Journal of Probability and Statistics

Journal of Probability and Statistics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 138450 | https://doi.org/10.1155/2012/138450

Ali Al-Kenani, Keming Yu, "New Bandwidth Selection for Kernel Quantile Estimators", Journal of Probability and Statistics, vol. 2012, Article ID 138450, 18 pages, 2012. https://doi.org/10.1155/2012/138450

New Bandwidth Selection for Kernel Quantile Estimators

Academic Editor: Junbin B. Gao
Received08 Aug 2011
Revised26 Sep 2011
Accepted10 Oct 2011
Published01 Feb 2012

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 š‘‹1,š‘‹2,ā€¦,š‘‹š‘›be independent and identically distributed random sample drawn from an absolutely continuous distribution function š¹with density š‘“. Further, let š‘‹(1)ā‰¤š‘‹(2)ā‹Æā‰¤š‘‹š‘›denote the corresponding order statistics. For (0<š‘<1) a quantile function š‘„(š‘)is defined as follows: š‘„(š‘)=inf{š‘„āˆ¶š¹(š‘„)ā‰„š‘}.(1.1) If īš‘„(š‘) denotes š‘th sample quantile, then īš‘„(š‘)=š‘„([š‘›š‘]+1) where [š‘›š‘] denotes the integral part ofš‘›š‘. 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: ī‚š‘„š¾(š‘)=š‘›ī“š‘–=1ī‚øī€œš‘–/š‘›š‘–āˆ’1/š‘›š¾ā„Žī‚¹š‘‹(š‘”āˆ’š‘)š‘‘š‘”(š‘–).(1.2) From (1.2) one can readily observe that ī‚š‘„š¾(š‘) puts most weight on the order statistics š‘‹(š‘–), for which š‘–/š‘› is close toš‘. In practice, the following approximation to ī‚š‘„š¾(š‘) is often used:ī‚š‘„š“š¾(š‘)=š‘›ī“š‘–=1ī€ŗš‘›āˆ’1š¾ā„Žī€»š‘‹(š‘–/š‘›āˆ’š‘)(š‘–).(1.3) Yang [3] proved that ī‚š‘„š¾(š‘) and ī‚š‘„š“š¾(š‘)are asymptotically equivalent in terms of mean square errors. Similarly, Falk [4] demonstrates that, from a relative deficiency perspective, the asymptotic performance of ī‚š‘„š“š¾(š‘) 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 ā„Ž in kernel quantile estimators forī‚š‘„š“š¾(š‘) given by (1.3). One of these methods use derivatives of the quantile density for ī‚š‘„š“š¾(š‘).

Building on Falk [4], Sheather and Marron [1] gave the MSE of ī‚š‘„š“š¾(š‘) as follows. If š‘“ is not symmetric or š‘“ is symmetric but š‘ā‰ 0.5, ī‚€ī‚š‘„AMSEš“š¾ī‚=1(š‘)4šœ‡2(š‘˜)2ī‚ƒš‘„ā€²ī…žī‚„(š‘)2ā„Ž4ī€ŗš‘„+š‘(1āˆ’š‘)ī…žī€»(š‘)2š‘›āˆ’1ī€ŗš‘„āˆ’š‘…(š¾)ī…žī€»(š‘)2š‘›āˆ’1ā„Ž,(2.1) where āˆ«š‘…(š¾)=2āˆžāˆ’āˆžš‘¢š¾(š‘¢)š¾āˆ’1(š‘¢)š‘‘š‘¢, šœ‡2āˆ«(š‘˜)=āˆžāˆ’āˆžš‘¢2š¾(š‘¢)š‘‘š‘¢and š¾āˆ’1 is the antiderivative of š¾.

If š‘„ī…ž>0 then ā„Žopt=š›¼(š¾)ā‹…š›½(š‘„)ā‹…š‘›āˆ’1/3,(2.2) where š›¼(š¾)=[š‘…(š¾)/šœ‡2(š‘˜)2]1/3, š›½(š‘„)=[š‘„ā€²(š‘)/š‘„ī…žī…ž(š‘)]2/3.

There is no single optimal bandwidth minimizing the ī‚š‘„AMSE(š“š¾(š‘)) when š¹ is symmetric andš‘=0.5. Also, If š‘ž=0, we need higher terms and the ī‚š‘„AMSE(š“š¾(š‘)) can be shown to beī‚€ī‚š‘„AMSEš“š¾(ī‚=ī‚€1š‘)4āˆ’1š‘›ī‚ā„Ž4ī‚ƒš‘„ā€²ī…ž(ī‚„š‘)2šœ‡2(š‘˜)2+2š‘›āˆ’1ā„Ž2ī€ŗš‘„ī…žī…ž(ī€»š‘)2ī€œ(š‘žāˆ’ā„Žš‘”)š‘”š¾(š‘”)š‘—(š‘”)š‘‘š‘”,(2.3)where āˆ«š‘—(š‘”)=š‘”āˆ’āˆžš‘„š¾(š‘„)š‘‘š‘„,see Cheng and Sun [6].

