Quantile Function

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P G Sankaran - One of the best experts on this subject based on the ideXlab platform.

Alan D. Hutson - One of the best experts on this subject based on the ideXlab platform.

  • An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage
    Communications in statistics: theory and methods, 2014
    Co-Authors: Lai Wei, Dongliang Wang, Alan D. Hutson
    Abstract:

    In this article, we investigate the limitations of traditional Quantile Function estimators and introduce a new class of Quantile Function estimators, namely, the semi-parametric tail-extrapolated Quantile estimators, which has excellent performance for estimating the extreme tails with finite sample sizes. The smoothed bootstrap and direct density estimation via the characteristic Function methods are developed for the estimation of confidence intervals. Through a comprehensive simulation study to compare the confidence interval estimations of various Quantile estimators, we discuss the preferred Quantile estimator in conjunction with the confidence interval estimation method to use under different circumstances. Data examples are given to illustrate the superiority of the semi-parametric tail-extrapolated Quantile estimators. The new class of Quantile estimators is obtained by slight modification of traditional Quantile estimators, and therefore, should be specifically appealing to researchers in estima...

  • A fractional order statistic towards defining a smooth Quantile Function for discrete data
    Journal of Statistical Planning and Inference, 2011
    Co-Authors: Dongliang Wang, Alan D. Hutson
    Abstract:

    Abstract This work is motivated in part by a recent publication by Ma et al. (2011) who resolved the asymptotic non-normality problem of the classical sample Quantiles for discrete data through defining a new mid-distribution based Quantile Function. This work is the motivation for defining a new and improved smooth population Quantile Function given discrete data. Our definition is based on the theory of fractional order statistics. The main advantage of our definition as compared to its competitors is the capability to distinguish the uth Quantile across different discrete distributions over the whole interval, u ∈ ( 0 , 1 ) . In addition, we define the corresponding estimator of the smooth population Quantiles and demonstrate the convergence and asymptotic normal distribution of the corresponding sample Quantiles. We verify our theoretical results through a Monte Carlo simulation, and illustrate the utilization of our Quantile Function in a Q–Q plot for discrete data.

  • Exact Nonparametric Bootstrap Confidence Bands for the Quantile Function Given Censored Data
    Communications in Statistics - Simulation and Computation, 2004
    Co-Authors: Alan D. Hutson
    Abstract:

    Abstract In this note we develop an exact bootstrap algorithm for generating confidence bands for a single Quantile Function or joint confidence bands for k independent Quantile Functions given censored data.The method utilizes the classic product-limit estimator [Kaplan, E. L., Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Am. Statist. Assoc. 52:457–481] in conjunction with what is now referred to as Steck's determinant [Steck, G. P. (1971). Rectangle probabilities for uniform order statistics and the probability that the empirical distribution Function lies between two distribution Functions. Ann. Math. Statist. 42:1–11]. The method is termed an exact bootstrap method in the sense that no resampling is required. Unlike other bootstrap methods based on censored data the exact bootstrap method presented here is consistent.

  • utilizing a Quantile Function approach to obtain exact bootstrap solutions
    Statistical Science, 2003
    Co-Authors: Michael D Ernst, Alan D. Hutson
    Abstract:

    The popularity of the bootstrap is due in part to its wide applicability and the ease of implementing resampling procedures on modern computers. But careful reading of Efron (1979) will show that at its heart, the bootstrap is a “plug-in” procedure that involves calculating a Functional θ(F ) from an estimate of the c.d.f. F . Resampling becomes invaluable when, as is often the case, θ(F ) cannot be calculated explicitly. We discuss some situations where working with the sample Quantile Function, Q, rather than F , can lead to explicit (exact) solutions to θ(F ).

  • A Semi-Parametric Quantile Function Estimator for Use in Bootstrap Estimation Procedures
    Statistics and Computing, 2002
    Co-Authors: Alan D. Hutson
    Abstract:

    In this note we develop a new Quantile Function estimator called the tail extrapolation Quantile Function estimator. The estimator behaves asymptotically exactly the same as the standard linear interpolation estimator. For finite samples there is small correction towards estimating the extreme Quantiles. We illustrate that by employing this new estimator we can greatly improve the coverage probabilities of the standard bootstrap percentile confidence intervals. The method does not reqiure complicated calculations and hence it should appeal to the statistical practitioner.

