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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Nonparametric Estimation of Quantile and Quantile Density Function

Yang X1*, Hutson AD1,2 and Wang D3

1Department of Biostatistics, University at Buffalo, Buffalo, NY 14214, USA

2Roswell Park Cancer Institute, Buffalo, NY 14263, USA

3Department of Public Health and Preventive Medicine, SUNY Upstate Medical University, Syracuse, NY 13210, USA

*Corresponding Author:
Yang X
Department of Biostatistics
University at Buffalo
Buffalo, NY 14214-3000, USA
Tel: (716) 845-1300
E-mail: [email protected]

Received date: May 26, 2017; Accepted date: June 16, 2017; Published date: June 20, 2017

Citation: Yang X, Hutson AD, Wang D (2017) Nonparametric Estimation of Quantile and Quantile Density Function. J Biom Biostat 8: 356. doi: 10.4172/2155-6180.1000356

Copyright: © 2017 Yang X, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this article, we derive a new and unique method of estimating quantile and quantile density function, which is based on moments of fractional order statistics. A comparison of the proposed estimators is made with existing popular nonparametric quantile and quantile density estimators, in terms of mean squared error (MSE) for censored and uncensored data. Recommendations for the choice of quantile and/or quantile density estimators are given.

Keywords

Quantile function estimators; Quantile density function estimators; Order statistics; Kernel function

Introduction

The quantile function

Q(u) = inf {x : F(x) ≥ u} , (1)

where F(.) is the cumulative distribution function (CDF) of a continuous random variable X and 0<u<1, is an alternative to the probability density function (PDF), the CDF and the characteristic function for describing a probability distribution. The estimation of Q(u) is of great interest, especially when one is unwilling to assume the distribution as parametric or when the underlying distribution is skewed.

The use of quantile function estimation has been around for decades in exploratory data analysis, statistical analysis, reliability and medical studies [1-11]. A more recent application can be found in Jeong and Fine [12] and Sankaran et al. [13] for competing risk models.

Many nonparametric estimators of the quantile function have been proposed and studied extensively. For uncensored data, the simplest method is the empirical quantile (EQ) estimator based on a single order statistic. It is a piecewise constant function that does not provide a useful quantile density function estimation. Details about advantages of smoothed quantile estimators can be found in Cheng and Parzen [7]. Numerous smoothed quantile function estimators have been introduced. Here, only the most representative ones are outlined. The most commonly used estimator is the linear interpolation of successive order statistics, which is employed in applications, for example, Q-Q plots and popular software packages such as SAS, BMDP, and MINITAB. Parzen [1] developed kernel smoothing of the EQ estimator, which is well known as the kernel quantile estimator. It has been extensively studied and analyzed [3-5,8]. More complete literature reviews on kernel-based quantile estimators can be found in Sheather and Marron [8] and Cheng and Parzen [7]. However, kernel quantile estimators in general are complicated and analytically intractable. Their performance in the sense of MSEs is very sensitive to the choice of bandwidth. In addition, the approximations to kernel estimators may violate the monotonicity requirement as described by Yang [5] and this issue is also shown in our simulation study and the real data application in Section 5 and Section 6. Generalized order statistics were considered as alternatives to sample quantiles by Harrell and Davis [14] and Kaigh and Lachenbruch [15]. Huang [16] proposed a modification of the Harrell-Davis (HD) estimator based on developing a weighting scheme through the use of the level crossing empirical distribution function.

In the presence of right-censored data, the product-limit quantile (PLQ) estimator proposed by Sander [17] and the general kernel smoothing version of PLQ estimator by Padgett [18], which share similar problems as their parallels for uncensored data, have gained the most popularity in the literature. More recently, Wang et al. [19] extended the HD quantile function estimator for censored data and proposed an exact bootstrap procedure for optimization in terms of MSE related criteria.

In the same way that the CDF can be differentiated to give the PDF, Parzen [1] and Jones [20] defined the derivative of Q(u) as the quantile density function. That is, q(u)=Q'(u). Common applications of q(u) include but are not limited to constructing the asymptotic confidence interval of sample quantiles, inference procedures based on linear rank statistics in Hettmansperger [21], and quantile density based approach in the location scale problem, see Eubank [22].

To estimate the quantile density function q(u), given either censored or uncensored data, two main approaches can be applied. One is the mathematical derivative of quantile estimators (if differentiable), the other is the reciprocal of density quantile function f(Q(u)) obtained by differentiating on both sides of the equation F(Q(u))=u. The former way is more advantageous over the latter in terms of efficiency; see Jones [20] for more information pertaining to the comparison of these two methods. In addition, kernel smoothing of the reciprocal of density quantile function has also been considered [23]. As mentioned before, PLQ and EQ estimators fail to provide useful quantile density function estimation. However, the linear interpolation of two successive order statistics is differentiable and the resulting quantile density function estimator is a histogram type estimator by Siddiqui [24]. The derivative of the kernel quantile estimator by Parzen [1] was introduced by Falk [25]. Xiang [26] proposed a natural derivative of the quantile estimator by Padgett [18]. More reviews on quantile density estimators can be found in Cheng and Parzen [7].