In order to obtain ā„Žopt we need to estimate š‘„ā€²=š‘ž and š‘„ī…žī…ž=š‘žī…ž. It follows from (1.3) that the estimator of š‘„ī…ž=š‘ž can be constructed as follows:Ģƒš‘žš“š¾ī‚š‘„(š‘)=ī…žš“š¾(š‘)=š‘›ī“š‘–=1š‘‹(š‘–)ī‚øš¾š‘Žī‚µ(š‘–āˆ’1)š‘›ī‚¶āˆ’š‘āˆ’š¾š‘Žī‚€š‘–š‘›ī‚ī‚¹āˆ’š‘.(2.4) Jones [7] derived that the AMSE(Ģƒš‘žš“š¾(š‘)) as ī€·AMSEĢƒš‘žš“š¾ī€ø=š‘Ž(š‘)44šœ‡2(š‘˜)2ī‚ƒš‘žā€²ī…žī‚„(š‘)2+1[]š‘›š‘Žš‘ž(š‘)2ī€œš¾2(š‘¦)š‘‘š‘¦.(2.5) By minimizing (2.5), we obtain the asymptotically optimal bandwidth for ī‚š‘„ī…žš“š¾(š‘): š‘Žāˆ—opt=āŽ”āŽ¢āŽ¢āŽ¢āŽ£ī‚ƒš‘„ā€²ī‚„(š‘)2āˆ«š¾2(š‘¦)š‘‘š‘¦š‘›ī€ŗš‘„ī…žī…žī…žī€»(š‘)2šœ‡2(š‘˜)2āŽ¤āŽ„āŽ„āŽ„āŽ¦1/5.(2.6) To estimate š‘„ī…žī…ž=š‘žī…ž in (2.2), we employ the known result ī‚š‘„ī…žī…žš“š¾š‘‘(š‘)=ī‚š‘„š‘‘š‘ī…žš“š¾1(š‘)=š‘Ž2š‘›ī“š‘–=1š‘‹(š‘–)ī‚øš¾ī…žī‚µ(š‘–āˆ’1)/š‘›āˆ’š‘š‘Žī‚¶āˆ’š¾ī…žī‚µš‘–/š‘›āˆ’š‘š‘Žī‚¶ī‚¹,(2.7) and it readily follows that š‘Žāˆ—āˆ—opt=āŽ”āŽ¢āŽ¢āŽ¢āŽ£3ī‚ƒš‘„ā€²ī‚„(š‘)2āˆ«š¾ī…ž2(š‘„)š‘‘š‘„š‘›ī€ŗš‘„ī…žī…žī…žī…žī€»(š‘)2šœ‡2(š‘˜)2āŽ¤āŽ„āŽ„āŽ„āŽ¦1/7(2.8) which represents the asymptotically optimal bandwidth for ī‚š‘„ī…žī…žš“š¾(š‘). By substituting š‘Ž=š‘Žāˆ—opt in (2.4) and š‘Ž=š‘Žāˆ—āˆ—opt in (2.7) we can compute ā„Žopt.

3. Cross-Valdation Bandwidth Selection

When measuring the closeness of an estimated and true function the mean integrated squared (MISE) defined as ī€œMISE(ā„Ž)=šø10ī‚†ī‚ī‚‡š‘„(š‘)āˆ’š‘„(š‘)2š‘‘š‘(3.1) is commonly used as a global measure of performance.

The value which minimises MISE(ā„Ž) is the optimal smoothing parameter, and it is unknown in practice. The following ASE(ā„Ž) is the discrete form of error criterion approximatingMISE(ā„Ž):1ASE(ā„Ž)=š‘›š‘›ī“š‘–=1ī‚†ī‚š‘„ī‚€š‘–š‘›ī‚ī‚€š‘–āˆ’š‘„š‘›ī‚ī‚‡2.(3.2)

The unknown š‘„(š‘) is replaced by īš‘„(š‘) and a function of cross-validatory procedure is created as:1š‘›š‘›ī“š‘–=1ī‚†ī‚š‘„āˆ’š‘–ī‚€š‘–š‘›ī‚āˆ’īš‘„ī‚€š‘–š‘›ī‚ī‚‡2,(3.3)where ī‚š‘„āˆ’š‘–(š‘–/š‘›) denotes the kernel estimator evaluated at observation š‘„š‘–, but constructed from the data with observation š‘„š‘– 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 š›æ(š‘„āˆ’š‘„š‘–), whose expectation is š‘“(š‘„), 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 š¼(š‘„āˆ’š‘„š‘–)whose expectation is š¹(š‘„). This implies that the kernel method for density estimation can be expressed asī‚1š‘“(š‘„)=š‘›š‘›ī“š‘–=1š¾ā„Žī€·š‘„āˆ’š‘„š‘–ī€ø,(3.4)

when ā„Žā†’0š¾ā„Ž(š‘„āˆ’š‘„š‘–)ā†’š›æ(š‘„āˆ’š‘„š‘–).

