Statistical modeling of skewed data using newly formed parametric distributions

Several newly formed continuous parametric distributions are introduced to analyze skewed data. Firstly, a two-parameter smooth continuous lognormal-Pareto composite distribution is introduced for modeling highly positively skewed data. The new density is a lognormal density up to an unknown threshold value and a Pareto density for the remainder. The resulting density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation methods and the goodness-of-fit criterion for the new distribution are presented. A large actuarial data set is analyzed to illustrate the better fit and applicability of the new distribution over other leading distributions. Secondly, the Odd Weibull family is introduced for modeling data with a wide variety of hazard functions. This three-parameter family is derived by considering the distributions of the odds of the Weibull and inverse Weibull families. As a result, the Odd Weibull family is not only useful for testing goodness-of-fit of the Weibull and inverse Weibull as submodels, but it is also convenient for modeling and fitting different data sets, especially in the presence of censoring and truncation. This newly formed family not only possesses all five major hazard shapes: constant, increasing, decreasing, bathtub-shaped and unimodal failure rates, but also has wide variety of density shapes. The model parameters for exact, grouped, censored and truncated data are estimated in two different ways due to the fact that the inverse transformation of the Odd Weibull family does not change its density function. Examples are provided based on survival, reliability, and environmental sciences data to illustrate the variety of density and hazard shapes by analyzing complete and incomplete data. Thirdly, the two-parameter logistic-sinh distribution is introduced for modeling highly negatively skewed data with extreme observations. The resulting family provides not only negatively skewed densities with thick tails, but also variety of monotonic density shapes. The advantages of using the proposed family are demonstrated and compared by illustrating well-known examples. Finally, the folded parametric families are introduced to model the positively skewed data with zero data values

Some recent developments in microeconometrics: A survey

This paper summarizes some recent developments in rnicroeconometrics with respect to methods for estimation and inference in non-linear models based on cross-section and panel data. In particular we discuss recent progress in estimation with conditional moment restrictions, simulation methods, serniparametric methods, as well as specification tests. We use the binary cross-section and panel probit model to illustrate the application of some of the theoretical results. --

계층적 가능도를 이용한 불완전 자료 분석

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 통계학과, 2022. 8. 임요한.The h-likelihood has been proposed as an extension of Fisher's likelihood to allow the maximum likelihood estimation for statistical models including unobserved latent variables of recent interest. However, the current h-likelihood approach does not allow maximum likelihood estimators (MLEs) of variance components as Henderson’s joint likelihood (1959) does not in linear mixed models. In this thesis, we discuss how to form the canonical scale for the h-likelihood in order to facilitate joint maximization for MLEs of whole parameters. To show the usefulness of the h-likelihood for analyzing incomplete data, various types of unobserved latent variables are examined; missing data, random effect and censored data. As we shall see, a statistical model for unobserved latent variables may not be identifiable based on the observed data. Thus, we also present how to make robust inferences against various assumptions on statistical models.계층적 가능도는 고정된 모수만을 취급하던 기존의 가능도를 확장하여 관측되지 않은 잠재 변수를 포함하는 통계 모형에 대해 최대 가능도 추정을 허락하기 위해 제안되었다. 하지만, 기존의 계층적 가능도는 분산 성분을 포함한 모든 추정의 대상에 대해 최대 가능도 추정을 허락하지 못한다는 한계가 있었다. 본 학위논문에서는 계층적 가능도의 정준 척도의 성질을 살펴본 뒤, 이를 바탕으로 모든 모수들의 최대 가능도 추정량을 얻는 방법에 대하여 논의하였다. 불완전 자료의 예로는 결측 자료, 변량 효과, 중도 절단 자료 등이 있다. 이러한 불완전 자료에 대하여, 계층적 가능도를 이용한 통계적 추론의 유용성을 살펴보았다. 하지만, 관측되지 않은 잠재 변수에 대한 통계 모형의 경우, 관측된 자료로부터 항상 식별 가능하지 않을 수 있다. 따라서, 본 학위논문에서는 통계 모형에 사용되는 다양한 가정들에 대해 로버스트한 추론을 허락하는 방법도 함께 제시하였다.1 Introduction (1) 1.1 Maximum Likelihood Imputation (2) 1.2 Robust Imputation under Missing at Random (5) 1.3 Enhanced Laplace Approximation (8) 1.4 Accelerated Failure Time Random Effect Model with GEV Distribution (10) 2 Maximum Likelihood Imputation (12) 2.1 Basic Setup (14) 2.2 H-likelihood (16) 2.2.1 MLE of Fixed Parameter (20) 2.2.2 MLE of Random Parameter (21) 2.3 Scale for Joint Maximization (30) 2.4 ML Imputation (32) 2.5 Illustrative Examples (36) 2.5.1 Normal Regression Model (36) 2.5.2 Exponential Regression Model (37) 2.5.3 Tobit Regression Model (37) 2.6 Conclusion (40) 3 Robust Imputation under Missing at Random (46) 3.1 Basic Setup (48) 3.2 Semiparametric Outcome Regression Model (50) 3.3 Misspecification of Propensity Score Model (53) 3.4 Outliers in Outcome Regression Model (57) 3.5 Simulation Study (59) 3.5.1 Robustness against Model Misspecification (60) 3.5.2 Robustness against Outliers (63) 3.6 Conclusion (63) 4 Enhanced Laplace Approximation (71) 4.1 Review of the LA (73) 4.2 ELA (75) 4.3 Restricted Likelihood (79) 4.4 Salamander Mating Data (82) 4.4.1 Summer Data (83) 4.4.2 Pooled Data (86) 4.5 Rongelap Spatial Data (88) 4.6 Conclusion (94) 5 AFT Random Effect Model with GEV Distribution (98) 5.1 Model (100) 5.1.1 GEV Distribution (100) 5.1.2 AFT Random Effect Model with GEV Distribution (100) 5.2 Estimation Procedure (102) 5.3 Simulation Study (104) 5.4 Real Data Analysis: COHRI Data (107) 5.5 Conclusion (110) Bibliography (117) Abstract (in Korean) (130)박