In this article we take a new and novel approach to quantile and quantile density function estimation based on estimating moments of fractional order statistics and solving a set of simultaneous equations pertaining to a series of moment expansions. We studied and compared the performance of our estimators with EQ estimator, PLQ estimator, the kernel smoothing of EQ and PLQ estimators, piecewise linear estimator and their corresponding quantile density estimators (if exist) for censored and uncensored data. The competing estimators were considered simply because they are commonly used for quantile and quantile density estimation. The advantages of our method are as follows: First, it does not require a selection of the optimal bandwidth and therefore can be more stable compared to the common kernelbased methods. Second, it at least outperforms the PL and the piecewise linear quantile estimators across all possible simulation parameters we considered in terms of MSEs and also appears to preserve the monotonicity of the quantile function curve. Third, the associated quantile density function estimator is shown to yield the smallest MSE among all quantile density estimators considered for both censored and uncensored data.

In Sections 2 and 3, we outline the existing methods of quantile and quantile density estimation considered in this investigation. In Section 4, our new quantile and quantile density estimators are introduced. The performance of our estimators is illustrated in terms of MSE by a Monte Carlo simulation study in Section 5. This is followed by an application of the switch life data reported by Nair [27] in Section 6. Recommendations of the choice of quantile, and/or quantile density estimators are summarized in Section 7.

Estimation of Q(u)

Let, equation be the order statistics from a random sample with a continuous distribution F(⋅). And letequation be the order statistics corresponding to the i.i.d. sample of randomly right-censored times equation and equation are censoring indicators corresponding to the ordered T(i)’s, respectively. A value of δ(i)=1.

indicates that T(i) is uncensored, while a value of δ(i)=0 indicates that T(i) is censored. The methods described below can be readily applied for uncensored data by setting all δ(i)=1.

Then the well-known PLQ estimator by Sander [17], is defined as

equation (2)

where equation is the common PL estimator of the survival function by Kaplan and Meier [28]:

equation (3)

Defining equation has been studied by Xiang [26] with respect to convergence properties for a class of kernel quantile function estimators, and was shown to provide a more technically suitable definition in term of the large sample theory as compared to setting equation as undefined.

In the absence of censoring, equation reduces to the EQ estimator,

equation (4)

where equation is the empirical distribution,equation

The linear interpolation estimator of the quantile function given uncensored data is denoted as

equation (5)

where equation andequation

Padgett(1986) defined the kernel smoothing version of equation as

equation (6)

where Si is the (3) at T(i) and K(.) is a symmetric kernel function. If no censoring, the estimator in eqn. (6) reduces to the general kernel smoothing of the EQ estimator by Parzen [1],

equation (7)

Estimation of q(u)

Let, equation the first order derivative of equation is:

equation (8)

This is called the spacings of the sample, see Pyke [29,30], or a histogram type estimator by the fact that equation whereequation is the density quantile function estimator based on finite differences introduced by Siddiqui [24].

Again, the PLQ estimator and the EQ estimator do not have the corresponding quantile density estimators. The natural derivative of (6),equation was established by Xiang [26] as

equation (9)

where si denotes the jump of equation , that is,

equation (10)

When no censoring is present, eqn. (9) reduces to the kernel quantile density estimator equation, which is the complementary of Parzen [1] estimator in eqn. (7).

Fractional Order Statistic-Based Quantile and Quantile Density Estimator

In order to develop our new quantile and quantile density function estimators, we need to derive asymptotic expansions corresponding to the kth non-central moment of the fractional order statistic, equation where the fractional order statisticequation for an i.i.d. uniform sample jointly follow a particular Dirichlet process equation see Stigler [31]. Even though fractional order statistics do not exist in the empirical sense, their respective expectations may be calculated.

In deriving the expansion we assume that the first three derivatives of Q are bounded in a neighborhood of u and denote them as equation and equation . We also assume, similar to the results pertaining to the ith order statistics as in Section 3.1 and Section 3.2 of W. R. van Zwet (1964), that the expectationequation of the equation uth order statistic exists for some u and n. Then the asymptotic expansion is as the following:

Theorem: For large samples the first three non-central moments of equation are given by

equation (11)

equation (12)

equation (13)

Proof: First note that equation. Expandingequation aboutequation and taking expectations of each term in the expansion based upon a Beta distribution equation proves the result. Note that the bounds in terms of equation is with respect to the remainder term of the expansion.