The kernel method for distribution functionī‚1š¹(š‘„)=š‘›š‘›ī“š‘–=1š‘Šī‚€š‘„āˆ’š‘„š‘–ā„Žī‚,(3.5)

where š‘Š is a distribution function, ā„Ž is the bandwidth controls the degree of smoothing. When ā„Žā†’0š‘Šī‚€š‘„āˆ’š‘„š‘–ā„Žī‚ī€·āŸ¶š¼š‘„āˆ’š‘„š‘–ī€ø,(3.6)

where š¼(š‘„āˆ’š‘„š‘–)is the indicator functionš¼ī€·š‘„āˆ’š‘„š‘–ī€ø=ī‚»1,ifš‘„āˆ’š‘„š‘–ā‰„0,0,otherwise.(3.7)

Now, from (1.3) when ā„Žā†’0ī‚š‘„AKī‚€š‘–(š‘)āŸ¶š›æš‘›ī‚š‘‹āˆ’š‘(š‘–),(3.8) and thus a cross-validation function can be written as 1CV(ā„Ž)=š‘›š‘›ī“š‘–=1ī€œ10ī‚†š›æī‚€š‘–š‘›ī‚š‘‹āˆ’š‘(š‘–)āˆ’ī‚š‘„āˆ’š‘–ī‚€š‘–š‘›ī‚ī‚‡2š‘‘š‘.(3.9) The smoothing parameter ā„Ž is then chosen to minimise this function. By subtracting a term that characterise the performance of the true (š‘) we have1š»(ā„Ž)=CV(ā„Ž)āˆ’š‘›š‘›ī“š‘–=1ī€œ10ī‚†š›æī‚€š‘–š‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚€š‘–āˆ’š‘„š‘›ī‚ī‚‡2š‘‘š‘(3.10) which does not involve ā„Ž. By expanding the braces and taking expectation, we obtain 1š»(ā„Ž)=š‘›š‘›ī“š‘–=1ī€œ10ī‚†ī‚š‘„2āˆ’š‘–ī‚€š‘–š‘›ī‚ī‚€š‘–āˆ’2š›æš‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚š‘„āˆ’š‘–ī‚€š‘–š‘›ī‚ī‚€š‘–+2š›æš‘›ī‚š‘‹āˆ’š‘(š‘–)š‘„ī‚€š‘–š‘›ī‚āˆ’š‘„2ī‚€š‘–š‘›ī‚ī‚‡š‘‘š‘.(3.11) When š‘›ā†’āˆž the (š‘›š‘)th order statistic š‘„(š‘›š‘) is asymptotically normally distributed š‘„(š‘›š‘)ī‚µāˆ¼ANš‘„(š‘),š‘(1āˆ’š‘)š‘›[]š‘“(š‘„(š‘))2ī‚¶,īƒ¬1šø{š»(ā„Ž)}=šøš‘›š‘›ī“š‘–=1ī€œ10ī‚†ī‚š‘„2āˆ’š‘–ī‚€š‘–š‘›ī‚ī‚€š‘–āˆ’2š›æš‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚š‘„āˆ’š‘–ī‚€š‘–š‘›ī‚ī‚€š‘–+2š›æš‘›ī‚š‘‹āˆ’š‘(š‘–)š‘„ī‚€š‘–š‘›ī‚āˆ’š‘„2ī‚€š‘–š‘›īƒ­,1ī‚ī‚‡š‘‘š‘šø{š»(ā„Ž)}=š‘›š‘›ī“š‘–=1ī€œ10ī‚ƒšøī‚†ī‚š‘„2āˆ’š‘–ī‚€š‘–š‘›ī‚€š‘–ī‚ī‚‡āˆ’2š›æš‘›ī‚š‘„ī‚€š‘–āˆ’š‘š‘›ī‚šøī‚†ī‚š‘„āˆ’š‘–ī‚€š‘–š‘›ī‚€š‘–ī‚ī‚‡+2š›æš‘›ī‚š‘„āˆ’š‘2ī‚€š‘–š‘›ī‚āˆ’š‘„2ī‚€š‘–š‘›ī€œī‚ī‚„š‘‘š‘,šø{š»(ā„Ž)}=šø10ī‚†ī‚š‘„š‘›āˆ’1ī‚€š‘–š‘›ī‚ī‚€š‘–āˆ’š‘„š‘›ī‚ī‚‡2š‘‘š‘,(3.12) where the notation ī‚š‘„š‘›āˆ’1(š‘–/š‘›) with positive subscript denotes a kernel estimator based on a sample size of š‘›āˆ’1. The proceeding arguments demonstrate that CV(ā„Ž)provides an asymptotic unbiased estimator of the true MISE(ā„Ž) curve for a sample size š‘›āˆ’1. The identity at (3.12) strongly suggests that crossvalidation should perform well.

