Huber Estimation for Uncertain Autoregressive Model

2021 ◽  
pp. 2150010
Author(s):  
Zhe Liu
Keyword(s):  
Autoregressive Model ◽  
Prediction Interval ◽  
Real Data ◽  
Least Square ◽  
Time Analysis ◽  
Unknown Parameters ◽  
Least Square Estimation ◽  
Uncertain Variables ◽  
Carbon Dioxide Concentrations ◽  
Future Value

Traditional time analysis deals with observations in chronological order assuming the observations are precise numbers under the framework of probability theory, whereas data are imprecisely collected in many cases. This paper characterizes the imprecisely observed data as uncertain variables and estimates the unknown parameters in the uncertain autoregressive model using Huber loss function, which is more flexible compared with other robust estimations for a pre-given [Formula: see text] that regulates the amount of robustness. Then prediction value and prediction interval of the future value are given. What is more, a method to choose [Formula: see text] by cross-validation is proposed. At last, numerical examples show our methods in detail and illustrate the robustness of Huber estimation by comparing it with the least square estimation. black Our methods are also applied to a set of real data with carbon dioxide concentrations.

Mathematical properties of the Kumaraswamy-Lindley distribution and its applications

2017 ◽  
Vol 5 (1) ◽  
pp. 17
Author(s):  
Hamdy Salem ◽  
Abd-Elwahab Hagag
Keyword(s):  
Hazard Function ◽  
Likelihood Estimation ◽  
Real Data ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Least Square ◽  
Estimation Of Parameters ◽  
Least Square Estimation ◽  
Mathematical Properties ◽  
Composite Distribution

In this paper, a composite distribution of Kumaraswamy and Lindley distributions namely, Kumaraswamy-Lindley Kum-L distribution is introduced and studied. The Kum-L distribution generalizes sub-models for some widely known distributions. Some mathematical properties of the Kum-L such as hazard function, quantile function, moments, moment generating function and order statistics are obtained. Estimation of parameters for the Kum-L using maximum likelihood estimation and least square estimation techniques are provided. To illustrate the usefulness of the proposed distribution, simulation study and real data example are used.

scholarly journals Transmutation of the two parameters Rayleigh distribution

2016 ◽  
Vol 4 (2) ◽  
pp. 95
Author(s):  
Ehsan Ullah ◽  
Mirza Shahzad
Keyword(s):  
Real Data ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Rayleigh Distribution ◽  
Least Square ◽  
Least Square Estimation ◽  
Proposed Model ◽  
Method Of Least Square ◽  
Two Parameters ◽  
Mean Variance

In this study, transmuted two parameters Rayleigh distribution is proposed using quadratic rank transmutation map. This proposed distribution is more flexible and versatile than two parameters Rayleigh distribution. Some properties of the proposed distribution are derived such as moments, moment generating function, mean, variance, median, quantile function, reliability, and hazard function. The parameter estimation is approached through the method of least square estimation. The th and joint order statistics are also derived for the proposed distribution. The application of proposed model illustrated and compared using real data.

scholarly journals Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Wael S. Abu El Azm ◽  
Ehab M. Almetwally ◽  
Abdulaziz S. Alghamdi ◽  
Hassan M. Aljohani ◽  
Abdisalam Hassan Muse ◽  
...  
Keyword(s):  
Real Life ◽  
Distance Estimation ◽  
Least Square ◽  
Likelihood Method ◽  
Interval Length ◽  
Unknown Parameters ◽  
Least Square Estimation ◽  
Monte Carlo Simulation Study ◽  
Von Mises ◽  
Percentile Bootstrap

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters θ 1 , θ 2 , and β and the system of stress-strength reliability R = P Y < X for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute-log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.

