A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q（p） = inf{t：F（t） ≥ p}, 0 ≤ p ≤ 1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.

Single-Index Quantile Regression with Left Truncated Data

The purpose of this paper is two fold.First,the authors investigate quantile regression(QR)estimation for single-index QR models when the response is subject to random left truncation.The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed.The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained.Further,by combining the QR loss function and the adaptive LASSO penalization,a variable selection procedure for the index parameter is introduced and its oracle property is established.Second,a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution.Furthermore,confidence regions of the index parameter can be constructed.The finite sample performance of the proposed methods are demonstrated.A real data analysis is also conducted to show the usefulness of the proposed approaches.

ESTIMATING A DISTRIBUTION FUNCTION WITH TRUNCATED DATA

Let (Xi, Yi), i=1, 2,...,be i. i. d. vector valued random variables with unknown common marginal distribution functions F(x) and G(x). One model of incomplete observations studied in the literature is the truncated model, where both Xi and Yi are observed if, and nothing can be observed otherwise. From this kind of observations, if any, we describe the modified nonparametric maximum likelihood estimators of F(x). The law of the iterated logarithm for the uniform covergence is proved.

KERNEL ESTIMATION OF HIGHER DERIVATIVES OF DENSITY AND HAZARD RATE FUNCTION FOR TRUNCATED AND CENSORED DEPENDENT DATA

Based on left truncated and right censored dependent data, the estimators of higher derivatives of density function and hazard rate function are given by kernel smoothing method. When observed data exhibit α-mixing dependence, local properties including strong consistency and law of iterated logarithm are presented. Moreover, when the mode estimator is defined as the random variable that maximizes the kernel density estimator, the asymptotic normality of the mode estimator is established.

STRONG REPRESENTATIONS OF THE SURVIVAL FUNCTION ESTIMATOR ON INCREASING SETS FOR TRUNCATED AND CENSORED DATA

In this paper, based on random left truncated and right censored data, the authors derive strong representations of the cumulative hazard function estimator and the product-limit estimator of the survival function. which are valid up to a given order statistic of the observations. A precise bound for the errors is obtained which only depends on the index of the last order statistic to be included.

Sedimentary Microfacies and Porosity Modeling of Deep-Water Sandy Debris Flows by Combining Sedimentary Patterns with Seismic Data： An Example from Unit I of Gas Field A, South China Sea

Sandy debris flow deposits are present in Unit I during Miocene of Gas Field A in the Baiyun Depression of the South China Sea. The paucity of well data and the great variability of the sedimentary microfacies make it difficult to identify and predict the distribution patterns of the main gas reservoir, and have seriously hindered further exploration and development of the gas field. Therefore, making full use of the available seismic data is extremely important for predicting the spatial distribution of sedimentary microfacies when constructing three-dimensional reservoir models. A suitable reservoir modeling strategy or workflow controlled by sedimentary microfacies and seismic data has been developed. Five types of seismic attributes were selected to correlate with the sand percentage, and the root mean square （RMS） amplitude performed the best. The relation between the RMS amplitude and the sand percentage was used to construct a reservoir sand distribution map. Three types of main sedimentary microfacies were identified： debris channels, fan lobes, and natural levees. Using constraints from the sedimentary microfacies boundaries, a sedimentary microfacies model was constructed using the sequential indicator and assigned value simulation methods. Finally, reservoir models of physical properties for sandy debris flow deposits controlled by sedimentary microfacies and seismic inversion data were established. Property cutoff values were adopted because the sedimentary microfacies and the reservoir properties from well-logging interpretation are intrinsically different. Selection of appropriate reservoir property cutoffs is a key step in reservoir modeling when using simulation methods based on sedimentary microfacies control. When the abnormal data are truncated and the reservoir properties probability distribution fits a normal distribution, microfacies-controlled reservoir property models are more reliable than those obtained from the sequence Gauss simulation method. The cutoffs for effective porosity of the debris channel, fan lobe, and natural levee facies were 0.2, 0.09, and 0.12, respectively; the corresponding average effective porosities were 0.24, 0.13, and 0.15. The proposed modeling method makes full use of seismic attributes and seismic inversion data, and also makes the property data of single-well depositional microfacies more conformable to a normal distribution with geological significance. Thus, the method allows use of more reliable input data when we construct a model of a sandy debris flow.