4. Theoretical Properties

From (3.1), we can write āˆ«MISE(ā„Ž)=10bias2(ī‚š‘„š¾āˆ«(š‘))š‘‘š‘+10ī‚š‘„var(š¾(š‘))š‘‘š‘.

Sheather and Marron [1] have shown that ī‚€ī‚š‘„biasš¾ī‚=1(š‘)2ā„Ž2šœ‡2(š‘˜)š‘„ā€²ī…žī€·ā„Ž(š‘)+02ī€ø.(4.1) while Falk [4, page 263] proved thatī‚€ī‚š‘„varš¾ī‚ī‚ƒš‘„(š‘)=š‘(1āˆ’š‘)ā€²ī‚„(š‘)2š‘›āˆ’1ī€ŗš‘„āˆ’š‘…(š¾)ī…žī€»(š‘)2š‘›āˆ’1ī€·š‘›ā„Ž+0āˆ’1ā„Žī€ø.(4.2)

On combining the expressions for bias and variance we can express the mean integrated square error as1MISE(ā„Ž)=4ā„Ž4šœ‡2(š‘˜)2ī€œ10ī‚ƒš‘„ā€²ī…žī‚„(š‘)2ī€œš‘‘š‘+š‘(1āˆ’š‘)10ī€ŗš‘„ī…žī€»(š‘)2š‘‘š‘š‘›āˆ’1ī€œāˆ’š‘…(š¾)10ī€ŗš‘„ī…žī€»(š‘)2š‘‘š‘š‘›āˆ’1ī€·ā„Žā„Ž+04+š‘›āˆ’1ā„Žī€ø,(4.3)

and for š¶1āˆ«=š‘(1āˆ’š‘)10[š‘„ī…ž(š‘)]2š‘‘š‘,š¶2=š‘…(š¾)āˆ«10[š‘„ī…ž(š‘)]2š‘‘š‘ and š¶3=šœ‡2(š‘˜)2āˆ«10[š‘„ī…žī…ž(š‘)]2š‘‘š‘ the MISE can be expressed as MISE(ā„Ž)=š¶1š‘›āˆ’1āˆ’š¶2š‘›āˆ’11ā„Ž+4š¶3ā„Ž4ī€·ā„Ž+04+š‘›āˆ’1ā„Žī€ø.(4.4) Therefore, the asymptotically optimal bandwidth is ā„Ž0=š¶š‘›āˆ’1/3, whereš¶={š¶2/š¶3}1/3.

We can see from (3.12) that š»(ā„Ž) may be a good approximation to MISE(ā„Ž) or at least to that function evaluated for a sample of size š‘›āˆ’1rather than š‘›. Additionally, this is true if we adjusted š»(ā„Ž) by adding the quantityš½š‘›=ī€œ10ī‚»ī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2ī‚€īī‚āˆ’šøš‘„(š‘)āˆ’š‘„(š‘)2ī‚¼.(4.5) This quantity is demean and does not depend on ā„Ž which makes it attractive for obtaining a particularly good approximation to MISE(ā„Ž).

Theorem 4.1. Suppose that š‘„(š‘) is bounded on [0,1] and right continuous at the point 0, and that š¾ is a compactly supported density and symmetric about 0. Then, for each š›æ,šœ€,š¶>0, š»(ā„Ž)+š½=MISE(ā„Ž)+02š‘›ī€½ī€·āˆ’3/2+š‘›āˆ’1ā„Ž3/2+š‘›āˆ’1/2ā„Ž3ī€øš‘›š›æī€¾(4.6) with probability1, uniformly in 0ā‰¤ā„Žā‰¤š¶š‘›š›æ, as š‘›ā†’āˆž.

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

From the above theorem, we can conclude that minimisation of š»(ā„Ž) produces a bandwidth that is asymptotically equivalent to the bandwidth ā„Ž0 that minimises MISE(ā„Ž).