We may utilize equations 11-13 to find an approximation for Q(u) as a function o equation = 1,2,3. Hence, we propose:

Definition: The moment quantile estimator, equation is given by solving the set of simultaneous nonlinear equations

equation (14)

equation (15)

equation (16)

at a fixed u in terms of Q(u), Q'(u)and Q''(u)in order to estimate Q(u), where equation is given by the exact bootstrap kth moment estimator of the uth fraction order statistic by Hutson and Ernst [32],

equation (17)

The weight wi is given as

equation (18)

for uncensored data and

equation (19)

for censored data in Wang et al. [19], where equation dt is the incomplete beta function. As an aside, estimates of Q'(u) and Q″(u) are also available as part of this process. In this investigation, we only interested in the performance of the first-order derivative of Q(u), i.e., the quantile density function q(u), and let’s denote the quantile density estimator asequation. In the later simulation, we will show that equation is at least more reliable thanequation, and equation in terms of MSE.

If we are only interested in an estimate for Q(u) then the numerical solution with respect to Q(u) in terms of our system of equations 17- 19 is relatively straightforward and reduces to solving the simple cubic equation

equation (20)

with respect to Q(u), u fixed, where E* denotes the exact bootstrap moment estimator of the quantity at (4.10). Alternatively, we can reformulate (20) and define it as follows:

Definition: The cubic quantile estimator, equation is given by minimizing

equation (21)

with respect to Q(u), where the weights wi are defined at (19) for censored data or at (18) for uncensored data. The associated quantile density function equation can still be estimated by taking the numerical derivative of an interpolated function from cubic quantile estimates at a proper number of u values. As motivated by this, an alternative way of estimating the moment quantile and quantile density functions can be interpolating a quantile function curve from the moment quantile estimates at a proper number of u values and taking the numerical first-order derivative of the interpolated function at interested u points, respectively. To distinguish with equation andequation which are obtained by solving the equation system (14-16) simultaneously, we denote the alternatives as equation and equation. The simulation study in Section 5 shows that equation andequation yield basically the same quantile estimations. Even though equation andequation are different, they are both significantly better than equation, equation and equation in terms of MSE criteria.

Simulation Results

For the purpose of illustrating the behavior of our estimators, a straight forward simulation study was carried out for samples of size n=30, 50, 100 for Weibull distribution with the quantile function, Q(u) = (−log(1− u))θ , θ=0.5,1,1.5 and across the standardized normal, exponential, extreme value, and logistic distributions. The censoring distribution was given as a uniform distribution uniform (0,T) with T=2,5 and uncensored case was also considered. We utilized fixed quantiles of u=0.25, 0.5 and 0.75. For each combination, 2,000 Monte Carlo simulations were utilized.

For censored data, a comparison of equation andequation was made in terms of MSEs in Tables 1-4. For uncensored data, quantile estimators we considered are equation andequation as shown in Tables 5 and 6. MSEs of the corresponding quantile density estimators, if exist, were also summarized in Tables 7-12. Epanechnikov kernel function, equation was utilized here for the considered kernel estimators. Note that even though the triangular kernel is more commonly used for kernel quantile estimators in literature, see Padgett [18], Nair and Sankaran [33], and Soni [23], it fails to provide a useful derivative when calculating the K1(.) in the kernel quantile density estimators. The Epanechnikov kernel, which gives the optimal kernel, see Prakasa Rao [34], was studied by Soni [23] for comparing non-parametric quantile density estimators and our simulation showed that Q(u) estimation behaviors under Epanechnikov kernel and triangular kernel were quite close, which was also confirmed in the study of Soni [23]. For simplicity, only the result of the Epanechnikov kernel was presented here. Bandwidth h for kernel estimators was chosen based on minimizing the bootstrap MSE, in which 300 bootstrap samples with replacement at each value of u for each sample size, distribution and censoring combination were used [35-41].

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 13.247 18.632 36.021 14.756 45.677 173.423 12.858 70.408 571.736
30 equation 5.413 13.528 64.776 16.357 32.939 651.067 10.903 36.465 1200.693
30 equation 11.117 16.527 72.228 14.665 44.066 527.885 14.576 66.225 1309.573
30 equation 11.117 16.527 72.226 14.665 44.066 527.905 14.576 66.225 1309.574
30 equation 9.492 14.319 22.232 13.03 33.949 107.402 13.362 47.344 389.307
50 equation 7.456 11.262 23.24 8.372 30.323 129.169 5.771 43.967 497.303
50 equation 3.247 8.085 28.04 9.741 17.729 360.658 5.642 87.146 1218.629
50 equation 6.433 10.034 24.182 8.021 28.461 318.258 6.907 45.937 1162.492
50 equation 6.433 10.034 24.181 8.021 28.461 318.258 6.907 45.937 1162.487
50 equation 5.697 8.746 16.202 7.177 24.194 54.206 6.263 36.878 240.124
100 equation 3.681 5.323 10.703 3.935 13.744 75.536 2.591 21.057 378.403
100 equation 2.584 5.629 10.762 12.72 11.341 157.495 16.762 16.339 794.296
100 equation 3.198 4.763 9.519 3.61 13.204 138.78 2.739 22.763 787.276
100 equation 3.198 4.763 9.513 3.61 13.204 138.779 2.739 22.763 787.276
100 equation 2.929 4.4 8.781 3.317 11.965 27.973 2.468 20.106 111.144

