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# Pramita Bagchi

**Title:** Shape Restricted Inference for Dependent Data

**Advisors: **Associate Professor Moulinath Banerjee, Associate Professor Stilian Stoev

**Committee Members: **Professor Xuming He, Professor George Michailidis

**Abstract: **Isotonic regression is an important problem arising in many applications such as climate studies, economics, current-status data in biostatistics, among many others. When the data are independent and Gaussian, the well-known Pooled Adjacent Violators Algorithm provides the maximum likelihood estimate (MLE) of the monotone function we want to estimate. Its asymptotic properties are well understood. In many applications, however, the data are dependent. Motivated by the work of Banerjee & Wellner (2001), our goal is to study the pseudo likelihood ratio statistic (LRS) and a certain L2-distance statistic for dependent data. The limit distributions of these two statistics for independent identically distributed regression errors do not involve the slope of the regression function, which is difficult to estimate and arises in the asymptotic distribution of the MLE. Here, we present some results, which show that these statistics with suitably modified scaling will have the same asymptotic distribution in the case when the noise is weakly (short-range) dependent. We have also some preliminary results on the limit distributions of these statistics under strong (long-range) dependence. As these two statistics converge jointly, their ratio has a limit distribution that is free of the slope of the regression function and other scaling factors under both kinds of dependence. We expect to use this new ratio statistic to construct point-wise confidence intervals for the regression function, which will be of great interest in applications where the independence assumption on the errors is unwarranted and/or the estimation of the scaling factors is difficult.