Corollary 4.2. Suppose that the conditions of previous theorem hold. If īā„Ž denotes the bandwidth that minimises CV(ā„Ž) in the range 0ā‰¤ā„Žā‰¤š¶š‘›š›æ, for any š¶>0 and any 0ā‰¤šœ€ā‰¤1/3, then īā„Žā„Ž0āŸ¶1(4.7) with probability 1 as š‘›ā†’āˆž.

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(1), Log-normal(0,1) and double exponential distributions and we calculate 18 quantiles ranging from š‘=0.05 to š‘=0.95. 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). ī‚MSE(ā„Ž)=šø{š‘„(š‘)āˆ’š‘„(š‘)}2. And the relative efficiency (R.E)īƒ¬R.E=MISEMethod2ī€·ā„ŽMethod2,optī€øMISEMethod1ī€·ā„ŽMethod1,optī€øīƒ­.(5.1)

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).


š‘ š‘› = 1 0 0 š‘› = 2 0 0 š‘› = 5 0 0

0.05method 10.348419560.3210738700.298936771
method 20.296367580.1643647380.090598082
0.10method 10.076459560.0654405750.054697205
method 20.049477450.0228463550.015566907
0.15method 10.022915010.0139206680.007384189
method 20.029397080.0133862340.005005849
0.20method 10.018919190.0092737460.003152866
method 20.022288280.0100941720.003812209
0.25method 10.015969480.0085813980.003000777
method 20.018359120.0088806390.003568772
0.30method 10.016149810.0080356670.003208531
method 20.016391480.0082998380.003375445
0.35method 10.014618800.0076775670.003534028
method 20.015447900.0077636290.003012045
0.40method 10.012794740.0073754280.002899081
method 20.014945060.0072484970.002661230
0.45method 10.012242680.0061288170.002183302
method 20.014441530.0067904900.002295830
0.55method 10.014140500.0063488930.001922013
method 20.013732580.0067024300.002099446
0.60method 10.013753730.0063927210.002007274
method 20.013417630.0067627980.002254869
0.65method 10.013447730.0060635020.002589679
method 20.012905690.0068019010.002507202
0.70method 10.013208320.0063941020.002456085
method 20.012339480.0070010640.002691678
0.75method 10.015032640.0070118670.002789939
method 20.012198290.0072163260.002679609
0.80method 10.016048470.0072466050.002715445
method 20.013278360.0076023460.002791240
0.85method 10.017571710.0092395890.004770755
method 20.017409310.0095221810.003848474
0.90method 10.031923790.0232929750.019942754
method 20.037027740.0180539760.012250413
0.95method 10.153238930.1477739630.150811561
method 20.248251880.1468401770.092517440

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


š‘ š‘› = 1 0 0 š‘› = 2 0 0 š‘› = 5 0 0

0.05method 10.0016870250.00146999900.0014107454
method 20.00060232360.0002476745 8 . 1 2 2 8 7 3 š‘’ āˆ’ 0 5
0.10method 10.0013062110.00092293380.0007744410
method 20.00082252540.0004075822 1 . 7 4 9 1 5 0 š‘’ āˆ’ 0 4
0.15method 10.0015896460.00089404860.0006237375
method 20.00129635760.0006938287 3 . 1 8 6 5 9 7 š‘’ āˆ’ 0 4
0.20method 10.0021879900.00114770630.0006801504
method 20.00191881720.0010358272 4 . 7 4 6 9 0 9 š‘’ āˆ’ 0 4
0.25method 10.0029164170.00158056780.0008156225
method 20.00268386590.0014096523 6 . 3 0 3 5 3 8 š‘’ āˆ’ 0 4
0.30method 10.0038275110.00197242070.0010289166
method 20.00365426880.0018358956 7 . 9 4 8 9 4 0 š‘’ āˆ’ 0 4
0.35method 10.0049196180.00255403230.0012720751
method 20.00483016570.0023318358 9 . 7 2 4 7 9 2 š‘’ āˆ’ 0 4
0.40method 10.0058681130.00319323550.0016253398
method 20.00600922430.0028998751 1 . 1 7 0 0 3 8 š‘’ āˆ’ 0 3
0.45method 10.0072677830.00399624260.0021094081
method 20.00727856410.0035363816 1 . 4 1 7 2 6 9 š‘’ āˆ’ 0 3
0.55method 10.0117769760.00651482220.0039208447
method 20.01105991560.0055548552 2 . 1 5 4 1 3 0 š‘’ āˆ’ 0 3
0.60method 10.0128645210.00703666990.0026965785
method 20.01385853650.0070359561 2 . 6 2 6 1 3 7 š‘’ āˆ’ 0 3
0.65method 10.0181730970.00864763490.0031472559
method 20.01697094130.0088832263 3 . 2 5 5 1 1 4 š‘’ āˆ’ 0 3
0.70method 10.0211255320.01116075010.0041235720
method 20.02010497200.0114703180 4 . 2 0 1 7 4 0 š‘’ āˆ’ 0 3
0.75method 10.0240258360.01507852890.0057215181
method 20.02297639520.0149490250 5 . 8 1 2 5 2 6 š‘’ āˆ’ 0 3
0.80method 10.0373673440.02046763680.0081595071
method 20.04071068850.0181647976 8 . 0 2 0 7 8 7 š‘’ āˆ’ 0 3
0.85method 10.0577855390.03174048710.0098128398
method 20.08386576810.0300656149 1 . 1 3 4 8 6 1 š‘’ āˆ’ 0 2
0.90method 10.0787973790.04264184100.0152139697
method 20.18784568520.1117820016 2 . 1 5 6 9 8 7 š‘’ āˆ’ 0 2
0.95method 10.1212391020.08101354500.0284524316
method 20.66683238360.4923732684 1 . 4 7 8 6 7 9 š‘’ āˆ’ 0 1