Table 1: Mean squared error of Q(´1000),T=2.

n   Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 61.4 56.205 80.77 14.972 52.288 164.457 52.086 90.331 246.574 183.639 151.429 336.384
30 equation 38.166 20.288 58.405 12.275 36.824 525.884 39.379 42.463 511.026 168.324 97.933 302.179
30 equation 59.2 65.269 69.11 14.62 46.719 514.84 49.776 75.557 414.817 203.765 175.495 343.174
30 equation 59.2 65.269 69.107 14.62 46.719 514.832 49.776 75.557 414.815 203.765 175.495 343.174
30 equation 47.021 41.931 51.586 12.758 35.976 102.749 40.936 57.167 129.728 154.008 106.64 165.655
50 equation 37.597 33.418 47.666 7.63 29.643 120.266 30.762 50.907 166.823 109.202 85.275 206.402
50 equation 37.597 13.125 113.547 9.81 28.552 332.367 22.934 27.343 246.909 113.405 57.43 175.97
50 equation 34.73 40.438 43.743 7.477 28.994 314.704 27.369 45.607 248.009 112.425 99.965 173.61
50 equation 34.73 40.438 43.743 7.477 28.994 314.699 27.369 45.607 248.009 112.425 99.965 173.61
150 equation 29.669 26.818 35.7 6.778 24.635 55.111 25.406 39.059 67.723 93.358 67.633 85.413
100 equation 18.866 14.112 23.3 3.874 12.969 75.43 15.696 23.431 95.473 53.101 36.898 105.636
100 equation 15.029 7.782 37.868 14.438 11.071 297.886 11.924 56.015 121.979 63.618 40.671 89.408
100 equation 17.434 19.795 21.456 3.554 12.626 140.229 13.324 21.08 96.522 50.908 48.445 71.062
100 equation 17.434 19.795 21.455 3.554 12.626 140.229 13.324 21.08 96.522 50.908 48.445 71.062
1100 equation 15.817 13.582 18.817 3.268 11.485 27.836 12.953 19.473 34.1 45.648 36.278 44.123

Table 2: Mean squared error of Q(´1000),Weibull,T = 5.

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 10.859 13.27 22.34 14.146 37.672 142.768 11.943 64.733 457.705
30 equation 3.597 8.835 17.584 13.142 23.019 120.039 15.396 105.723 324.724
30 equation 9.309 12.372 20.406 13.291 42.257 164.929 13.767 105.495 414.619
30 equation 9.309 12.372 20.408 13.291 42.257 164.929 13.767 105.495 414.619
30 equation 8.127 10.686 17.804 11.777 35.497 124.617 12.512 79.644 278.786
50 equation 6.414 8 13.462 7.714 22.729 79.774 5.831 39.046 282.241
50 equation 2.257 5.733 11.657 6.183 20.284 157.402 5.211 23.261 171.749
50 equation 5.674 7.417 12.188 7.277 23.464 90.408 6.619 48.853 283.646
50 equation 5.674 7.417 12.196 7.277 23.464 90.408 6.619 48.853 283.646
50 equation 5.031 6.714 11.039 6.559 20.486 76.107 5.905 40.317 217.301
100 equation 3.245 4.175 6.825 3.701 11.242 38.589 2.492 17.671 133.052
100 equation 1.329 2.931 5.81 3.637 12.15 35.236 2.394 14.675 254.84
100 equation 2.837 3.85 6.242 3.393 10.937 38.892 2.675 19.171 154.735
100 equation 2.837 3.85 6.239 3.393 10.937 38.892 2.675 19.171 154.735
100 equation 2.62 3.617 5.779 3.127 10.109 34.754 2.385 17.257 128.109

Table 3: Mean squared error of Q(´1000), T = 5.

    Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 61.4 53.732 66.443 14.609 41.123 141.977 51.487 78.876 190.112 183.639 142.001 223.828
30 equation 37.694 16.412 42.86 10.783 21.891 113.932 94.098 31.522 137.074 193.813 57.821 201.164
30 equation 59.191 63.566 66.781 13.215 45.305 156.931 49.248 78.867 199.736 203.257 181.754 241.536
30 equation 59.191 63.566 66.781 13.215 45.305 156.931 49.248 78.867 199.736 203.257 181.754 241.536
30 equation 46.989 40.512 52.121 11.473 36.475 121.961 40.547 60.993 148.648 153.82 109.909 173.994
50 equation 37.597 32.471 39.993 7.47 23.868 80.226 30.748 47.044 100.269 109.202 81.942 127.889
50 equation 35.783 10.17 33.218 6.697 17.278 112.964 23.829 40.997 260.688 72.528 35.711 92.7
50 equation 34.73 40.047 39.113 7.079 23.934 89.937 27.281 41.682 108.988 112.424 98.252 135.912
50 equation 34.73 40.047 39.113 7.079 23.934 89.937 27.281 41.682 108.988 112.424 98.252 135.912
50 equation 29.667 26.336 32.94 6.392 20.717 74.968 25.359 36.81 88.194 93.345 66.871 108.154
100 equation 18.866 14.212 20.272 3.654 11.216 37.26 15.703 21.681 47.944 53.101 36.621 62.186
100 equation 16.904 4.886 25.055 3.358 10.189 54.867 10.007 9.765 38.076 65.78 18.422 57.201
100 equation 17.434 19.655 18.3 3.376 10.709 36.847 13.393 19.592 47.157 50.909 47.86 60.004
100 equation 17.434 19.655 18.3 3.376 10.709 36.848 13.393 19.592 47.156 50.909 47.86 60.004
100 equation 15.817 13.452 16.346 3.099 9.797 33.184 12.935 18.394 41.918 45.647 35.72 52.355