N.n. Midhu - One of the best experts on this subject based on the ideXlab platform.

  • Nonparametric estimation of mean residual Quantile Function under right censoring
    Journal of Applied Statistics, 2016
    Co-Authors: P G Sankaran, N.n. Midhu
    Abstract:

    In this paper, we develop non-parametric estimation of the mean residual Quantile Function based on right-censored data. Two non-parametric estimators, one based on the empirical Quantile Function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.

  • A new Quantile Function with applications to reliability analysis
    Communications in Statistics - Simulation and Computation, 2015
    Co-Authors: P G Sankaran, N. Unnikrishnan Nair, N.n. Midhu
    Abstract:

    In this article, we propose a new class of distributions defined by a Quantile Function, which nests several distributions as its members. The Quantile Function proposed here is the sum of the Quantile Functions of the generalized Pareto and Weibull distributions. Various distributional properties and reliability characteristics of the class are discussed. The estimation of the parameters of the model using L-moments is studied. Finally, we apply the model to a real life dataset.

  • A software reliability model using mean residual Quantile Function
    Journal of Applied Statistics, 2015
    Co-Authors: Bijamma Thomas, N.n. Midhu, P G Sankaran
    Abstract:

    In this paper, we propose a class of distributions with the inverse linear mean residual Quantile Function. The distributional properties of the family of distributions are studied. We then discuss the reliability characteristics of the family of distributions. Some characterizations of the class of distributions are also discussed. The parameters of the class of distributions are estimated using the method of L-moments. The proposed class of distributions is applied to a real data set.

  • Testing exponentiality using mean residual Quantile Function
    Statistical Papers, 2014
    Co-Authors: P G Sankaran, N.n. Midhu
    Abstract:

    Abstract In the present paper, a non-parametric test is developed to test exponentiality using mean residual Quantile Function. Asymptotic distribution of the test statistic is derived. Simulation studies are carried out to assess the efficiency of the test. We also compare the power of the proposed test with the existing tests. We apply the proposed test to two real life data sets.

  • a class of distributions with linear hazard Quantile Function
    Communications in Statistics-theory and Methods, 2014
    Co-Authors: N.n. Midhu, P G Sankaran, Unnikrishnan N Nair
    Abstract:

    In this article, we introduce and study a class of distributions that has linear hazard Quantile Function. Various distributional properties and reliability characteristics of the class are studied. Some characterizations of the class of distributions are presented. The method of L-moments is employed to estimate parameters of the class of distributions. Finally, we apply the proposed class to a real data set.

Mahesh D. Pandey - One of the best experts on this subject based on the ideXlab platform.

  • derivation of sample oriented Quantile Function using maximum entropy and self determined probability weighted moments
    Environmetrics, 2010
    Co-Authors: Jian Deng, Mahesh D. Pandey
    Abstract:

    The paper proposes a new distribution free method for deriving the Quantile Function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints in terms of the self-determined probability-weighted moments estimated from observed sample data. The principle of MaxEnt constrained by probability weighted moments (PWMs) was utilized to estimate the Quantile Function. For correct estimation of a Quantile Function, outliers must be rationally considered in the analysis. However, conventional PWM was criticized for assigning non-exceedance probabilities to sample points based on only their rank number in an ordered series rather than the magnitude of the points themselves, hereby being unable to satisfactorily accommodate outlier in a finite sample. The difficulty in obtaining accurate PWM estimates from samples has been the main impediment to the application of the MaxEnt Principle in extreme Quantile estimation. This paper is an attempt to circumvent this difficulty by the use of self-determined probability-weighted moments, which are completely decided by the distribution itself and sample data's magnitude. By interpreting the SD-PWM as moment of Quantile Function, the paper derives a more rigorous Quantile Function using MaxEnt principle, which is extraordinarily suitable for cases with small samples containing outliers. An efficient algorithm is presented to estimate the unknown parameters of this sample oriented MaxEnt QF. Comparative studies and numerical analysis are performed to assess the accuracy of the proposed QF estimation method. Copyright © 2009 John Wiley & Sons, Ltd.

