Two-piece Normal Distribution Research Articles

A flexible two-piece normal dynamic linear model

We construct a flexible dynamic linear model for the analysis and prediction of multivariate time series, assuming a two-piece normal initial distribution for the state vector. We derive a novel Kalman filter for this model, obtaining a two components mixture as predictive and filtering distributions. In order to estimate the covariance of the error sequences, we develop a Gibbs-sampling algorithm to perform Bayesian inference. The proposed approach is validated and compared with a Gaussian dynamic linear model in simulations and on a real data set.

A Flexible Class of Two-Piece Normal Distribution with a Regression Illustration to Biaxial Fatigue Data

Using a two-piece normal distribution for modeling univariate data that exhibits symmetry, and uni/bimodality is notably effective. In this respect, the shape parameter value determines whether unimodality or bimodality is present. This paper proposes a flexible uni/bimodal distribution with platykurtic density, which can be used to simulate a variety of data. The concept is based on the transforming of a random variable into a folded distribution. Further, the proposed class includes the normal distribution as a sub-model. In the current study, the maximum likelihood method is considered for deriving the main structural properties and for the estimation of parameters. In addition, simulation experiments are presented to evaluate the behavior of estimators. Finally, fitting and regression applications are presented to illustrate the usefulness of the proposed distribution for data modeling in different real-life scenarios.

Visualizing Ignorance

The Bank of England fan chart of inflation visualizes the uncertainty of the bank’s inflation projections. Visualization is a way to tame uncertainty, in the sense that uncertainty is brought under the measure of a probabilistic distribution, in this case a (two-piece) normal distribution. As such, taming is a process of homogenization, that is, a process of translating various heterogeneous items into a common medium. In this case, the common medium is a specific curve, the shape of which is determined by principles of ignorance: one starts with a simple symmetrical and smooth shape, and deviates from it if there is reason to. These principles of visualization work epistemologically in the same way that gestalt principles are used to perceptually structure visual information. This article shows that the principles of symmetry, proximity, and smoothness are the underlying heuristics that shapes the unknown future into a fan chart.

A two-piece normal measurement error model

In the context of measurement error models, the true unobservable covariates are commonly assumed to have a normal distribution. This assumption is replaced here by a more flexible two-piece normal distribution, which allows for asymmetry. After setting-up a general formulation for two-piece distributions, we focus on the case of the normal two-piece construction. It turns out that the joint distribution of the actual observations (the multivariate observed covariates and the response) is a two-component mixture of multivariate skew-normal distributions. This connection facilitates the construction of an EM-type algorithm for performing maximum likelihood estimation. Some numerical experimentation with two real datasets indicates a substantial improvement of the present formulation with respect to the classical normal-theory construction, which greatly compensates the introduction of a single parameter for regulation of skewness.

Two-piece power normal distribution

In this article, we introduce a new flexible distribution which generalizes the normal and the two-piece normal distribution. Some properties of the introduced distribution are derived, such are the cumulative distribution and the probability density function, moments, skewness, and kurtosis. The unknown parameters are estimated by the maximum likelihood method. Illustrative example of applicability and flexibility of the introduced distribution is given. At the end, we compare the flexibility of the introduced distribution and various distributions using skewness and kurtosis.

A Control Chart Based on Two-piece Normal Distribution Using Repetitive Sampling

In this manuscript, a control chart is designed for two-piece normal distribution using repetitive sampling. The necessary measures to determine the average run lengths for in control and out-of-control process are given. The average run lengths are presented for various specified parameters and shift constants. The efficiency of the proposed chart is compared with the existing control chart using single sampling. The application of the proposed chart is given with the help of an example.

Fan Chart A Tool for NBP's Monetary Policy Making

In this note we describe, in substantial detail, the methodology behind the construction of NBP’s fan chart. This note is meant to help the readers interpret information contained in the mid-term projection, understand the differences in the predicted uncertainty between the projection rounds, and the usefulness of the projection for the monetary policy conduct. We describe the process which leads to the final projection, the methodology of estimation of the variance of the final forecast probability distribution, the method used for quantifying asymmetry of the fan chart and the role the two-piece normal distribution plays in it. Finally, we describe the analysis of the changes in the fan charts between the projection rounds and explain how the narrative associated with the projection is consistent with its assessment of risk.

Tests for symmetry of two-piece normal distribution

ABSTRACTIn this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.