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


š‘ š‘› = 1 0 0 š‘› = 2 0 0 š‘› = 5 0 0

0.05method 10.0016630320.00100985730.0006568989
method 20.0023841360.00072704410.0003613541
0.10method 10.0018631410.00084383330.0002915013
method 20.0026019940.00083614750.0002981938
0.15method 10.0026331530.00134928700.0004451506
method 20.0026235520.00119431440.0003738508
0.20method 10.0037534580.00199223560.0006866399
method 20.0031073510.00147245250.0005685022
0.25method 10.0049566350.00271408780.0009886053
method 20.0045643820.00229520790.0008557756
0.30method 10.0064801950.00356031710.0015897314
method 20.0064369670.00315742640.0011938924
0.35method 10.0088588500.00479723720.0023446072
method 20.0084431290.00386261050.0015443970
0.40method 10.0100539690.00559891430.0022496198
method 20.0108933980.00517357210.0017579579
0.45method 10.0129989400.00690583620.0030102466
method 20.0136079310.00636067580.0019799551
0.55method 10.0196878500.01154314730.0051386226
method 20.0205811100.01008288100.0029554466
0.60method 10.0238818830.01292279020.0046644050
method 20.0258454190.01290811380.0040301844
0.65method 10.0321555370.01604761260.0056732073
method 20.0357370080.01671474690.0056528658
0.70method 10.0450279650.02495768360.0077709058
method 20.0426813150.02239363020.0077616346
0.75method 10.0607156760.03188911760.0121926243
method 20.0592761980.03237387490.0104119217
0.80method 10.0876947540.04508149110.0165993582
method 20.0907046300.05303747100.0168162426
0.85method 10.1405373740.08402903730.0311728395
method 20.1938571960.11319499070.0350218855
0.90method 10.2899444170.16422360620.0679038026
method 20.5520926890.27633018180.1112433633
0.95method 11.1197171370.47640266160.1984216218
method 22.3066726681.31590086680.2217620895

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


š‘ š‘› = 1 0 0 š‘› = 2 0 0 š‘› = 5 0 0

0.05method 10.353724200.2882077420.251339747
method 20.454588190.3157043200.051385372
0.10method 10.071230720.0436841600.029307307
method 20.148689520.0978710720.023601368
0.15method 10.050817690.0253589460.009241326
method 20.093772440.0352071510.010910214
0.20method 10.024890790.0153602420.007647199
method 20.049973480.0248643590.008013159
0.25method 10.018638020.0122049040.004401402
method 20.031179420.0191010330.006247279
0.30method 10.018696110.0120311620.004145965
method 20.025169320.0146803350.004847191
0.35method 10.015622790.0095608730.003235724
method 20.020174040.0113558080.003513386
0.40method 10.014300680.0078607750.002493813
method 20.016695050.0091652030.002621345
0.45method 10.013863310.0075877050.002485022
method 20.015296640.0082215010.002104265
0.55method 10.015014580.0078010510.002013993
method 20.012806130.0077964110.002227569
0.60method 10.017122030.0090769220.002233672
method 20.013944540.0094756050.002791236
0.65method 10.019462410.0111298700.003521070
method 20.018408940.0125589980.003628169
0.70method 10.020983940.0119974050.003255335
method 20.023330920.0157924660.004534060
0.75method 10.027919430.0168854710.004419826
method 20.029374570.0198521220.005469359
0.80method 10.035328060.0213197140.005471649
method 20.042946340.0247578040.007270187
0.85method 10.054638900.0304899510.011338629
method 20.084411440.0353064150.012054182
0.90method 10.091886210.0585871640.030485192
method 20.147554440.0838442320.024399440
0.95method 10.281849450.2244323720.180645893
method 20.514622090.3191474350.076406491

We can compute and summarize the relative efficiency of ā„ŽMethod1,opt for the all previous distributions in Table 5.


