The Fitting Analysis of Skew-Normal and Skew-t Distribution to the Catastrophe Loss Data

2022 ◽  
Vol 11 (01) ◽  
pp. 1-9
Author(s):  
鹏 徐
Keyword(s):  
Loss Data ◽  
T Distribution ◽  
Skew Normal

scholarly journals Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions

Biostatistics ◽  
2010 ◽  
Vol 11 (2) ◽  
pp. 317-336 ◽  
Author(s):  
Sylvia Frühwirth-Schnatter ◽  
Saumyadipta Pyne
Keyword(s):  
Bayesian Inference ◽  
Mixture Models ◽  
Random Effects ◽  
Data Augmentation ◽  
Heterogeneous Data ◽  
High Dimensional ◽  
Finite Mixtures ◽  
Truncated Normal ◽  
T Distribution ◽  
Skew Normal

Abstract Skew-normal and skew-t distributions have proved to be useful for capturing skewness and kurtosis in data directly without transformation. Recently, finite mixtures of such distributions have been considered as a more general tool for handling heterogeneous data involving asymmetric behaviors across subpopulations. We consider such mixture models for both univariate as well as multivariate data. This allows robust modeling of high-dimensional multimodal and asymmetric data generated by popular biotechnological platforms such as flow cytometry. We develop Bayesian inference based on data augmentation and Markov chain Monte Carlo (MCMC) sampling. In addition to the latent allocations, data augmentation is based on a stochastic representation of the skew-normal distribution in terms of a random-effects model with truncated normal random effects. For finite mixtures of skew normals, this leads to a Gibbs sampling scheme that draws from standard densities only. This MCMC scheme is extended to mixtures of skew-t distributions based on representing the skew-t distribution as a scale mixture of skew normals. As an important application of our new method, we demonstrate how it provides a new computational framework for automated analysis of high-dimensional flow cytometric data. Using multivariate skew-normal and skew-t mixture models, we could model non-Gaussian cell populations rigorously and directly without transformation or projection to lower dimensions.

scholarly journals A Comparison of Different Nonnormal Distributions in Growth Mixture Models

2019 ◽  
Vol 79 (3) ◽  
pp. 577-597
Author(s):  
Sookyoung Son ◽  
Hyunjung Lee ◽  
Yoona Jang ◽  
Junyeong Yang ◽  
Sehee Hong
Keyword(s):  
Mixture Models ◽  
Outcome Variable ◽  
Normal Distributions ◽  
Nonnormal Distributions ◽  
Growth Mixture Models ◽  
Time Points ◽  
Growth Mixture ◽  
Distal Outcome ◽  
T Distribution ◽  
Skew Normal

The purpose of the present study is to compare nonnormal distributions (i.e., t, skew-normal, skew- t with equal skew and skew- t with unequal skew) in growth mixture models (GMMs) based on diverse conditions of a number of time points, sample sizes, and skewness for intercepts. To carry out this research, two simulation studies were conducted with two different models: an unconditional GMM and a GMM with a continuous distal outcome variable. For the simulation, data were generated under the conditions of a different number of time points (4, 8), sample size (300, 800, 1,500), and skewness for intercept (1.2, 2, 4). Results demonstrate that it is not appropriate to fit nonnormal data to normal, t, or skew-normal distributions other than the skew- t distribution. It was also found that if there is skewness over time, it is necessary to model skewness in the slope as well.

scholarly journals A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or t distributions

2019 ◽  
Vol 29 (6) ◽  
pp. 1542-1562 ◽  
Author(s):  
Yongqiang Tang
Keyword(s):  
Repeated Measures ◽  
Data Augmentation ◽  
Mixed Effects ◽  
Mixed Effects Model ◽  
Monte Carlo Algorithm ◽  
Sensitivity Analyses ◽  
Special Cases ◽  
Monotone Data ◽  
T Distribution ◽  
Skew Normal

The mixed effects model for repeated measures has been widely used for the analysis of longitudinal clinical data collected at a number of fixed time points. We propose a robust extension of the mixed effects model for repeated measures for skewed and heavy-tailed data on basis of the multivariate skew-t distribution, and it includes the multivariate normal, t, and skew-normal distributions as special cases. An efficient Markov chain Monte Carlo algorithm is developed using the monotone data augmentation and parameter expansion techniques. We employ the algorithm to perform controlled pattern imputations for sensitivity analyses of longitudinal clinical trials with nonignorable dropouts. The proposed methods are illustrated by real data analyses. Sample SAS programs for the analyses are provided in the online supplementary material.