Bayesian analysis of two-piece location–scale models under reference priors with partial information

Bayesian estimators are developed and compared with the maximum likelihood estimators for the two-piece location–scale models, which contain several well-known distributions such as the asymmetric Laplace distribution, the two-piece normal distribution, and the two-piece Student-t distribution. For the validity of Bayesian analysis, it is essential to use priors that could lead to proper posterior distributions. Specifically, reference priors with partial information have been considered. A sufficient and necessary condition is established to guarantee the propriety of the posterior distribution under a general class of priors. The performance of the proposed approach is illustrated through extensive simulation studies and real data analysis.

A Unimodal Model for Double Observer Distance Sampling Surveys.

Distance sampling is a widely used method to estimate animal population size. Most distance sampling models utilize a monotonically decreasing detection function such as a half-normal. Recent advances in distance sampling modeling allow for the incorporation of covariates into the distance model, and the elimination of the assumption of perfect detection at some fixed distance (usually the transect line) with the use of double-observer models. The assumption of full observer independence in the double-observer model is problematic, but can be addressed by using the point independence assumption which assumes there is one distance, the apex of the detection function, where the 2 observers are assumed independent. Aerially collected distance sampling data can have a unimodal shape and have been successfully modeled with a gamma detection function. Covariates in gamma detection models cause the apex of detection to shift depending upon covariate levels, making this model incompatible with the point independence assumption when using double-observer data. This paper reports a unimodal detection model based on a two-piece normal distribution that allows covariates, has only one apex, and is consistent with the point independence assumption when double-observer data are utilized. An aerial line-transect survey of black bears in Alaska illustrate how this method can be applied.

Best Estimate Reporting with Asymmetric Loss

This paper considers the problem of point prediction based on a predictive distribution, representing the uncertainty about the outcome. The issue explored is the reporting of a single characteristic, typically the mean, the median or the mode, in the context of a skewed distribution and asymmetric loss. Special attention is given to the two-piece normal distribution and asymmetric piecewise linear and quadratic loss. The practical context for the issue is the yearly reporting of remaining petroleum resources given by the authorities to stakeholders that may ask for just a single number.

The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its Origin and Rediscoveries

This paper traces the history of the two-piece normal distribution from its origin in the posthumous Kollektivmasslehre (1897) of Gustav Theodor Fechner to its rediscoveries and generalisations. The denial of Fechner’s originality by Karl Pearson, reiterated a century later by Oscar Sheynin, is shown to be without foundation.

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.

Asymptotic properties of sample quantiles of discrete distributions

The asymptotic distribution of sample quantiles in the classical definition iswell-knowntobenormalforabsolutelycontinuousdistributions.However,thisisno longertruefordiscretedistributionsorsampleswithties.Weshowthatthedefinitionof sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample quantiles from absolutely continuous and from discrete distributions. We demonstrate that the same asymptotic normal distribution result as for the classical sample quantiles holds at differentiable points, whereas a more general form arises for distributions whose cumulative dis- tribution function has only one-sided differentiability. For discrete distributions with finite support, the same type of asymptotics holds and the sample quantiles based on mid-distribution functions either follow a normal or a two-piece normal distribution. We also calculate the exact distribution of these sample quantiles for the binomial and Poisson distributions. We illustrate the asymptotic results with simulations.

Improving density forecast by modeling asymmetric features: An application to S&P500 returns

A density forecast is an estimate of the probability distribution of the possible future values of a random variable. From the current literature, an economic time series may have three types of asymmetry: asymmetry in unconditional distribution, asymmetry in conditional distribution, volatility asymmetry. In this paper, we propose three density forecasting methods under two-piece normal assumption to capture these asymmetric features. A GARCH model with two-piece normal distribution is developed to capture asymmetries in the conditional distributions. In this approach, we first estimate parameters of a GARCH model by assuming normal innovations, and then fit a two-piece normal distribution to the empirical residuals. Block bootstrap procedure, and moving average method with two-piece normal distribution are presented for volatility asymmetry and asymmetry in the conditional distributions. Application of the developed methods to the weekly S&P500 returns illustrates that forecast quality can be significantly improved by modeling these asymmetric features.

An Application of the Truncated Two-Piece Normal Distribution to the Measurement of Depths of Arsenic Implants in Silicon

The truncated two-piece normal distribution is applied to data obtained from backscattering experiments in order to investigate the depth of arsenic implants in silicon.

Methods for the two-piece normal distribution

This note is concerned with statistical methods for the two-piece normal (TPN) distribution, A short description of the history and applications of TPN is given and then likelihood-based methods of estimation and testing are discussed, Maximum likelihood estimation for the truncated TPN distribution is also dealt with briefly.