Table 4: Mean squared error of Q(´1000),Weibull,uncensored.

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 9.62 11.277 18.28 11.899 33.147 109.119 9.352 57.895 380.881
30 equation 10.7 12.471 20.634 14.213 36.217 128.025 11.423 61.913 437.653
30 equation 3.091 22.452 33.251 11.977 16.033 91.378 9.72 58.631 516.648
30 equation 8.589 10.724 16.607 12.523 37.139 124.11 12.983 87.72 605.788
30 equation 8.589 10.724 16.611 12.523 37.139 124.11 12.983 87.72 605.788
30 equation 7.489 9.382 14.54 11.086 31.1 97.904 11.767 67.257 418.041
50 equation 5.707 6.859 10.929 6.851 19.865 63.43 4.967 33.457 212.619
50 equation 6.126 7.556 12.106 7.656 22.171 69.311 5.712 36.968 231.98
50 equation 1.947 4.382 8.281 6.367 12.328 49.203 5.31 40.178 231.919
50 equation 5.169 6.47 9.996 6.963 20.852 66.823 6.426 42.851 280.134
50 equation 5.169 6.47 9.993 6.963 20.852 66.823 6.426 42.851 280.134
50 equation 4.633 5.869 9.074 6.245 18.353 56.985 5.653 35.693 217.612
100 equation 2.955 3.553 5.385 3.464 9.998 30.3 2.363 16.108 97.287
100 equation 3.092 3.874 5.97 3.617 10.569 33.835 2.414 17.005 111.129
100 equation 0.939 2.171 4.534 7.287 8.746 21.288 4.971 38.55 64.046
100 equation 2.649 3.354 4.859 3.307 9.873 29.37 2.649 17.553 106.727
100 equation 2.649 3.354 4.859 3.307 9.873 29.37 2.649 17.553 106.727
100 equation 2.458 3.16 4.536 3.046 9.188 26.55 2.351 15.903 91.397

Table 5: Mean squared error of Q(´1000),uncensored.

    Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 60.84 48.796 59.372 12 34.299 107.76 49.799 66.839 146.884 184.702 126.487 186.103
30 equation 61.4 52.766 63.226 14.046 37.151 121.397 51.428 74.166 158.894 183.639 138.679 196.871
30 equation 57.226 15.829 48.744 10.009 16.269 67.007 37.638 26.696 193.783 154.978 53.183 365.382
30 equation 59.183 62.896 59.396 12.596 38.484 126.397 48.815 70.709 160.351 203.093 176.888 207.935
30 equation 59.183 62.896 59.396 12.596 38.484 126.397 48.815 70.709 160.351 203.093 176.888 207.935
30 equation 46.952 40.059 47.837 10.863 31.528 100.705 40.383 56.499 123.693 153.751 108.082 157.518
50 equation 37.025 30.908 36.429 6.674 20.995 61.35 30.17 42.189 83.558 109.424 76.321 107.885
50 equation 37.597 32.098 38.787 7.303 22.73 67.931 30.748 45.133 89.506 109.202 80.965 116.022
50 equation 28.952 9.827 31.515 7.042 26.832 81.404 21.237 16.118 112.673 71.916 31.587 82.484
50 equation 34.731 39.671 34.94 6.875 21.648 64.674 27.272 40.574 85.399 112.424 97.44 109.596
50 equation 34.731 39.671 34.94 6.875 21.648 64.674 27.272 40.574 85.399 112.424 97.44 109.596
50 equation 29.667 26.029 30.048 6.172 18.916 55.55 25.325 36.138 71.398 93.345 66.339 91.895
100 equation 18.621 15.299 18.813 3.403 10.172 30.263 15.356 20.735 41.36 53.474 39.596 55.149
100 equation 18.866 14.595 19.944 3.53 10.749 33.872 15.704 20.715 44.078 53.101 36.926 58.093
100 equation 28.152 4.781 17.073 3.264 4.833 35.748 11.863 7.954 28.928 42.272 15.797 73.102
100 equation 17.433 19.602 17.241 3.266 9.831 29.664 13.381 18.979 39.8 50.907 47.674 52.174
100 equation 17.433 19.602 17.241 3.266 9.831 29.664 13.381 18.979 39.8 50.907 47.674 52.174
100 equation 15.817 13.405 15.562 2.998 9.01 26.708 12.933 17.925 35.485 45.646 35.566 46.98

Table 6: Mean squared error of q(´1000),Weibull, T = 2.