š‘› Standard normal dist.Exponential dist.Log normal dist.Double exponential dist.

100 1.0372762.6362501.8060821.520903
200 0.69863242.9528082.0963071.308667
500 0.44558282.3244231.1735470.4519134

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 ā„ŽMethod1,opt we can see ā„ŽMethod1,opt more efficient from ā„ŽMethod2,opt for all the cases except the standard normal distribution cases with š‘›=200,500 and double exponential distribution cases with š‘›=500.

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 š‘›š»=š‘†1āˆ’2š‘†2, where š‘†1=ī“š‘–ī€œ10ī‚€ī‚š‘„āˆ’š‘–ī‚(š‘)āˆ’š‘„(š‘)2,š‘†2=ī“š‘–ī€œ10ī‚€š›æī‚€š‘–š‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚š‘„āˆ’š‘„(š‘)ī‚ī‚€āˆ’š‘–ī‚(š‘)āˆ’š‘„(š‘).(A.1)

Step 2. With š·š‘–(š‘)=š¾ā„Ž(š‘–/š‘›āˆ’š‘)š‘‹(š‘–)āˆ’š‘„(š‘) and š·0š‘–(š‘)=š›æ(š‘–/š‘›āˆ’š‘)š‘‹(š‘–)āˆ’š‘„(š‘)š‘†1=(š‘›āˆ’1)āˆ’2š‘›2ī€œ(š‘›āˆ’2)10ī‚€ī‚ī‚š‘„(š‘)āˆ’š‘„(š‘)2+(š‘›āˆ’1)š‘›āˆ’2ī“š‘–=1ī€œ10š·2š‘–š‘†(š‘),2=(š‘›āˆ’1)āˆ’1š‘›2ī€œ10ī‚€īī‚ī‚š‘„(š‘)āˆ’š‘„(š‘)ī‚ī‚€š‘„(š‘)āˆ’š‘„(š‘)+(š‘›āˆ’1)š‘›āˆ’1ī“š‘–=1ī€œ10š·š‘–š·0š‘–(š‘).(A.2)

Step 3. This step combines Steps 1 and 2 to prove that ī€½š»=1āˆ’(š‘›āˆ’1)āˆ’2ī€¾ī€œ10ī‚€ī‚ī‚š‘„(š‘)āˆ’š‘„(š‘)2+1š‘›(š‘›āˆ’1)2š‘›ī“š‘–=1ī€œ10š·2š‘–ī€½(š‘)āˆ’21+(š‘›āˆ’1)āˆ’1ī€¾ī€œ10ī‚€īī‚ī‚+2š‘„(š‘)āˆ’š‘„(š‘)ī‚ī‚€š‘„(š‘)āˆ’š‘„(š‘)š‘›(š‘›āˆ’1)š‘›ī“š‘–=1ī€œ10š·š‘–(š‘)š·0š‘–(š‘).(A.3)

Step 4. This step establishes that šøī‚»ī€œ10ī‚€ī‚ī‚š‘„(š‘)āˆ’š‘„(š‘)2ī‚¼2ī‚»ī€œ+šø10ī‚€īī‚ī‚ī‚¼š‘„(š‘)āˆ’š‘„(š‘)ī‚ī‚€š‘„(š‘)āˆ’š‘„(š‘)2ī€·š‘›=0āˆ’2+ā„Ž8ī€ø,šøīƒÆš‘›š‘›āˆ’3ī“š‘–=1ī€œ10š·2š‘–īƒ°(š‘)2īƒ©š‘›+varš‘›āˆ’2ī“š‘–=1ī€œ10š·š‘–š·0š‘–īƒŖ(š‘)2ī€·š‘›=0āˆ’3ī€ø.(A.4)

Step 5. This step combines Steps 3 and 4, concluding that ī€œš»+10ī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2=ī€œ10ī‚€ī‚īī‚š‘„(š‘)āˆ’š‘„(š‘)2+2(š‘›āˆ’1)āˆ’1šœ‡(ā„Ž)+02ī€·š‘›āˆ’3/2+š‘›āˆ’1ā„Ž4ī€ø,(A.5) where āˆ«šœ‡(ā„Ž)=10šø(š·š‘–(š‘)š·0š‘–(š‘)).
Let š‘ˆ=02(šœ‰), for a random variable š‘ˆ=š‘ˆ(š‘›) and a positive sequence šœ‰=šœ‰(š‘›)šøī€·š‘ˆ2ī€øī€·šœ‰=02ī€ø.(A.6)