  • Derivation of sample oriented Quantile Function using maximum entropy and self‐determined probability weighted moments
    Environmetrics, 2009
    Co-Authors: Jian Deng, Mahesh D. Pandey
    Abstract:

    The paper proposes a new distribution free method for deriving the Quantile Function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints in terms of the self-determined probability-weighted moments estimated from observed sample data. The principle of MaxEnt constrained by probability weighted moments (PWMs) was utilized to estimate the Quantile Function. For correct estimation of a Quantile Function, outliers must be rationally considered in the analysis. However, conventional PWM was criticized for assigning non-exceedance probabilities to sample points based on only their rank number in an ordered series rather than the magnitude of the points themselves, hereby being unable to satisfactorily accommodate outlier in a finite sample. The difficulty in obtaining accurate PWM estimates from samples has been the main impediment to the application of the MaxEnt Principle in extreme Quantile estimation. This paper is an attempt to circumvent this difficulty by the use of self-determined probability-weighted moments, which are completely decided by the distribution itself and sample data's magnitude. By interpreting the SD-PWM as moment of Quantile Function, the paper derives a more rigorous Quantile Function using MaxEnt principle, which is extraordinarily suitable for cases with small samples containing outliers. An efficient algorithm is presented to estimate the unknown parameters of this sample oriented MaxEnt QF. Comparative studies and numerical analysis are performed to assess the accuracy of the proposed QF estimation method. Copyright © 2009 John Wiley & Sons, Ltd.

  • estimation of minimum cross entropy Quantile Function using fractional probability weighted moments
    Probabilistic Engineering Mechanics, 2009
    Co-Authors: Jian Deng, Mahesh D. Pandey
    Abstract:

    Abstract The principle of minimum cross-entropy provides a systematic approach to derive the posterior distribution of a random variable given a prior and additional information in terms of its product moments. This approach can be extended to derive directly the Quantile Function by using probability weighted moments (PWMs) as constraints in the cross-entropy minimization approach, as shown in a previous study [Pandey MD. Extreme Quantile estimation using order statistics with minimum cross-entropy principle. Probabilistic Engineering Mechanics 2001;16(1):31–42]. The objective of the present paper is to extend and improve the previous method by incorporating the use of the fractional probability weighted moments (FPWMs) in the place of conventional integer-order PWMs. A new and general estimation method is proposed in which the Monte Carlo simulations and optimization algorithms are combined to estimate FPWMs that would subsequently lead to the best-fit Quantile Function. The numerical examples presented in the paper show a substantial improvement in accuracy by the use of the proposed method over the conventional approach.

  • Estimation of the maximum entropy Quantile Function using fractional probability weighted moments
    Structural Safety, 2008
    Co-Authors: Jian Deng, Mahesh D. Pandey
    Abstract:

    Abstract In a previous study, the conventional or integral-order probability weighted moments (IPWM) and the principle of maximum entropy were combined to derive an analytical form of the Quantile Function of a random variable [Pandey MD. Direct estimation of Quantile Functions using the maximum entropy principle. Struct Safety 2000;22(1):61–79]. This method is extended and improved in the present paper by utilizing the concept of fractional probability weighted moments (FPWMs). A general estimation method is proposed in which the Monte Carlo simulations and optimization algorithms are combined to estimate fractionals of FPWM that would lead to the best-fit Quantile Function. The numerical examples presented in the paper illustrate that the accuracy of the proposed FPWM based Quantile Function is superior to that estimated from the use of conventional IPWMs.

  • Direct estimation of Quantile Functions using the maximum entropy principle
    Structural Safety, 2000
    Co-Authors: Mahesh D. Pandey
    Abstract:

    The paper presents a distribution free method for estimating the Quantile Function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints specified in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density Function under specified moment constraints. The density Function is then integrated to obtain the cumulative distribution Function, which needs to be inverted to obtain a Quantile corresponding to some specified probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme Quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of Quantile Function, the paper derives an analytical form of Quantile Function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt Quantile estimates computed from small samples.

Cheng Cheng - One of the best experts on this subject based on the ideXlab platform.

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