Hazard Rate Estimation in Nonparametric Regression with Censored Data

Variance function estimation in multivariate nonparametric regression with fixed design

Pricing of American options in discrete time by means of regression-based Monte Carlo methods is considered in this thesis. Thereby, the options are allowed to be based on several underlyings (so-called basket options). It is assumed that the underlyings price processes satisfy the Markov property and consequently are Markov processes. In this dissertation, Monte Carlo methods are used to generate artificial sample paths of these price processes, and subsequently adaptive nonparametric least squares regression estimates are used to estimate the so-called continuation values from these data. The continuation values describe the mean values of the American options for given values of the underlyings at time t, subject to the constraint that the options are not exercised at time t, but optimally exercised in the future. Adaptive least squares neural networks as well as adaptive least squares spline estimates are used as nonparametric regression estimates. In conjunction with the pricing of American options, these estimates are theoretically analysed, and results with respect to consistency and rate of convergence are derived. Finally, the pricing of American options on simulated data by means of the estimates is illustrated.

In the present paper, we extend the work of Slaoui (Stat Sin 30:417–437, 2020) in the case of strong mixing data. Since, we are interested in nonparametric regression estimation, we focus on well adapted dependence structures based on mixing type conditions. We study the properties of these regression estimators and compare them with the nonparametric non-recursive regression estimator. The bias, variance and mean squared error are computed explicitly. We showed that using a selected wild bootstrap bandwidth procedure and a special stepsize, our proposed recursive regression estimators allowed us to obtain quite similar results compared to the non-recursive regression estimator under $$\alpha $$ -mixing condition in terms of estimation error and much better in terms of computational costs.

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.

Objectives: To investigate the Modified Inverse Weibull Distribution (MIWD) to determine its effectiveness in modelling real customer transaction data, which is often skewed and exhibits heavy-tailed characteristics. Methods: Parameters of the model were estimated using the Maximum Likelihood Estimation (MLE) technique. The model's performance was evaluated using goodness-of-fit measures, such as the log-likelihood (ℓ), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Kolmogorov–Smirnov (K–S) statistic. The probability density function (PDF), cumulative distribution function (CDF), hazard function, and survival function were derived analytically and visualized with Python to assess model behavior and data alignment. Findings: The MIWD demonstrated a better fit for the customer transaction data than its sub-models and other competing distributions. The estimated probability density function (PDF), cumulative distribution function (CDF), hazard, and survival functions highlighted the model's adaptability in representing both skewness and heavy-tailed characteristics in actual transaction data. Novelty: The main contribution of this study is the empirical validation of the four-parameter MIWD using actual customer transaction data, demonstrating its superiority over established sub-models based on various quantitative criteria. The findings provide compelling evidence that the MIWD is a robust and adaptable alternative for modelling skewed, heavy-tailed transaction and lifetime data, with practical applications in customer behavior analysis, reliability engineering, and risk modelling. Keywords: Modified Inverse Weibull Distribution, Maximum Likelihood Estimation, Customer transaction data, Parameters, Rayleigh distribution

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a Hölder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.

Nonparametric curves occur in a wide variety of contexts, including repeated measurements mean profiles, survivor functions and in different types of smoothing techniques such as nonparametric regression and density estimation. In some cases confidence bands can be attached to these curves as an indication of the variability of estimation. This is more difficult in the case of nonparametric regression and density estimation where bias is present. In the important case of the comparison of two curves, attention can be focused instead on whether there are differences between the curves. In this paper, the idea of a reference band for the comparison of two curves is introduced The band is derived from the standard error of the difference between the two curves at each point. The bands have a simple hypothesis testing interpretation which applies equally well to smoothing methods where bias occurs. It does not remove the need for a global test of effects of interest, but it can be very useful as a graphical means of exploring where any identified differences might lie, or of explaining why apparent differences do not actually contribute strong evidence of statistical significance to the global comparison of curves. Reference bands for equality are derived and explored in a variety of settings. Reference bands for parallelism are also derived for nonparametric regression models. Reference bands for parametric models, against which nonparametric regression and density estimates can be compared, are derived for linearity and for normality.

Large sample results under biased sampling when covariables are present

Nonparametric regression models offer a way to understand and quantify relationships between variables without having to identify an appropriate family of possible regression functions. Although many estimation methods for these models have been proposed in the literature, most of them can be highly sensitive to the presence of a small proportion of atypical observations in the training set. A review of outlier robust estimation methods for nonparametric regression models is provided, paying particular attention to practical considerations. Since outliers can also influence negatively the regression estimator by affecting the selection of bandwidths or smoothing parameters, a discussion of robust alternatives for this task is also included. Using many of the “classical” nonparametric regression estimators (and their robust counterparts) can be very challenging in settings with a moderate or large number of explanatory variables, so recently proposed robust nonparametric regression methods that scale well with a growing number of covariates are also discussed.

Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, the Weighted Least Square method and a pivotal quantity method were used for unknown population variance cases. The result of the theoretical study was that pivotal quantity distributed student-t. The shortest parameter interval estimation of semi parametric spline truncated regression model was obtained through the optimization process using the method of Lagrange.Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, t...

Nonparametric M-regression with free knot splines

In this paper, we have derived and studied new probability distributions by extending the 2‐dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3‐dimensional Rayleigh distribution (3‐DRD) and k‐dimensional Rayleigh distribution (k‐DRD), respectively. For both 3‐DRD and k‐DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi‐square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3‐DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3‐DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh‐based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.

Hazard function plays a crucial role in survival analysis. Its estimation has garnered a lot of attention when the survival time variable suffers from right-censoring. Most of the existing works focus on the cases that failure information is complete or missing at random (MAR). When the censoring information is missing not at random (MNAR), statistical inferences on hazard function are very challenging. In this study, estimation of hazard function is addressed under the MNAR mechanism of the failure information. Three estimators are proposed by employing the techniques of correcting biases and adjusting weighting probabilities. These estimators are validated to be consistent and asymptotically normal. Simulation studies and two real-data analyses are performed to assess the proposed methods.

In this paper, we propose the method of Maximum product of spacings for point estimation of parameter of generalized inverted exponential distribution (GIED). The aim of this paper is to analyse the small sample behaviour of proposed estimators. Further, we have also proposed asymptotic confidence interv als of the parameters and the estimates of reliability and ha zard function using Maximum Product Spacings (MPS) method and compared with corresponding asymptotic confidence intervals and the es timates of reliability and hazard function of Maximum Likelihood estimation (MLEs). A comparative study among the method of MLE, method of least square (LSE) and the method of maximum product of spacings (MPS) is performed on the basis of simulated sample of GIED. The MPS method outperforms the method of MLE and the method ofLSE. Furthermore, comparison of different estimation method have been proposed on the basis of K-S distance and AIC. For numerical illustration one real data set has been considered.

The burr XII and weibull distributions are the most important and the most widely used distributions for the life time distribution as well as the wealth distribution. Burr XII distribution is mainly used to explain the allocation of wealth and wages among the people of the particular society. And weibull distribution is mostly used in the extreme value theory, failure analysis and the reliability engineering. In this paper, the classical properties of the mixture burr XII weibull distribution have been discussed. The estimation of the parameters and the construction of the information matrix are introduced. The var-cov matrix and the interval estimation of the parameters are also proposed in the paper. Cumulative distribution function, hazard rate, failure rate, inverse hazard function, odd function and the cumulative hazard function, rth moment, moment generating function characteristic function, moments, mean and variance, Renyi and Beta entropies, mean deviation from mean and mean time between failures(MTBF)have also been discussed.