A Family of Bayesian Estimators for the Two-Parametric Burr Type II Distribution

This study discusses the posterior estimation for the parameters of the Burr type II distribution (BIID). The informative and noninformative priors along with different loss functions have also been assumed for the posterior estimation. The applicability of the proposed distribution has also been discussed. The modeling capability of the proposed model has been compared with seven classes of the lifetime distributions using real data. The generalizations of Weibull, exponential, Rayleigh, gamma, log normal, Pareto, Maxwell, Levy, Laplace, inverse gamma, Gompertz, chi-square, inverse chi-square, half normal, and log-logistic distributions have been considered for the comparison. The comparison has been made based on different goodness-of-fit criteria, such as Akaike information criteria (AIC), Bayesian information criteria (BIC), and Kolmogorov-Smirnov (KS) test. Based on the results from the study, it can be suggested that the BIID can efficiently replace commonly used lifetime distributions and their modifications. The results under this model were comparable with different conventional/modified distributions having up to six parameters.

Assessing volumetric change distributions and scaling relations of retrogressive thaw slumps across the Arctic

Arctic ice-rich permafrost is becoming increasingly vulnerable to terrain-altering thermokarst, and among the most rapid and dramatic of these changes are retrogressive thaw slumps (RTSs). They initiate when ice-rich soils are exposed and thaw, leading to the formation of a steep headwall which retreats during the summer months. The impacts and the distribution and scaling laws governing RTS changes within and between regions are unknown. Using TanDEM-X-derived digital elevation models, we estimated RTS volume and area changes over a 5-year time period from winter 2011/12 to winter 2016/17 and used for the first time probability density functions to describe their distributions. We found that over this time period all 1853 RTSs mobilized a combined volume of 17×106 m3 yr−1, corresponding to a volumetric change density of 77 m3 yr−1 km−2. Our remote sensing data reveal inter-regional differences in mobilized volumes, scaling laws, and terrain controls. The distributions of RTS area and volumetric change rates follow an inverse gamma function with a distinct peak and an exponential decrease for the largest RTSs. We found that the distributions in the high Arctic are shifted towards larger values than at other study sites We observed that the area-to-volume scaling was well described by a power law with an exponent of 1.15 across all study sites; however the individual sites had scaling exponents ranging from 1.05 to 1.37, indicating that regional characteristics need to be taken into account when estimating RTS volumetric changes from area changes. Among the terrain controls on RTS distributions that we examined, which included slope, adjacency to waterbodies, and aspect, the latter showed the greatest but regionally variable association with RTS occurrence. Accounting for the observed regional differences in volumetric change distributions, scaling relations, and terrain controls may enhance the modelling and monitoring of Arctic carbon, nutrient, and sediment cycles.

Confidence Intervals for the Ratio of the Coefficients of Variation of Inverse-Gamma Distributions

Herein, we present four methods for constructing confidence intervals for the ratio of the coefficients of variation of inverse-gamma distributions using the percentile bootstrap, fiducial quantities, and Bayesian methods based on the Jeffreys and uniform priors. We compared their performances using coverage probabilities and expected lengths via simulation studies. The results show that the confidence intervals constructed with the Bayesian method based on the uniform prior and fiducial quantities performed better than those constructed with the Bayesian method based on the Jeffreys prior and the percentile bootstrap. Rainfall data from Thailand was used to illustrate the efficacies of the proposed methods.

Bayesian and Classical Inference under Type-II Censored Samples of the Extended Inverse Gompertz Distribution with Engineering Applications

In this article, we introduce a new three-parameter distribution called the extended inverse-Gompertz (EIGo) distribution. The implementation of three parameters provides a good reconstruction for some applications. The EIGo distribution can be seen as an extension of the inverted exponential, inverse Gompertz, and generalized inverted exponential distributions. Its failure rate function has an upside-down bathtub shape. Various statistical and reliability properties of the EIGo distribution are discussed. The model parameters are estimated by the maximum-likelihood and Bayesian methods under Type-II censored samples, where the parameters are explained using gamma priors. The performance of the proposed approaches is examined using simulation results. Finally, two real-life engineering data sets are analyzed to illustrate the applicability of the EIGo distribution, showing that it provides better fits than competing inverted models such as inverse-Gompertz, inverse-Weibull, inverse-gamma, generalized inverse-Weibull, exponentiated inverted-Weibull, generalized inverted half-logistic, inverted-Kumaraswamy, inverted Nadarajah–Haghighi, and alpha-power inverse-Weibull distributions.

Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Bayesian Inference under Small Sample Sizes Using General Noninformative Priors

This paper proposes a Bayesian inference method for problems with small sample sizes. A general type of noninformative prior is proposed to formulate the Bayesian posterior. It is shown that this type of prior can represent a broad range of priors such as classical noninformative priors and asymptotically locally invariant priors and can be derived as the limiting states of normal-inverse-Gamma conjugate priors, allowing for analytical evaluations of Bayesian posteriors and predictors. The performance of different noninformative priors under small sample sizes is compared using the likelihood combining both fitting and prediction performances. Laplace approximation is used to evaluate the likelihood. A realistic fatigue reliability problem was used to illustrate the method. Following that, an actual aeroengine disk lifing application with two test samples is presented, and the results are compared with the existing method.

Association of Population Density and Distance to the City with the Risks of COVID-19: A Bayesian Spatial Analysis

The outbreak of Coronavirus disease-2019 (Covid-19) poses a severe threat around the world. Although several studies of modelling Covid-19 cases have been done, there appears to have been limited research into modelling Covid-19 using Bayesian hierarchical spatial models. This study aims to examine the most suitable Bayesian spatial CAR Leroux models in modelling the number of confirmed Covid-19 cases without and with covariates namely distance to the capital city and population density. Data on the number of confirmed positive cases of Covid-19 (March 20, 2020 - August 30, 2021) in 15 sub-districts in Makassar City, the number of populations, population density, and distance to the city are used. The best model selection is based on several criteria, namely Deviance Information Criteria (DIC), Watanabe Akaike Information Criteria (WAIC), residuals from Moran’s I Modification (MMI), and the 95% credible interval does not contain zero. The results showed that the best model in modelling Covid-19 is spatial CAR Leroux with hyperprior Inverse-Gamma (0.5, 0.05) model with the incorporation of distance to the capital city. It is found that there was a negative correlation between the distance to the capital city and Covid-19 risk, but the association between population density and the relative risk of Covid-19 was not statistically significant. Ujung Pandang district and Sangkarrang Island have the highest and the lowest relative risk respectively.