Step 6. This step notes that āˆ«10(ī‚īš‘„(š‘)āˆ’š‘„(š‘))2=š‘†+š‘‡, where š‘†=š‘›āˆ’2ī“ī“š‘–ā‰ š‘—š‘”ī€·š‘‹š‘–,š‘‹š‘—ī€ø,š‘‡=š‘›š‘›āˆ’2ī“š‘–=1š‘”ī€·š‘‹š‘–,š‘‹š‘–ī€ø,š‘”ī€·š‘‹š‘–,š‘‹š‘—ī€ø=ī€œ10ī‚†š¾ā„Žī‚€š‘–š‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚€š‘–āˆ’š›æš‘›ī‚š‘‹āˆ’š‘(š‘–)ī‚‡ī‚»š¾ā„Žī‚µš‘—š‘›ī‚¶š‘‹āˆ’š‘(š‘—)ī‚µš‘—āˆ’š›æš‘›ī‚¶š‘‹āˆ’š‘(š‘—)ī‚¼š‘‘š‘,(A.7) and thatš‘†=š‘†(1)+š‘†(2)+(1āˆ’š‘›āˆ’1)š‘”0, where š‘†(1)=š‘›āˆ’2ī“ī“š‘–ā‰ š‘—ī€½š‘”ī€·š‘‹š‘–,š‘‹š‘—ī€øāˆ’š‘”1ī€·š‘‹š‘–ī€øāˆ’š‘”1ī€·š‘‹š‘—ī€ø+š‘”0ī€¾,š‘†(2)=2š‘›āˆ’1ī€·1āˆ’š‘›āˆ’1ī€øš‘›ī“š‘–=1ī€½š‘”1ī€·š‘‹š‘–ī€øāˆ’š‘”0ī€¾,š‘”1ī€½š‘”ī€·(š‘„)=šøš‘„,š‘‹1ī€øī€¾,š‘”0ī€½š‘”=šø1ī€·š‘‹1.ī€øī€¾(A.8)

Step 7. Shows that šø{š‘”(š‘‹1,š‘‹1)2}=0(1),šø{š‘”(š‘‹1,š‘‹2)2}=0(ā„Ž3),šø{š‘”1(š‘‹1)2}=0(ā„Ž6)var{š‘‡}=0(š‘›āˆ’3),šø(š‘†(1))2=0(š‘›āˆ’2ā„Ž3)andšø(š‘†(2))2=0(š‘›āˆ’1ā„Ž6).

Step 8. This step combines the results of Steps 5, 6, 7, obtaining ī€œš»+10ī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2ī€·=šø(š‘‡)+1āˆ’š‘›āˆ’1ī€øš‘”0+2(š‘›āˆ’1)āˆ’1šœ‡(ā„Ž)+02ī€·š‘›āˆ’3/2+š‘›āˆ’1ā„Ž3/2+š‘›āˆ’1/2ā„Ž3ī€ø=ī€œ10šøī‚€ī‚īī‚š‘„(š‘)āˆ’š‘„(š‘)2+2(š‘›āˆ’1)āˆ’1šœ‡(ā„Ž)+02ī€·š‘›āˆ’3/2+š‘›āˆ’1ā„Ž3/2+š‘›āˆ’1/2ā„Ž3ī€ø.(A.9)

Step 9. This step notes that šœ‡(ā„Ž)=0(ā„Ž) and ī€œ10šøī‚€ī‚īī‚š‘„(š‘)āˆ’š‘„(š‘)2=ī€œ10šøī‚€ī‚ī‚š‘„(š‘)āˆ’š‘„(š‘)2+ī€œ10šøī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2āˆ’2š‘›āˆ’1šœ‡(ā„Ž).(A.10)

Step 10. This step combines Steps 8 and 9, establishing that ī€œš»+10ī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2āˆ’ī€œ10šøī‚€īī‚š‘„(š‘)āˆ’š‘„(š‘)2=ī€œ10šøī‚€ī‚š‘„ī‚(š‘)āˆ’š‘„(š‘)2+02ī€·š‘›āˆ’3/2+š‘›āˆ’1ā„Ž3/2+š‘›āˆ’1/2ā„Ž3ī€ø.(A.11) This means that šø{š»+š½āˆ’MISE(ā„Ž)}2=02ī€·š‘›āˆ’3/2+š‘›āˆ’1ā„Ž3/2+š‘›āˆ’1/2ā„Ž3ī€ø.(A.12)

References

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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.


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