Log-Skew-Normal and Log-Skew-t Distributions as Models for Family Income Data

2002 ◽  
Author(s):  
Adelchi Azzalini ◽  
Thomas Del Cappello ◽  
Samuel Kotz
Keyword(s):  
Goodness Of Fit ◽  
Family Income ◽  
Sample Survey ◽  
Type I ◽  
Variable Degree ◽  
Numerical Comparisons ◽  
Skewed Normal Distribution ◽  
Income Data ◽  
T Distribution ◽  
Skew Normal

The U.S. family income data for the years 1970, 1975, 1978, 1980, 1985 and 1990 was fitted using the log-normal, Gamma, Singh-Maddala, Dagum type I and generalized Beta of second kind distributions, among others in earlier publications. Here we supplement these fittings by adding the log-skew-normal and log-skew-t distributions. In addition, we have performed similar numerical comparisons using 1997 income data collected in a sample survey from several European countries. The overall picture emerging from these numerical comparisons indicates that, while the log-skewed normal distribution provides a somewhat variable degree of goodness-of-fit, the log-skewed-t distribution seems to fit the data satisfactorily in a quite consistent way, and on the par with most creditable distributions.

scholarly journals Bayesian Estimation of Random Parameter Models of Responses with Normal and Skew-t Distibutions Evidence from Monte Carlo Simulation

2018 ◽  
Vol 24 (1) ◽  
pp. 27-50
Author(s):  
Mohammad Masjkur ◽  
Henk Folmer
Keyword(s):  
Monte Carlo ◽  
Dose Response ◽  
Random Parameter ◽  
Response Models ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Response Modeling ◽  
T Distribution ◽  
Valuation Criteria ◽  
Skew Normal

Random parameter models have been found to outperform xed pa-rameter models to estimate dose-response relationships with independent errors. Amajor restriction, however, is that the responses are assumed to be normally andsymmetrically distributed. The purpose of this paper is to analyze Bayesian infer-ence of random parameter response models in the case of independent responseswith normal and skewed, heavy-tailed distributions by way of Monte Carlo simu-lation. Three types of Bayesian estimators are considered: one applying a normal,symmetrical prior distribution, a second applying a Skew-normal prior and, a thirdapplying a Skew-t-distribution. We use the relative bias (RelBias) and Root MeanSquared Error (RMSE) as valuation criteria. We consider the commonly applied lin-ear Quadratic and the nonlinear Spillman-Mitscherlich dose-response models. Onesimulation examines the performance of the estimators in the case of independent,normally and symmetrically distributed responses; the other in the case of indepen-dent responses following a heavy-tailed, Skew-t-distribution. The main nding isthat the estimator based on the Skew-t prior outperforms the alternative estima-tors applying the normal and Skew-normal prior for skewed, heavy-tailed data. Fornormal data, the Skew-t prior performs approximately equally well as the Skew-normal and the normal prior. Furthermore, it is more ecient than its alternatives.Overall, the Skew-t prior seems to be preferable to the normal and Skew-normal fordose-response modeling.

scholarly journals SKEW NORMAL AND SKEW STUDENT-T DISTRIBUTIONS ON GARCH(1,1) MODEL

2021 ◽  
Vol 14 (1) ◽  
pp. 21-32
Author(s):  
Didit Budi Nugroho ◽  
Agus Priyono ◽  
Bambang Susanto
Keyword(s):  
Heavy Tails ◽  
Time Series Data ◽  
Financial Time Series ◽  
Linear Method ◽  
Real Data ◽  
Series Data ◽  
Model Parameters ◽  
Ratio Test ◽  
T Distribution ◽  
Skew Normal

The Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) type models have become important tools in financial application since their ability to estimate the volatility of financial time series data. In the empirical financial literature, the presence of skewness and heavy-tails have impacts on how well the GARCH-type models able to capture the financial market volatility sufficiently. This study estimates the volatility of financial asset returns based on the GARCH(1,1) model assuming Skew Normal and Skew Student-t distributions for the returns errors. The models are applied to daily returns of FTSE100 and IBEX35 stock indices from January 2000 to December 2017. The model parameters are estimated by using the Generalized Reduced Gradient Non-Linear method in Excel’s Solver and also the Adaptive Random Walk Metropolis method implemented in Matlab. The estimation results from fitting the models to real data demonstrate that Excel’s Solver is a promising way for estimating the parameters of the GARCH(1,1) models with non-Normal distribution, indicated by the accuracy of the estimation of Excel’s Solver. The fitting performance of models is evaluated by using log-likelihood ratio test and it indicates that the GARCH(1,1) model with Skew Student-t distribution provides the best fitting, followed by Student-t, Skew-Normal, and Normal distributions.