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 8149.28 21114.3 302430 5279.13 81367 10753.5 184243 30426.8 4376781
30 equation 190.06 138.632 1773.02 357.404 416.272 19065.2 554.12 1016.54 60623.1
30 equation 346.405 454.448 7033.58 375.337 1521.34 58892.9 472.144 2854.32 131391
30 equation 140.399 164.01 588.519 233.121 461.381 8392.33 334.335 808.027 35986.6
50 equation 5365.27 9166.75 79146.1 2799.42 31653.6 3.2E+08 8938081 6779.78 33467.5
50 equation 171.934 115.401 546.322 263.807 186.036 16876.7 334.348 589.967 71765.9
50 equation 218.037 234.301 3712.65 246.717 960.017 70666.7 258.313 1914.88 161146
50 equation 107.444 130.532 398.723 169.025 404.488 6448.35 220.841 596.617 32809.8
100 equation 3457.73 5540.73 39824.2 3866.26 76229.6 4700535 1754.15 96355.8 2.5E+07
100 equation 143.995 100.197 214.914 185.4 131.304 14898.8 180.215 272.313 78301
100 equation 138.296 141.534 546.366 149.814 426.399 68805.3 180.709 922.613 225776
100 equation 73.674 83.005 314.818 101.28 290.091 3769.66 110.307 600.577 27189.1

Table 7: Mean squared error of q(´1000), T = 2.

    Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.75 u = 0.25 u = 0.25 u = 0.50 u = 0.75
30 equation 32923.4 8200.43 389215 93756.5 141038 47665.2 136389 68124.8 37944.4 54747.8 278660 786858
30 equation 1253.95 711.916 1832.75 336.029 361.954 18553.9 770.943 820.779 20900.7 14222.9 1631.87 23260.3
30 equation 1504.34 2822.31 4729.17 357.54 1678.35 55976.9 1874.88 2758.74 57398.9 3792.06 7567.05 48062.3
30 equation 814.682 469.436 1176.31 234.564 433.048 8338.58 596.582 662.985 9350.68 3022.53 1230.23 10283.8
50 equation 1.8E+09 8264.46 21800.3 2806.18 9350.96 350462 111190 77323 2.8E+07 602767 177417 1255108
50 equation 1055.82 550.683 1276.25 273.726 176.808 16926.5 605.194 508.001 18993.3 14291.5 1229.24 18865.5
50 equation 960.359 1667.24 2538.14 249.059 742.928 69260 1087.41 1370.22 56219.3 2663.91 4035.12 43938.8
50 equation 594.841 343.731 904.337 168.349 416.507 6608.79 450.908 581.489 6998.28 1939.95 957.065 7112.76
100 equation 33449.9 8325.39 30866.9 5382.12 101981 51028.4 167707 10893.5 526053 434490 143301 618886
100 equation 917.332 365.142 977.522 187.784 125.629 14339 427.641 365.059 16055.1 14354 888.459 16108.1
100 equation 619.263 822.268 1107.62 149.805 392.538 64085.2 536.566 703.808 50992.1 1658.33 1804.27 33753.7
100 equation 375.799 215.821 659.105 100.431 278.377 3788.13 309.846 402.304 3641.83 1199.3 613.156 3578.73

Table 8: Mean squared error of q(´1000),Weibull, T = 5.

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 3254.05 5038.94 6887.1 17748.6 7139.63 866352 20238.5 48852.4 2328631
30 equation 158.016 81.839 213.034 300.251 183.126 7409.52 482.898 513.333 33910.9
30 equation 281.619 245.544 463.94 292.757 782.814 5512.51 381.838 3165.88 24081.6
30 equation 110.409 107.756 341.551 194.985 516.728 2790.31 324.463 1675.97 9230.97
50 equation 3241.09 4910.1 69152.1 33010.5 168218 16064.3 169396 7966.03 431404
50 equation 144.37 83.057 204.728 226.47 145.642 7452.77 302.343 365.374 34197.4
50 equation 182.975 164.569 331.176 203.681 474.967 3657.56 221.098 1538.72 14659.9
50 equation 85.272 83.814 234.323 138.365 338.525 2387 195.897 976.238 6065.82
100 equation 3221.26 5363.8 3759.22 24398.1 73960.8 2697764 1631024 8019.2 58598
100 equation 122.228 73.692 174.536 168.461 111.735 7390.44 164.246 239.365 34434
100 equation 118.142 96.882 225.505 135.451 295.26 1856.85 166.299 714.2 9585.25
100 equation 61.637 56.189 151.94 89.949 197.94 1333.92 100.662 456.847 6206.19

