scholarly journals Mathematical problems associated with errors in estimating distances to galaxies

2020 ◽  
pp. 25-27
Author(s):  
S. Parnovsky
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood Method ◽  
Least Squares Method ◽  
Hubble Constant ◽  
Likelihood Method ◽  
Original Sample ◽  
Separate Paper ◽  
The Monte Carlo Method ◽  
Mathematical Problems ◽  
Math Problems

I generate many mock samples for applying the Monte Carlo method in order to estimate the bias of the Hubble constant because of the use of estimates of distances to galaxies determined from statistical dependences. I add errors to the original sample generated according to the Hubble law. In doing so, I use two possible options for generating errors in distance, having a constant relative error. Both are practical, but there are some math problems with them. I discuss their effect on the properties of the mock sample. The application of the standard least squares method is discussed and shown that it leads to an underestimation of the slope in the Hubble law. A formula is derived for calculating this slope using the maximum likelihood method and it is shown that it is applicable only for one of the variants of the sample noising. All estimates were obtained theoretically, without using the results of mock samples processing, which are described in a separate paper.

Nonparametric Versus Parametric Reasoning Based on Contingency Tables

2019 ◽  
Vol 65 (3) ◽  
pp. 314-349
Author(s):  
Piotr Sulewski
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood Method ◽  
Flow Parameter ◽  
Contingency Tables ◽  
Likelihood Method ◽  
The Monte Carlo Method ◽  
Divergence Statistics ◽  
Probability Flow ◽  
Power Divergence ◽  
Nonparametric Statistical

This paper proposes scenarios of generating two-way and three way contingency tables (CTs). A concept of probability flow parameter (PFP) plays a crucial role in these scenarios. Additionally, measures of untruthfulness of H0 are defined. The power divergence statistics and the |X| statistics are used. This paper is a simple attempt to replace a nonparametric statistical inference from CTs by the parametric one. Maximum likelihood method is applied to estimate PFP and instructions of generating CTs according to scenarios in question are presented. The Monte Carlo method is used to carry out computer simulations.

A Monte Carlo Maximum Likelihood Method for Estimating Uncertainty Arising from Shared Errors in Exposures in Epidemiological Studies of Nuclear Workers

2007 ◽  
Vol 168 (6) ◽  
pp. 757-763 ◽  
Author(s):  
Leslie Stayner ◽  
Martine Vrijheid ◽  
Elisabeth Cardis ◽  
Daniel O. Stram ◽  
Isabelle Deltour ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Maximum Likelihood Method ◽  
Epidemiological Studies ◽  
Likelihood Method ◽  
Nuclear Workers ◽  
Monte Carlo Maximum Likelihood

scholarly journals مقارنة طرائق تقدير معلمات توزيع كاما ذي المعلمتين في حالة البيانات المفقودة باستخدام المحاكاة

2008 ◽  
Vol 14 (51) ◽  
Author(s):  
ظافر حسين رشيد ◽  
اوات سردار وادي
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Shrinkage Method

المستخلص تم في هذا البحث تقدير معلمات توزيع كاما ذي المعلمتين في حالة البيانات المفقودة وذلك باستخدام اثنين من الطرائق المهمة وهما: طريقة الامكان الأعظم (Maximum Likelihood Method) والتي تضمنت ثلاث طرائق لحل معادلات الإمكان غير الخطية التي يتم الحصول من خلالها على ثلاث مقدرات للإمكان الأعظم وهي: طريقة نيوتن- رافسن وطريقتين تم تطويرهما في هذا البحث لتلائم حالة البيانات المفقودة وهما تطوير طريقة (Thom) وتطوير طريقة (Sinha)، فضلاً عن تطوير طريقة أخرى تعتمد على توزيع كاما ذي المعلمات الثلاث في إيجاد مقدرات الإمكان الأعظم وهي تطوير طريقة (Bowman, Shenton and Lam) وطريقة التقلص (Shrinkage Method). وتم إجراء مقارنة بين أفضلية هذه الطرائق في الجانب التجريبي من خلال أسلوب المحاكاة باستخدام طريقة مونت كارلو (Monte Carlo) وإجراء عدة تجارب مستخدمين المقياس الإحصائي متوسط مربعات الخطأ (MSE) لغرض الحصول على افضل طريقة تقدير.

scholarly journals Comparing Parameter Estimation of Random Coefficient Autoregressive Model by Frequentist Method

