Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.

In the sampling approaches framework, the combination of the information on the sizes of the nonsampled units can help to attain better estimators by using semiparametric models. Sometimes, the design variables that have an important role in the sampling mechanism are not available. Hence predictions require to be adapted for the consequence of selection. To infer the population mean in a sample survey, we study Bayesian nonparametric model with Dirichlet process prior by considering the inverse-probability weights as the only available information. Indeed, we present a Bayesian nonparametric mixture of regression models for the survey outcomes with the weights as predictors and impute the nonsampled units. Finally, the model-based estimators that are derived from the Bayesian (parametric and nonparametric) methods are compared with the design-based estimator based on the simulation approaches.

Recently, the Bayesian nonparametric approach in survival studies attracts much more attentions. Because of multi modality in survival data, the mixture models are very common in this field. One of the famous priors on Bayesian nonparametric models is Dirichlet process prior. In this paper we introduce a Bayesian nonparametric mixture model with Burr distribution(Burr type XII) as the kernel of mixture model. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time data sets, we show the preference of this model and compare it with other Dirichlet process mixture models with different kernels. And also we show that this model can be applied for the right censored data. For calculating the posterior of the parameters for inference and modeling, we used the MCMC simulation methods, especially Gibbs sampling.

Nonlinear regression models are extensively applied across various scientific disciplines. It is vital to accurately fit the optimal nonlinear model while considering the biases of the Bayesian optimal design. We present a Bayesian optimal design by utilising the Dirichlet process as a prior. The Dirichlet process serves as a fundamental tool in the exploration of Nonparametric Bayesian inference, offering multiple representations that are well-suited for application. This research paper introduces a novel one-parameter model, referred to as the ’Unit-Exponential distribution’, specifically designed for the unit interval. Additionally, we employ a stick-breaking representation to approximate the D-optimality criterion considering the Dirichlet process as a functional tool. Through this approach, we aim to identify a Nonparametric Bayesian optimal design.

Kernel density estimation has been applied in many computational subjects. In this paper, we propose a density estimation procedure from a Bayesian nonparametric perspective using Dirichlet process prior for the length-biased data under an unknown kernel function. In this situation, the kernel within the Dirichlet process mixture model will be approximated by the kernel density estimator. We present a Bayesian nonparametric method for finding the bandwidth parameter in the kernel density estimation using a Markov chain Monte Carlo approach. Then, this approach is used to the simulated and real data set. Finally, we compare the proposed bandwidth estimation with the other estimations like cross-validation and Bayes based on the mean integrated squared error criterion.