In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear **kernel** **estimator** proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this **estimator** from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also pro- vided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

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We can further refer to some large sample properties of nonparameter estimate based on WOD samples. For instance, Wang et al. ([11], 2013) studied the strong consistency of **estimator** of ﬁxed design regression model for WOD samples. Shi and Wu ([12], 2014) discussed the strong consistency of **kernel** density **estimator** for identically distributed WOD samples. Li et al. ([13], 2015) studied the pointwise strong consistency for a kind of recursive **kernel** **estimator** based WOD samples.

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For the strong stationary sequences the nonparametric counterpart of the optimal equation was constructed in the theory of nonparametric signal processing. This approach was developed when the state equation and the probability distribution of an unobservable signal are unknown, and the stochastic observation equation is known completely. The estimation equation includes the **kernel** **estimator** of the logarithmic density derivative, which depends on bandwidths of density estimates and its derivatives.

of MSE for orthogonal **kernel** estimators. Article [2] introduced geo- metric extrapolation of nonnegative kernels, while [3] discussed the number of vanishing moments of **kernel** or- der using Fourier transformation. Variable **kernel** estimation in [4] successfully reduced the bias by employing larger smoothing parameters in low density regions, while [5] introduced the idea of inadmissible kernels which also results in reduced bias. On the other hand, [6] proposed an **estimator** using some probabilistic arguments which achieves the goal of bias reduction. Article [7] suggested a locally parametric density **estimator**, a semi- parametric technique, which effectively reduces the order of bias. Article [8] proposed algorithms relevant to quadratic polynomial and β cumulative distribution function (c.d.f.) which accommodates possible poles at boundaries and in consequence reduces the bias at boundaries. Article [9] introduced a bias reduction method using estimated c.d.f. via smoothed **kernel** transformations. Article [10] introduced a two-stage multiplicative bias corrected **estimator**. Article [11] developed a skewing method to reduce the bias while the variance is only increased by a moderate constant factor. In addition, some recent works discussed approaches of obtaining smaller bias of the **estimator** via several other methods. Article [12] worked out a bias reduced **kernel** relative to the classical **kernel** **estimator** via Lipschitz condition. Article [13] introduced an adjusted **kernel** density estima- tor in which the **kernel** is adapted to the data but not fixed. This method naturally leads to an adaptive-choice of the smoothing parameters which can reduce the bias.

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We should stress that in any practical application the choice of origin (“cut-point”) should be chosen dependent on u to minimise the width of the interval to be estimated. This adjustment is important to obtain meaningful inter- pretations, especially if the conditional mean can take values close to π. We note that the double-**kernel** **estimator** has two tuning parameters (λ and κ) whereas the circular check function **estimator** has only one. Maybe, it is for this rea- son that, in some simulation experiments with small samples, we have observed a slightly better performance for the double-**kernel** **estimator** in most settings. However, this comes with the cost of more computational effort, as well as the need to select good smoothing choices. This effort can be reduced by first selecting a good value of λ in the check function, and then using this value, together with an appropriate κ, in the double-**kernel** **estimator**. Cross-validation can be used in this process, though we have found a joint selection of both κ and λ to be problematic.

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Statistical analysis of extremes is conducted for predicting large return periods events. LQ-moments that are based on linear combinations are reviewed for characterizing the upper quantiles of distributions and larger events in data. The LQ-moments method is presented based on a new quick **estimator** using five points quantiles and the weighted **kernel** **estimator** to estimate the parameters of the generalized extreme value (GEV) distribution. Monte Carlo methods illustrate the performance of LQ-moments in fitting the GEV distribution to both GEV and non-GEV samples. The proposed estimators of the GEV distribution were compared with conventional L-moments and LQ-moments based on linear interpolation quantiles for various sample sizes and return periods. The results indicate that the new method has generally good performance and makes it an attractive option for estimating quantiles in the GEV distribution.

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Fan [3] studies the quadratic mean convergence rate of the **kernel** deconvolution **estimator**; it turns out that the convergence rate of the **estimator** depends heavily on the type of error distribution. In particular, it is determined by the tail behaviour of the modulus of the characteristic function of the error distribution: the faster the modulus function goes to zero in the tail, the slower the converge rate. The following lemma, which is from Van Es, Spreij, and Van Zanten [10], gives the tail behaviour of |φ k (t)|.

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This paper introduces a novel approach to clustering and semi-supervised classification which directly identifies low-density hyperplanes in the finite sample setting. In this ap- proach the density on a hyperplane criterion proposed by Ben-David et al. (2009) is directly minimised with respect to a **kernel** density **estimator** that employs isotropic Gaussian ker- nels. The density on a hyperplane provides a uniform upper bound on the value of the empirical density at points that belong to the hyperplane. This bound is tight and propor- tional to the density on the hyperplane. Therefore, the smallest upper bound on the value of the empirical density on a hyperplane is achieved by hyperplanes that minimise the den- sity on a hyperplane criterion. An important feature of the proposed approach is that the density on a hyperplane can be evaluated exactly through a one-dimensional **kernel** density **estimator**, constructed from the projections of the data sample onto the vector normal to the hyperplane. This renders the computation of minimum density hyperplanes tractable even in high dimensional applications.

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One of the ways for achieving the confidence interval for quantiles is direct use of a central limit theorem. In this approach, we require a good **estimator** of the quantile density function. In this paper, we consider the nonparametric **estimator** of the quantile density function from Soni et al. (2012) and we obtain confidence interval for quantiles. In the following, by using simulation, the coverage probability and mean square error of this confidence interval is calculated. Also, we compare our proposed approach with alternative approaches such as sectioning and jackknife.