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 62 ◽  
Author(s):  
Autcha Araveeporn
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Mean Square Error ◽  
Maximum Likelihood Method ◽  
Least Squares Method ◽  
Likelihood Function ◽  
Likelihood Method ◽  
Mean Square ◽  
Average Mean Square Error ◽  
Frequentist Method

This paper compares the frequentist method that consisted of the least-squares method and the maximum likelihood method for estimating an unknown parameter on the Random Coefficient Autoregressive (RCA) model. The frequentist methods depend on the likelihood function that draws a conclusion from observed data by emphasizing the frequency or proportion of the data namely least squares and maximum likelihood methods. The method of least squares is often used to estimate the parameter of the frequentist method. The minimum of the sum of squared residuals is found by setting the gradient to zero. The maximum likelihood method carries out the observed data to estimate the parameter of a probability distribution by maximizing a likelihood function under the statistical model, while this estimator is obtained by a differential parameter of the likelihood function. The efficiency of two methods is considered by average mean square error for simulation data, and mean square error for actual data. For simulation data, the data are generated at only the first-order models of the RCA model. The results have shown that the least-squares method performs better than the maximum likelihood. The average mean square error of the least-squares method shows the minimum values in all cases that indicated their performance. Finally, these methods are applied to the actual data. The series of monthly averages of the Stock Exchange of Thailand (SET) index and daily volume of the exchange rate of Baht/Dollar are considered to estimate and forecast based on the RCA model. The result shows that the least-squares method outperforms the maximum likelihood method.

Bayesian data analysis for agricultural experiments

2010 ◽  
Vol 90 (5) ◽  
pp. 575-603 ◽  
Author(s):  
X. Che ◽  
S. Xu
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Data Analysis ◽  
Markov Chain Monte Carlo ◽  
Linear Model ◽  
Generalized Linear Model ◽  
Bayesian Method ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Mcmc Algorithm

Data collected in agricultural experiments can be analyzed in many different ways using different models. The most commonly used models are the linear model and the generalized linear model. The maximum likelihood method is often used for data analysis. However, this method may not be able to handle complicated models, especially multiple level hierarchical models. The Bayesian method partitions complicated models into simple components, each of which may be formulated analytically. Therefore, the Bayesian method is capable of handling very complicated models. The Bayesian method itself may not be more complicated than the maximum likelihood method, but the analysis is time consuming, because numerical integration involved in Bayesian analysis is almost exclusively accomplished based on Monte Carlo simulations, the so called Markov Chain Monte Carlo (MCMC) algorithm. Although the MCMC algorithm is intuitive and straightforward to statisticians, it may not be that simple to agricultural scientists, whose main purpose is to implement the method and interpret the results. In this review, we provide the general concept of Bayesian analysis and the MCMC algorithm in a way that can be understood by non-statisticians. We also demonstrate the implementation of the MCMC algorithm using professional software packages such as the MCMC procedure in SAS software. Three datasets from agricultural experiments were analyzed to demonstrate the MCMC algorithm.Key words: Bayesian method, Generalized linear model, Markov Chain Monte Carlo, SAS, WinBUGS

scholarly journals مقارنة بين طريقة التقلص وطريقة الإمكان الأعظم لتقدير المعلمات ودالة المعولية لتوزيع ويبل بثلاثة معلمات باستخدام المحاكاة

2013 ◽  
Vol 19 (73) ◽  
pp. 414
Author(s):  
صباح هادي الجاسم ◽  
فراس صدام عبد
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Shrinkage Method

المستخلص في هذا البحث تم استعمال توزيع ويبل بثلاثة معلمات هي معلمة الشكل ، معلمة القياس ومعلمة الموقع . أن هذا التوزيع يعتبر من توزيعات الفشل الملائمة عندما تكون معدلات الفشل عاليه نسبياً في بداية   تشكيل المكائن ومن ثم تبدأ هذه المعدلات بالتناقص تدريجياً بزيادة الزمن. أما بالنسبة للجانب الرئيسي من هذا البحث وهو الجانب التجريبي ، فقد تم في هذا الجانب أجراء مقارنه بين مقدرات طريقة التقلص(Shrinkage Method) وطريقة الإمكان الأعظم(Maximum likelihood Method) لمعلمات ودالة المعوليه لهذا التوزيع باستعمال المقياسين الإحصائيين (MSE) و(MAPE) وذلك بعد أن تم توظيف طريقة (Monte – Carlo) مونت-كارلو للمحاكاة ،علماً بأن حجوم العينات المستعملة والملائمة لطبيعة البحوث من هذا النوع هي العينة الصغيرة n=20,30)) ، والعينة المتوسطة (n=50) والعينة الكبيرة (n=100)، ولقد تم التوصل في هذا البحث الى أفضلية طريقة التقلص لتقدير المعلمات في حين كانت الافضليه لطريقة الإمكان الأعظم لتقدير دالة المعوليه .

scholarly journals When Was Westerhus Churchyard in Use?

