009239589
0.004770755

method 2
0.01740931
0.009522181
0.003848474

0.90
method 1
0.03192379
0.023292975
0.019942754

method 2
0.03702774
0.018053976
0.012250413

0.95
method 1
0.15323893
0.147773963
0.150811561

method 2
0.24825188
0.146840177
0.092517440

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Left panel: plots of the quantile estimators for method 1 (solid line), method 2 (dotted line), and true quantile (dashed line) for different sample sizes and for data from a normal distribution. Right panel: box plots of mean squared errors for the quantile estimators for method 1 and method 2 for different sample sizes.

(b) Exponential distribution (see Table 2 and Figure 2).

Table 2: Mean squared errors results for bandwidth selection methods for different sample sizes and for data from an exponential distribution.

0.05
method 1
0.001687025
0.0014699990
0.0014107454

method 2
0.0006023236
0.0002476745

0.10
method 1
0.001306211
0.0009229338
0.0007744410

method 2
0.0008225254
0.0004075822

0.15
method 1
0.001589646
0.0008940486
0.0006237375

method 2
0.0012963576
0.0006938287

0.20
method 1
0.002187990
0.0011477063
0.0006801504

method 2
0.0019188172
0.0010358272

0.25
method 1
0.002916417
0.0015805678
0.0008156225

method 2
0.0026838659
0.0014096523

0.30
method 1
0.003827511
0.0019724207
0.0010289166

method 2
0.0036542688
0.0018358956

0.35
method 1
0.004919618
0.0025540323
0.0012720751

method 2
0.0048301657
0.0023318358

0.40
method 1
0.005868113
0.0031932355
0.0016253398

method 2
0.0060092243
0.0028998751

0.45
method 1
0.007267783
0.0039962426
0.0021094081

method 2
0.0072785641
0.0035363816

0.55
method 1
0.011776976
0.0065148222
0.0039208447

method 2
0.0110599156
0.0055548552

0.60
method 1
0.012864521
0.0070366699
0.0026965785

method 2
0.0138585365
0.0070359561

0.65
method 1
0.018173097
0.0086476349
0.0031472559

method 2
0.0169709413
0.0088832263

0.70
method 1
0.021125532
0.0111607501
0.0041235720

method 2
0.0201049720
0.0114703180

0.75
method 1
0.024025836
0.0150785289
0.0057215181

method 2
0.0229763952
0.0149490250

0.80
method 1
0.037367344
0.0204676368
0.0081595071

method 2
0.0407106885
0.0181647976

0.85
method 1
0.057785539
0.0317404871
0.0098128398

method 2
0.0838657681
0.0300656149

0.90
method 1
0.078797379
0.0426418410
0.0152139697

method 2
0.1878456852
0.1117820016

0.95
method 1
0.121239102
0.0810135450
0.0284524316

method 2
0.6668323836
0.4923732684

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Left panel: plots of the quantile estimators for method 1 (solid line), method 2 (dotted line) and true quantile (dashed line) for different sample sizes and for data from an exponential distribution. Right panel: box plots of mean squared errors for the quantile estimators for method 1 and method 2 for different sample sizes.

(c) Log-normal distribution (see Table 3 and Figure 3).

Table 3: Mean squared errors results for bandwidth selection methods for different sample sizes and for data from a Log-normal distribution.

0.05
method 1
0.001663032
0.0010098573
0.0006568989

method 2
0.002384136
0.0007270441
0.0003613541

0.10
method 1
0.001863141
0.0008438333
0.0002915013

method 2
0.002601994
0.0008361475
0.0002981938

0.15
method 1
0.002633153
0.0013492870
0.0004451506

method 2
0.002623552
0.0011943144
0.0003738508

0.20
method 1
0.003753458
0.0019922356
0.0006866399

method 2
0.003107351
0.0014724525
0.0005685022

0.25
method 1
0.004956635
0.0027140878
0.0009886053

method 2
0.004564382
0.0022952079
0.0008557756

0.30
method 1
0.006480195
0.0035603171
0.0015897314

method 2
0.006436967
0.0031574264
0.0011938924

0.35
method 1
0.008858850
0.0047972372
0.0023446072

method 2
0.008443129
0.0038626105
0.0015443970

0.40
method 1
0.010053969
0.0055989143
0.0022496198

method 2
0.010893398
0.0051735721
0.0017579579

0.45
method 1
0.012998940
0.0069058362
0.0030102466

method 2
0.013607931
0.0063606758
0.0019799551

0.55
method 1
0.019687850
0.0115431473
0.0051386226

method 2
0.020581110
0.0100828810
0.0029554466

0.60
method 1
0.023881883
0.0129227902
0.0046644050

method 2
0.025845419
0.0129081138
0.0040301844

0.65
method 1
0.032155537
0.0160476126
0.0056732073

method 2
0.035737008
0.0167147469
0.0056528658

0.70
method 1
0.045027965
0.0249576836
0.0077709058

method 2
0.042681315
0.0223936302
0.0077616346

0.75
method 1
0.060715676
0.0318891176
0.0121926243

method 2
0.059276198
0.0323738749
0.0104119217

0.80
method 1
0.087694754
0.0450814911
0.0165993582

method 2
0.090704630
0.0530374710
0.0168162426

0.85
method 1
0.140537374
0.0840290373
0.0311728395

method 2
0.193857196
0.1131949907
0.0350218855

0.90
method 1
0.289944417
0.1642236062
0.0679038026

method 2
0.552092689
0.2763301818
0.1112433633

0.95
method 1
1.119717137
0.4764026616
0.1984216218

method 2
2.306672668
1.3159008668
0.2217620895

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Left panel: plots of the quantile estimators for method 1 (solid line), method 2 (dotted line) and true quantile (dashed line) for different sample sizes and for data from a Log-normal distribution. Right panel: box plots of mean squared errors for the quantile estimators for method 1 and method 2 for different sample sizes.