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in which (⋅) is a **kernel** function. is defined for > 0 and is defined for > for an arbitrary constant > 0. and are originally proposed and investigated respectively by Jones [8] and Bhattacharyya et al. [2]. In order to achieve the desired result, we need to present another versions of Jones and Bhattacharyya estimators of , which are denoted by and . These estimators are defined as bellow

We then kept the data ( Y i , T i ) such that Y i T i . Using this scheme, m 10 independent samples of size n were generated. For each sample, plug-in estimates n and for and were used respectively. The following figures represent the average of m density estimations and their the confidence bounds. The **kernel** function

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This paper considers the problem of testing for an explosive bubble in …nancial data in the presence of time-varying volatility. We propose a weighted least squares- based variant of the Phillips, Wu and Yu (2011) test for explosive autoregressive behaviour. We …nd that such an approach has appealing asymptotic power proper- ties, with the potential to deliver substantially greater power than the established OLS-based approach for many volatility and bubble settings. Given that the OLS- based test can outperform the weighted least squares-based test for other volatility and bubble speci…cations, we also suggested a union of rejections procedure that succeeds in capturing the better power available from the two constituent tests for a given alternative. Our approach involves a nonparametric **kernel**-based volatility function **estimator** for computation of the weighted least squares-based statistic, together with the use of a wild bootstrap procedure applied jointly to both individ- ual tests, delivering a powerful testing procedure that is asymptotically size-robust to a wide range of time-varying volatility speci…cations.

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financial econometrics to describe the evolution of financial returns. Model (23) incorpo- rates popular discrete-time SV models (e.g. Taylor (1982)) and discretized continuous- time SV model which assume the volatility process to be stationary as special cases (see e.g. Shephard (2005) for a review). Van Es, Spreij, and Van Zanten (2003) and Van Es, Spreij, and Van Zanten (2005) considered estimating the volatility density using **kernel** deconvolution **estimator** in this model.

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A. Assenza and al (2008) summarizes, in [3], some probability density estimation methods and affirms that density estimation gives better results than traditional tools of data analysis like Principal Component Analysis. In the same way, Adriano Z. Zambom and al (2013) adds, in [4], that **kernel** density with smoothing is the most used approach. All those approaches, treats mathematical aspects but did not discussed implementation sides and computational complexity aspects. A. Sinha and al (2008) discussed in [5] algorithmic cost in time of those methods and optimized computational complexity of **Kernel** density **estimator** using clustring... Normally, the estimation of complexity in time must includes costs of all functions in global expression. Exponential function cost must be included in **Kernel** complexity estimation whatever its complexity class.

distance **estimator** of Ai and Chen [2003] also has a hidden curse of dimensionality, since it requires progressively stronger smoothness properties of the unknown functions when the dimension of the vector X grows. Other methods, such as the **estimator** by Ichimura [1993], may not have this problem. Un- fortunately, the second-order asymptotic properties of Ichimura’s **estimator** are not known. It is likely though that strong smoothness assumptions will be needed for it to have the rate of convergence of order O n 1=2 for the error between the …nite-sample distribution of the **estimator** and the asymptotic normal

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The output presented in Table 1 shows that for the autoregressive **estimator** of spectral density, the Ng-Perron test statistics is below the critical value and unit root hypothesis should be considered rejected. On the other hand, for **Kernel** based **estimator** of spectral density, the Ng- Perron test statistics is far above the critical value and the null of unit root could not be rejected even at a loose significance level. Therefore the person applying the unit root test may be confused in the choice of result. Therefore this study is designed to compare the size and power properties of Ng-Perron test so that a practitioner may get some guidance for the right choice of spectral density **estimator**.

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the one-dimensional components of a multiplicative separable hazard. Given a local lin- ear pilot **estimator** of the d-dimensional hazard, the backfitting algorithm is motivated from a least squares criterion and converges to the closest multiplicative function. We show that the one dimensional components are estimated with a one-dimensional con- vergence rate, and hence do not suffer from the curse of dimensionality. The setting is very similar to Linton, Nielsen, and Van de Geer (2003), but has two significant im- provements. First, our approach works without the use of higher order kernels. With them, one can theoretically derive nearly n −1/2 –consistency (with growing order), but they often fail to show good performance in practice. Second, the support of the mul- tivariate hazard does not need to be rectangular. In the provided in-sample forecasting application of reserving in general insurance, the support is indeed triangular.

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target distribution by an m-fold mixture of models whose parameter functions are fitted using regression techniques. As a special case we can even regard the **estimator** in terms of a nonparametric **kernel** density in which the assumed con- ditional distribution is the **kernel**, and the scedasticity function determines the data-adaptive local bandwidth. From that perspective assuming homoscedasticity corresponds to using a common global bandwidth; the use of asymmetric condi- tional distributions corresponds to the case of applying special kernels typically used for boundary correction or asymmetric information (like the knn estima- tors). In each of these three interpretations the choice of the conditional density plays a relatively minor role, the scedasticity function is more important, and the choice of model for the mean regression mainly impacts on the variability of the final estimate.

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is used to construct a density **estimator**. It should be noticed that applied the Epanechnikov **kernel** (4.1) satisfies the assumption (1). Bandwidth parameter is chosen based on the least square cross validation method discussed in [1]. According to the Figure 1, the Jones’ **estimator** can estimate the density function ( ) f properly.

As mentioned above a bandwidth and a **kernel** (the order of the **kernel**) have big influence on quality of an estimate. It could not be excluded that using dif- ferent type of **kernel** function will bring better results (see Poměnková, 2004b; Poměnková, 2005; Koláček, 2005).