This article provides a new Inverted Exponential Teissier (IET) distribution to model an extreme value data set and explain temporal dependence in environmental statistics employing bi-variate probability distribution. We deduce its various statistical properties, including descriptive statistics, characterization, and different measurements of reliability. The model parameters are estimated using Bayesian and non-Bayesian frameworks. For exploring the dependency structures between two geographical Random Variables (RV), we extend the IET to bi-variate IET (BIET) distribution. We introduce a novel time series forecasting algorithm based upon copula assuming stationarity of the data set. We validate the proposed method using extensive simulation studies with different possible combinations of parameter values. This method is applied to the seasonal rainfall data of Kerala from 1901 to 2017. We estimate the monsoon rainfall using median regression derived from BIET, where summer rainfall data is used as an important covariate. We found the Mean Absolute Percentage Error (MAPE) is on the test data set.

Interpolating a skewed conditional spatial random field with missing data is cumbersome in the absence of Gaussianity assumptions. Copulas can capture different types of joint tail characteristics beyond the Gaussian paradigm. Maintaining spatial homogeneity and continuity around the observed random spatial point is also challenging. Especially when interpolating along a spatial surface, the boundary points also demand focus in forming a neighborhood. As a result, importing the concept of hierarchical clustering on the spatial random field is necessary for developing the copula model with the interface of the Expectation-Maximization algorithm and concurrently utilizing the idea of the Bayesian framework. This article introduces a spatial cluster-based C-vine copula and a modified Gaussian distance kernel to derive a novel spatial probability distribution. To make spatial copula interpolation compatible and efficient, we estimate the parameter by employing different techniques. We apply the proposed spatial interpolation approach to the air pollution of Delhi as a crucial circumstantial study to demonstrate this newly developed novel spatial estimation technique.

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