The Calculation of the Limiting Distribution of the Least-Squares Estimator in a Near-Integrated Model

1989 ◽  
Vol 5 (2) ◽  
pp. 241-255 ◽  
Author(s):  
Pierre Perron
Keyword(s):  
Least Squares ◽  
Probability Density ◽  
Moment Generating Function ◽  
Probability Density Functions ◽  
Least Squares Estimator ◽  
Cumulative Distribution ◽  
Density Functions ◽  
Autoregressive Processes ◽  
First Order ◽  
True Model

We tabulate the limiting cumulative distribution and probability density functions of the least-squares estimator in a first-order autoregressive regression when the true model is near-integrated in the sense of Phillips. The results are obtained using an exact numerical method which integrates the appropriate limiting moment generating function. The adequacy of the approximation is examined for various first-order autoregressive processes with a root close to unity.

Reformative translation model for probability density functions of amplitude processes of crosswind-excited super-tall buildings

2019 ◽  
Vol 23 (5) ◽  
pp. 847-856
Author(s):  
Wei Hao ◽  
Qingshan Yang
Keyword(s):  
Probability Distribution ◽  
Probability Density ◽  
Tall Buildings ◽  
Extreme Value ◽  
Probability Density Functions ◽  
Cumulative Distribution ◽  
Extreme Value Analysis ◽  
Density Functions ◽  
Translation Model ◽  
Non Gaussian

At the vicinity of vortex lock-in wind speed, the nonlinear aerodynamic damping effect of super-tall buildings is significant, which can greatly promote the surge of vortex-induced vibration in the crosswind direction, where the crosswind response characterized by harmonic amplitude shows narrow-band hardening non-Gaussian characteristic with the kurtosis well below 3, and the corresponding probability distribution of amplitude process distinctly differs from that of typical random buffeting response. Although the moment-based Hermite translation model has been widely used for estimating the extreme value distribution of non-Gaussian process, it fails to represent the probability distribution of hardening non-Gaussian amplitude process, notably for the response with a kurtosis close to 1.5. In this study, a new translation model based on orthogonal expansion of random processes is developed for obtaining the non-Gaussian amplitude process from an underlying Gaussian amplitude process, and the probability density function of the non-Gaussian amplitude process is derived by mapping the cumulative distribution function. The coefficients of translation model are determined by minimizing the errors between the estimated probability density functions and target values through nonlinear optimization, and the closed-form semi-empirical formulations, which connect the model coefficients with response kurtosis, are also proposed using least-square curve fitting. Moreover, the effectiveness and monotonicity of the proposed translation model are examined. This model can be readily incorporated into the extreme value analysis of crosswind response and facilitate the evaluation of wind-induced fatigue of super-tall buildings.

scholarly journals On the Performance of Transmuted Logistic Distribution: Statistical Properties and Application

2019 ◽  
Vol 1 (3) ◽  
pp. 34-42
Author(s):  
Adeyinka Femi Samuel
Keyword(s):  
Biological Sciences ◽  
Probability Density ◽  
Order Statistics ◽  
Likelihood Estimation ◽  
Moment Generating Function ◽  
Probability Density Functions ◽  
Logistic Distribution ◽  
Density Functions ◽  
Maximum Order ◽  
Method Of Maximum Likelihood

In this article we transmute the logistic distribution using quadratic rank transmutation map to develop a transmuted logistic distribution. The quadratic rank transmutation map enables the introduction of extra parameter into its parent model to enhance more flexibility in the analysis of data in various disciplines such as biological sciences, actuarial science, finance and insurance. The mathematical properties such as moment generating function, quantile, median and characteristic function of this distribution are discussed. The probability density functions of the minimum and maximum order statistics of the transmuted logistic distribution are established and the relationships between the probability density functions of the minimum and maximum order statistics of the parent model and the probability density function of the transmuted logistic distribution are considered. The parameter estimation is done by the method of maximum likelihood estimation. The flexibility of the model in statistical data analysis and its applicability is demonstrated by using it to fit relevant data. The study is concluded by demonstrating that the transmuted logistic distribution performs better than its parent model. 

scholarly journals Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
J. Javier Gorgoso-Varela ◽  
Rafael Alonso Ponce ◽  
Francisco Rodríguez-Puerta
Keyword(s):  
Probability Density ◽  
Linear Models ◽  
Pinus Halepensis ◽  
Beta Function ◽  
Probability Density Functions ◽  
Lidar Data ◽  
Density Functions ◽  
Minimum Diameter ◽  
Location Parameters ◽  
Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

scholarly journals A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

2021 ◽  
Vol 502 (2) ◽  
pp. 1768-1784
Author(s):  
Yue Hu ◽  
A Lazarian
Keyword(s):  
Magnetic Fields ◽  
Probability Density ◽  
Column Density ◽  
Mass Density ◽  
Probability Density Functions ◽  
Emission Lines ◽  
Density Functions ◽  
Star Forming ◽  
Velocity Gradients ◽  
Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

scholarly journals Interactive design of probability density functions for shape grammars

2015 ◽  
Vol 34 (6) ◽  
pp. 1-13 ◽  
Author(s):  
Minh Dang ◽  
Stefan Lienhard ◽  
Duygu Ceylan ◽  
Boris Neubert ◽  
Peter Wonka ◽  
...  
Keyword(s):  
Probability Density ◽  
Probability Density Functions ◽  
Interactive Design ◽  
Density Functions ◽  
Shape Grammars
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