Table 9: Mean squared error of q(´1000),T=5

    Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 38545 8553.6 54397.2 54012.1 7139.28 596465 24249.9 11207.7 500538 57260.4 21867.5 2E+07
30 equation 1253.5 654.914 1272.82 280.413 173.871 7443.92 752.581 592.599 10547.3 14228.1 1636.36 14070.1
30 equation 1495.8 2643.93 1774.43 270.254 773.599 4735.6 1747.54 1974.06 5364.8 3781.01 8183.3 5797.39
30 equation 812.46 417.749 964.279 185.737 500.469 2634.03 576.176 743.603 3127.67 3018.33 1283.02 3744.31
50 equation 2.4E+07 8598.16 240537 392563 5429.96 17970.9 154250 2741767 31780.1 109331 22086.3 280650
50 equation 1055.96 513.579 1116.81 235.759 154.3 7444.73 599.108 453.36 10658.6 14292.3 1137.33 14236.6
50 equation 960.447 1641.38 1224.61 206.582 496.723 3758.39 1059.6 1060.58 4119.6 2650.51 3747.45 4777.37
50 equation 594.953 311.538 721.009 140.666 315.854 2450.21 442.924 478.437 2876.64 1939.01 877.011 3303.95
100 equation 131996 8640.7 29730 1009626 5460.92 20570.2 73217.3 11335.3 74369.3 666840 22209.8 690653
100 equation 917.377 350.549 941.464 164.87 109.83 7394.43 427.406 331.716 10603.5 14354 850.491 14350
100 equation 618.658 818.937 771.579 130.775 281.672 1825.94 528.506 577.48 2267.99 1658.73 1721.42 2664.37
100 equation 375.79 205.667 445.164 88.231 197.073 1307.94 309.365 324.425 1596.78 1199.19 551.284 1823.81

Table 10: Mean squared error of q(´1000),Weibull,uncensored.

    q = 0.5 q = 1 q = 1.5
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 1639.43 1395.49 3123.14 1921.12 4130.41 19918.9 1525.46 7543.69 76720.2
30 equation 3272.54 2705.34 1062.83 359942 5781.29 19822.4 84394.5 283373 52233.3
30 equation 140.88 69.881 191.952 268.893 165.64 7502.06 437.996 474.492 34726.3
30 equation 247.23 214.761 314.455 253.467 574.395 3120.29 331.449 2256.15 22712.6
30 equation 96.881 89.108 243.226 177.204 378.479 2340.34 292.876 1232.07 15177.3
50 equation 1489.12 1381.23 3554.46 1729.66 4091.42 21371.5 1251.51 7266.19 75372.1
50 equation 3248.72 5506.37 4378.57 24746.6 6820.24 59502.6 52402.6 169446 151363
50 equation 129.451 72.218 182.486 210.65 136.673 7440.84 277.427 341.458 34565.5
50 equation 165.322 148.291 250.924 186.508 384.238 2033.78 210.659 1188.87 11177.7
50 equation 74.817 69.331 173.157 125.635 272.133 1428.3 176.52 763.439 7662.2
100 equation 1649.7 1469.24 2995.56 1908.75 4229.72 17156.8 1311.57 7029.24 56393.5
100 equation 3819.37 6093.39 3596.05 15641.8 7435.75 43767.3 21528.8 123924 82623.3
100 equation 110.794 61.735 163.644 158.097 104.196 7383.99 159.322 227.886 34432.2
100 equation 108.645 78.736 170.205 125.585 233.879 1168.04 164.729 571.317 5263.25
100 equation 54.822 46.171 113.761 84.617 159.967 823.231 97.201 379.635 3754.77

Table 11: Mean squared error of q(´1000),uncensored.

    Normal Exponential EVD Logistic
n method u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75 u = 0.25 u = 0.50 u = 0.75
30 equation 10326.1 6195.28 12227.3 1823.09 4070.72 19696.3 8110.1 7892.31 26530.6 31925.3 15847 35215.1
30 equation 22989.5 8603.5 269793 43385.8 5784.64 28342.2 107459 11724.5 34224.9 242847 22600.6 60192.5
30 equation 1253.29 613.463 1212.83 261.043 162.594 7508.28 740.556 550.844 10626.8 14229.4 1528.65 14108.8
30 equation 1494.93 2603.39 1358.93 247.874 583.998 3299.42 1719.6 1652.31 3832 3801.55 6976.98 4136.69
30 equation 811.772 396.356 757.417 168.081 372.244 2302.28 560.056 593.754 2675.48 3014.17 1132.32 3021.49
50 equation 10002.9 6245.06 10720 1909.79 4085.35 17228.1 7830.5 8447.68 25472.6 35739 15406.1 29693.7
50 equation 28853.1 8639.82 250605 21039.6 10344.1 287873 69511.4 11733.7 37261.5 96031.1 22701.2 92945.2
50 equation 1055.99 493.033 1065.56 213.744 139.433 7431.73 598.543 432.766 10609.9 14292.1 1101.03 14126
50 equation 960.538 1613.79 954.132 185.119 398.384 2051.36 1050.6 939.772 2359.12 2651.84 3552.06 2709.39
50 equation 595.079 299.404 555.958 129.118 259.409 1394.92 436.709 421.911 1727.29 1938.53 820.323 1977.23
100 equation 9909.31 6215.52 11217.7 1913.43 4137.78 16641.8 8307.05 8135.47 23534.7 29755.4 16260.1 31476.4
100 equation 25708.3 8660.66 235815 912186 5738.09 21414.9 168318 11737.4 57342 173060 22781.7 76443
100 equation 917.376 346.513 917.305 153.383 102.544 7386.75 427.416 301.712 10600 14354 830.98 14304.1
100 equation 618.594 812.926 646.631 123.076 234.206 1212.79 534.512 512.189 1523.53 1659.02 1678.65 1810.09
100 equation 375.762 202.429 379.963 82.181 160.428 853.959 309.609 288.904 1027.73 1199.03 531.529 1203.32