2021 ◽  
Vol 13 (1) ◽  
pp. 161-182
Author(s):  
Claes-Henric Siven
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Monte Carlo Simulations ◽  
Maximum Likelihood Estimator ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Likelihood Estimator ◽  
Raw Data ◽  
Mass Graves ◽  
Point Estimates

The period of use for the Swedish medieval churchyard of Westerhus has been estimated by the maximum likelihood method. Raw data consist of 30 calibrated '4C-dates of some of the skeletons from the site. Bias and other properties of the maximum likelihood estimator are analyzed via a number of Monte Carlo simulations. The point estimates imply that the site was used in the period 1073-1356, that is, a somewhat longer period than previously assumed. The estimated length of the period of use affects the interpretation ofthe great number ofburied children. Population calculations lead to the conclusion that the six agglomerations of children's graves cannot be interpreted as mass graves.

scholarly journals Analysis of selected mathematical models of high-cycle S-N characteristics

2017 ◽  
Vol 3 (20) ◽  
pp. 227-240
Author(s):  
Przemysław Strzelecki ◽  
Janusz Sempruch ◽  
Tomasz Tomaszewski
Keyword(s):  
Fatigue Life ◽  
Maximum Likelihood ◽  
Maximum Likelihood Method ◽  
Least Squares Method ◽  
Construction Materials ◽  
The Other ◽  
Likelihood Method ◽  
Staircase Method ◽  
Aisi 1045 ◽  
Fatigue Characteristics

The paper presents two approaches of determining S-N fatigue characteristics. The first is a commonly used and well-documented approach based on the least squares method and staircase method for limited fatigue life and fatigue limit, accordingly. The other approach employs the maximum likelihood method. The analysis of the parameters obtained through both approaches exhibited minor differences. The analysis was performed for four steel construction materials, i.e. C45+C, 45, SUS630 and AISI 1045. It should be noted that the quantity of samples required in the second approach is significantly smaller than with the first approach, which translates into lower duration and costs of tests.

scholarly journals Validation and Comparison of Different Statistical Models for the Prediction of Water Main Pipe Breaks in a Municipal Network in Québec, Canada

2016 ◽  
Vol 29 (1) ◽  
pp. 11-24 ◽  
Author(s):  
Sophie Duchesne ◽  
Babacar Toumbou ◽  
Jean-Pierre Villeneuve
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Method ◽  
Least Squares Method ◽  
Likelihood Method ◽  
Water Main ◽  
Pipe Replacement ◽  
The Impact ◽  
Over Time ◽  
Observation Period

In this study, three models for the simulation of the number of breaks in a water main network are presented and compared: linear regression, the Weibull-Exponential-Exponential (WEE), and the Weibull-Exponential-Exponential-Exponential (WEEE) models. These models were calibrated using a database of recorded breaks in a real water main network of a municipality in the province of Québec, for the observation period 1976 to 1996, with the least squares and the maximum likelihood methods. The ability of these models to predict breaks over time was then evaluated by comparing the predicted number of breaks for the years 1997 to 2007 with the observed breaks in the network over the same time period. Results show that if the period of observation is short (around 20 years), calibration of the WEE and WEEE models with the maximum likelihood method leads to estimates that are closer to the observations than when these models are calibrated with the least squares method. When the observation period is longer (around 30 years), the predictions obtained with the models calibrated using the maximum likelihood or the least squares methods are similar. However, the use of the maximum likelihood method for calibration is only possible when data for the occurrence of each break for each pipe of the network are available (a pipe being a homogeneous network segment between two adjacent street junctions). If this is not the case, a trend line will be sufficient to predict the number of breaks over time, though this type of curve does not allow to account for pipe replacement scenarios. If the only information available is the total number of breaks on the network each year, then the impact of replacement scenarios could be simulated with the WEE and WEEE models calibrated using the least squares method.

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