On some exact distributions in ranked set sampling

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4857-4864
Author(s):  
Ayyub Sheikhi
Keyword(s):  
Joint Distribution ◽  
Ranked Set Sampling ◽  
Normal Distributions ◽  
Ranked Set Sample ◽  
Numerical Example ◽  
The Family ◽  
Exact Distributions ◽  
Skew Normal Distributions ◽  
T Distribution ◽  
Skew Normal

In this article we obtain the exact joint distribution of a ranked set sample. We show that this distribution belongs to the family of unified multivariate skew normal distributions. We also investigate a multivariate skew-t distribution 62J12 using ranked set samples as an application of our results. A numerical example is also provided to illustrate our results.

scholarly journals Efficient Rank-Based Analysis of Multilevel Models for the Family of Skew-t Errors

2021 ◽  
pp. 89-98
Author(s):  
Sehar Saleem ◽  
Rehan Ahmad Khan Sherwani
Keyword(s):  
Fixed Effects ◽  
Multilevel Models ◽  
Score Function ◽  
Large Sample Size ◽  
Skew Normal Distribution ◽  
Monte Carlo Simulation Study ◽  
Underlying Distribution ◽  
Asymptotically Efficient ◽  
T Distribution ◽  
Skew Normal

Rank-based analysis of linear models is based on selecting an appropriate score function. The information about the shape of the underlying distribution is necessary for the optimal selection; leading towards asymptotically efficient analysis. In this study, we analyzed the multilevel model with cluster-correlated error terms following a family of skew-t distribution with the rank-based approach based on score function derived for the class of skew-normal distribution. The rank fit is compared with the Restricted Maximum Likelihood (REML) estimation in terms of validity and efficiency for different sample sizes. A Monte Carlo simulation study is carried out over skewed-t and contaminated-t distribution with a range of skewness parameter from moderately to highly skewed. The standard error of regression coefficients is significantly reduced in the rank-based approach and further reduces for a large sample size. Rank-based fit appeared asymptotically efficient than REML for each shape parameter of skewness in skew-t and contaminated-t distribution computed through a calculation of precision. The empirical validity of fixed effects is obtained up to the nominal level 0.95 in REML but not rank-based with skew-normal score function.

scholarly journals An Extension of the Birnbaum-Saunders Distribution Based on Skew-Normal t Distribution

2015 ◽  
Vol 12 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Ahad Jamalizadeh ◽  
Vahid Amirzadeh ◽  
Farzaneh Hashemi ◽  
◽  
◽  
...  
Keyword(s):  
T Distribution ◽  
Skew Normal

scholarly journals Skewed and Flexible Skewed Distributions: A Modern Look at the Distribution of BMI

2017 ◽  
Vol 14 (2) ◽  
Author(s):  
Thao Tran ◽  
Cara Wiskow ◽  
Mohammad Aziz
Keyword(s):  
Body Mass Index ◽  
Normal Distribution ◽  
Body Mass ◽  
Model Parameters ◽  
Skew Normal Distribution ◽  
Scale Mixture ◽  
Skewed Distributions ◽  
Goodness Of Fit Test ◽  
T Distribution ◽  
Skew Normal

The purpose of this study is to find distributions that best model body mass index (BMI) data. BMI has become a standard health indicator and numerous studies have been done to examine the distribution of BMI. Due to the skew and bimodal nature, we focus on modeling BMI with flexible skewed distributions. The distributions are fitted to University of Wisconsin–Eau Claire (UWEC) BMI data and to a data obtained from National Health and Nutrition Survey (NHANES). The model parameters are obtained using maximum likelihood estimation method. We compare flexible models to more conventional distributions, such as skew-normal, and skew-t distributions using AIC and BIC and Kolmogorov-Smirnov (K-S) goodness-of-fit test. Our results indicate that the skew-t and Alpha-Skew-Laplace distributions are reasonably competitive when describing unimodal BMI data whereas Alpha-Skew-Laplace and finite mixture of scale mixture of skew-normal and skew-t distributions are better alternatives to both unimodal and bimodal conventional distributions. The results we obtained are useful because we believe the models discussed in ours study will offer a framework for testing features such as bimodality, asymmetry, and robustness of the BMI data, thus providing a more detailed and accurate understanding of the distribution of BMI. KEYWORDS: Body Mass Index; Skew-normal distribution; Skew-t distribution; Flexible skewed distributions; Mixture distributions; Scale mixture of skew-normal distribution; K-S test

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