Table 12: Mean squared error of q(1000),uncensored.

From Tables 1-6, we conclude for nonparametric quantile estimators that:

equation or equation has the smallest MSE in the majority of cases except when u is large. But again, its performance is unstable or sensitive to the choice of h (e.g., A large MSE occurs in Table 1, n=100, θ=1, and u=0.25). In addition, the process of computing the optimal bandwidth based on minimizing MSE can be time-consuming.

When data are heavily censored and skewed, it seems that only equation and equation are good at tails. For example, in Tables 1 and 2, when u=0.75, MSEs of other estimators under Weibull, exponential, extreme value, and logistic distributions are at least twice as large as the MSE of equation. A further discussion on this is mentioned in Section 6.

Among estimators without the bandwidth selection, behaves the best for both censored and uncensored cases. And equation andequation are almost the same given fixed u.

1. And from Tables 7-12, we conclude for nonparametric quantile density estimators that:

equation Produces the smallest MSE in almost all cases.

equation orequation yields the largest MSE and is substantially larger than other quantile density estimators

equation is the second best quantile density estimator in terms of MSE, which also implies that it is better than equation.

Application

A real life test data set with n=40 mechanical switches by Nair [27] was used for the purpose of illustration. The observed data is: T=(1.151*, 1.667, 2.119*, 2.547,1.170*, 1.695, 2.135, 2.548, 1.248*, 1.710, 2.197, 2.738*, 1.331*, 1.955*, 2.199*, 2.794, 1.381*, 1.965, 2.227, 2.883*, 1.499 , 2.012*, 2.250*,2.883*, 1.508*, 2.051*, 2.254, 2.910, 1.543*, 2.076*, 2.261*, 3.015, 1.577*, 2.109, 2.349*, 3.017, 1.584*, 2.116*, 2.369, 3.793*), where * denotes a censored.

Observation: There is 57.5% censoring or 24 censored observations in this dataset. The choice of bandwidth h was borrowed from Padgett (1986) based on minimizing the bootstrap MSE: h=0.28 if 0 < u ≤ 0.25, h=0.34 if 0.25 < u ≤ 0.90, and h=0.40 if 0.90 < u ≤ 1.

equation andequation were not included here since they do not show any additional advantages compared with the moment quantile method and the cubic quantile method in Definition 4.1-2 as shown in the simulation study.

Figures 1 and 2 showed the estimates of quantile and quantile density for this example data. We see from Figure 1 that equation and equation do not differ much except at tails, and furthermore, only equation and equation preserve monotonicity of quantile function curves for this data. These matches with our finding in simulation study that equation and equation can be away from the true quantile function for large values of u when data is heavily censored. Some techniques for correction at tails have already been explored and a brief review can be found in Soni et al. [23]. In addition, as we found in the simulation study,equation performs the best among all quantile density estimators. The fluctuated curve of equation in Figure 2 may also give a little hint about how bad the KPL method may perform in estimating quantile density functions.

biometrics-biostatistics-quantile-estimators-switch

Figure 1: Quantile estimators for switch life data.

biometrics-biostatistics-quantile-density-estimators

Figure 2: Quantile density estimators for switch life data.

Conclusion

In this article, we proposed three types of smooth quantile and quantile density function estimators.equation are better thanequation andequation in terms of MSEs and computational efficiency. But the cubic method, equation performs better than the two moment quantile methods mentioned above for both Q(u)and q(u) estimation, and also shows an obvious advantage in MSE to all the other nonparametric estimators, especially when it comes to quantile density function estimation.

In summary, if one is only interested in quantile estimation, equation orequation with modification at tails if in need, is a good choice. But if one prefers a more stable estimator, equation orequation may also be considered. For the estimation of quantile density function, equation is clearly an optimal choice based on considerations of MSE, smoothness and simplicity. In addition, use equation andequation with care for quantile density estimation since the bias can be extremely large compared to other alternatives.

Acknowledgement

This work was supported by Roswell Park Cancer Institute and National Cancer Institute (NCI) grant P30CA016056 and NRG Oncology Statistical and Data Management Center grant U10CA180822.

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