Cone-beam local reconstruction based on a Radon inversion transformation

The local reconstruction from truncated projection data is one area of interest in image reconstruction for com- puted tomography （CT）, which creates the possibility for dose reduction. In this paper, a filtered-backprojection （FBP） algorithm based on the Radon inversion transform is presented to deal with the three-dimensional （3D） local recon- struction in the circular geometry. The algorithm achieves the data filtering in two steps. The first step is the derivative of projections, which acts locally on the data and can thus be carried out accurately even in the presence of data trun- cation. The second step is the nonlocal Hilbert filtering. The numerical simulations and the real data reconstructions have been conducted to validate the new reconstruction algorithm. Compared with the approximate truncation resistant algorithm for computed tomography （ATRACT）, not only it has a comparable ability to restrain truncation artifacts, but also its reconstruction efficiency is improved. It is about twice as fast as that of the ATRACT. Therefore, this work provides a simple and efficient approach for the approximate reconstruction from truncated projections in the circular cone-beam CT.

Sequential Confidence Bands for Quantile Densities Under Truncated and Censored Data

In this paper an asymptotic distribution is obtained for the maximaldeviation between the kernel quantile density estimator and the quantile density when the data aresubject to random left truncation and right censorship. Based on this result we propose a fullysequential procedure for constructing a fixed-width confidence band for the quantile density on afinite interval and show that the procedure has the desired coverage probability asymptotically asthe width of the band approaches zero.

Fixed Design Nonparametric Regression with Truncated and Censored Data

Abstract In this paper we consider a fixed design model in which the observations are subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covariate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator. Applications include trimmed mean and quantile function estimators.These applications demonstrate the usefulness of the new matrix products.

Asymptotic Behavior of the Lipschitz-1／2 Modulus of the PL-process for Truncated and Censored Data

In this paper, we give a detailed description of the local behavior of theLipschitz-1/2 modulus for cumulative hazard process and PL-process when the data are subject to lefttruncation and right censored observations. We establish laws of the iterated logarithm of theLipschitz-1/2 modulus of PL-process and cumulative hazard process. These results for the PL-processare sharper than other results found in the literature, which can be used to establish theasymptotic properties of many statistics.

Development and evaluation of robust tree biomass equations for rubber tree (Hevea brasiliensis) plantations in India

Background： In India, rubber（Hevea brasiliensis） plantations cover -0.8 million ha of land, emphasizing its significant role in the Earth＇s carbon dynamics. Therefore, it is important to estimate the biomass stocks of plantations precisely in the context of carbon management. Previous studies in India have focused on development of allometric equations for estimating aboveground biomass（AGB） through harvesting younger trees（up to 14 yr）only or on studies with small sample sizes without assessing model bias. The objective of this study was to develop biomass estimation models for different tree components in rubber plantations and assess model predictive performance at the stand level.Methods： A total of 67 trees were harvested from plantations of different ages（6, 15, 27 and 34 yr） in North East India and their diameter at 200 cm（D）, height and dry weights of different tree components were recorded. The data were used for evaluation of H-D and biomass estimation models at the stand level.Results： The Michaelis-Menten function was found to be the most appropriate model for estimating tree height among 10 commonly used H-D models. For estimation of AGB and coarse root biomass, a model that involves tree volume（i.e. D2 H） was found to provide better prediction than either D or H alone or a model that combines H, D and stand density. The estimated AGB varied from 28 Mg·ha-（-1） in 6 yr. old plantation to 169 Mg·ha-（-1） in 34 yr. old plantations.The coarse root biomass was estimated at 4 Mg·ha-（-1） for 6 yr. old plantation and 12 Mg·ha-（-1） for 34 yr. old stands.Conclusions： It is concluded that models involving tree volume are more appropriate for regional level biomass estimation than simple power-law models for individual stands. We recommend that the power-law model should not be used for estimation of AGB in plantations at different growth stages because power-law parameters can be biased due to data truncation.

Principle of MSD floating-point division based on Newton-Raphson method on ternary optical computer

The division operation is not frequent relatively in traditional applications, but it is increasingly indispensable and important in many modern applications. In this paper, the implementation of modified signed-digit （MSD） floating-point division using Newton-Raphson method on the system of ternary optical computer （TOC） is studied. Since the addition of MSD floating-point is carry-free and the digit width of the system of TOC is large, it is easy to deal with the enough wide data and transform the division operation into multiplication and addition operations. And using data scan and truncation the problem of digits expansion is effectively solved in the range of error limit. The division gets the good results and the efficiency is high. The instance of MSD floating-point division shows that the method is feasible.