(d) Double exponential distribution (see Table 4 and Figure 4).

Table 4: Mean squared errors results for bandwidth selection methods for different sample sizes and for data from a double exponential distribution.

0.05
method 1
0.35372420
0.288207742
0.251339747

method 2
0.45458819
0.315704320
0.051385372

0.10
method 1
0.07123072
0.043684160
0.029307307

method 2
0.14868952
0.097871072
0.023601368

0.15
method 1
0.05081769
0.025358946
0.009241326

method 2
0.09377244
0.035207151
0.010910214

0.20
method 1
0.02489079
0.015360242
0.007647199

method 2
0.04997348
0.024864359
0.008013159

0.25
method 1
0.01863802
0.012204904
0.004401402

method 2
0.03117942
0.019101033
0.006247279

0.30
method 1
0.01869611
0.012031162
0.004145965

method 2
0.02516932
0.014680335
0.004847191

0.35
method 1
0.01562279
0.009560873
0.003235724

method 2
0.02017404
0.011355808
0.003513386

0.40
method 1
0.01430068
0.007860775
0.002493813

method 2
0.01669505
0.009165203
0.002621345

0.45
method 1
0.01386331
0.007587705
0.002485022

method 2
0.01529664
0.008221501
0.002104265

0.55
method 1
0.01501458
0.007801051
0.002013993

method 2
0.01280613
0.007796411
0.002227569

0.60
method 1
0.01712203
0.009076922
0.002233672

method 2
0.01394454
0.009475605
0.002791236

0.65
method 1
0.01946241
0.011129870
0.003521070

method 2
0.01840894
0.012558998
0.003628169

0.70
method 1
0.02098394
0.011997405
0.003255335

method 2
0.02333092
0.015792466
0.004534060

0.75
method 1
0.02791943
0.016885471
0.004419826

method 2
0.02937457
0.019852122
0.005469359

0.80
method 1
0.03532806
0.021319714
0.005471649

method 2
0.04294634
0.024757804
0.007270187

0.85
method 1
0.05463890
0.030489951
0.011338629

method 2
0.08441144
0.035306415
0.012054182

0.90
method 1
0.09188621
0.058587164
0.030485192

method 2
0.14755444
0.083844232
0.024399440

0.95
method 1
0.28184945
0.224432372
0.180645893

method 2
0.51462209
0.319147435
0.076406491

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Left panel: plots of the quantile estimators for method 1 (solid line), method 2 (dotted line) and true quantile (dashed line) for different sample sizes and for data from a double exponential distribution. Right panel: box plots of mean squared error for the quantile estimators for method 1 and method 2 for different sample sizes.

We can compute and summarize the relative efficiency of for the all previous distributions in Table 5.

Table 5: The relative efficiency (R.E) of .

Standard normal dist.
Exponential dist.
Log normal dist.
Double exponential dist.

100
1.037276
2.636250
1.806082
1.520903

200
0.6986324
2.952808
2.096307
1.308667

500
0.4455828
2.324423
1.173547
0.4519134

From Tables 1, 2, 3, and 4, for the all distributions, it can be observed that in 52.3% of cases our method produces lower mean squared errors, slightly wins Sheather-Marron method.

Also, from Table 5 which describes the relative efficiency for we can see more efficient from for all the cases except the standard normal distribution cases with and double exponential distribution cases with .

So, we may conclude that in terms of MISE our bandwidth selection method is more efficient than Sheather-Marron for skewed distributions but not for symmetric distributions.

6. Conclusion

In this paper we have a proposed a cross-validation-based-rule for the selection of bandwidth for quantile functions estimated by kernel procedure. The bandwidth selected by our proposed method is shown to be asymptotically unbiased and in order to assess the numerical performance, we conduct a simulation study and compare it with the bandwidth proposed by Sheather and Marron [1]. Based on the four distributions considered the proposed bandwidth selection appears to provide accurate estimates of quantiles and thus we believe that the new bandwidth selection method is a practically useful method to get bandwidth for the quantile estimator in the form (1.3).

Appendix

Step 1. Let , where

Step 2. With and

Step 3. This step combines Steps 1 and 2 to prove that

Step 4. This step establishes that

Step 5. This step combines Steps 3 and 4, concluding that where Let , for a random variable and a positive sequence

Step 6. This step notes that , where and that, where

Step 7. Shows that .

Step 8. This step combines the results of Steps 5, 6, 7, obtaining

Step 9. This step notes that and

Step 10. This step combines Steps 8 and 9, establishing that This means that

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