Maximum likelihood estimation for uncertain autoregressive model with application to carbon dioxide emissions

2021 ◽  
Vol 40 (1) ◽  
pp. 1391-1399
Author(s):  
Dan Chen ◽  
Xiangfeng Yang
Keyword(s):  
Maximum Likelihood ◽  
Autoregressive Model ◽  
Carbon Dioxide Emissions ◽  
Goodness Of Fit ◽  
Mean Squared Error ◽  
Likelihood Estimation ◽  
Estimation Methods ◽  
Observation Data ◽  
Unknown Parameters ◽  
Uncertain Variables

The objective of uncertain time series analysis is to explore the relationship between the imprecise observation data over time and to predict future values, where these data are uncertain variables in the sense of uncertainty theory. In this paper, the method of maximum likelihood is used to estimate the unknown parameters in the uncertain autoregressive model, and the unknown parameters of uncertainty distributions of the disturbance terms are simultaneously obtained. Based on the fitted autoregressive model, the forecast value and confidence interval of the future data are derived. Besides, the mean squared error is proposed to measure the goodness of fit among different estimation methods, and an algorithm is introduced. Finally, the comparative analysis of the least squares, least absolute deviations, and maximum likelihood estimations are given, and two examples are presented to verify the feasibility of this approach.

scholarly journals THE LAST TIME ANALYSIS OF THE FOLLOWING FAMILY IN AL-QUR'AN PERSPECTIVE

2020 ◽  
Vol 2 (2) ◽  
pp. 18-20
Author(s):  
Rona Dwi Rahmah
Keyword(s):  
Natural Disasters ◽  
Correlation Coefficient ◽  
Real Data ◽  
Least Square Method ◽  
Least Square ◽  
Single Word ◽  
Time Analysis ◽  
The Earth ◽  
Relationship Of ◽  
The Relationship

Abstract. Earthquakes are natural disasters caused by shocks on the earth due to faults and the sudden movement of tectonic rocks that make up the earth's crust. This study of earthquakes will be interesting if explored further from the perspective of the Qur'an because in the Qur'an there are many verses that speak of earthquakes. As explained in the Qur'an Al-Zalzalah verses 1-8. On February 14 2016 to February 23 2016 aftershocks occurred in the Klagon Village Area, Saradan District, Madiun. To analyze when the end of aftershocks ends by using the Least Square Method and the relationship of frequency of aftershocks to times that include the Omori, Omogi 1, Omogi 2 and Utsu methods. The conclusion of this study is the Omogi 2 method which has obtained the correlation coefficient r = 0.195 from the correlation value -1 ≤ r ≤ 1, with the aftershocks ending on day 464 and from the comparison of aftershock frequency corresponding to the graph between the results data calculations with real data (actual data) namely the Omogi 2 method. And basically the term earthquake in the Qur'an can still be said not to make the verses interpreted as a single word containing the meaning of the earthquake as a brief explanation of aftershocks in the perspective of the Qur'an.

scholarly journals Estimation of density and distribution functions of a Burr X distribution

2018 ◽  
Vol 52 (1) ◽  
pp. 43-59
Author(s):  
AMULYA KUMAR MAHTO ◽  
YOGESH MANI TRIPATH ◽  
SANKU DEY
Keyword(s):  
Mean Squared Error ◽  
Real Data ◽  
Distribution Functions ◽  
Least Square ◽  
Efficient Estimation ◽  
Cumulative Distribution ◽  
Estimation Methods ◽  
Unbiased Estimation ◽  
Least Square Estimation ◽  
Data Set

Burr type X distribution is one of the members of the Burr family which was originally derived by Burr (1942) and can be used quite effectively in modelling strength data and also general lifetime data. In this article, we consider efficient estimation of the probability density function (PDF) and cumulative distribution function (CDF) of Burr X distribution. Eight different estimation methods namely maximum likelihood estimation, uniformly minimum variance unbiased estimation, least square estimation, weighted least square estimation, percentile estimation, maximum product estimation, Cremer-von-Mises estimation and Anderson-Darling estimation are considered. Analytic expressions for bias and mean squared error are derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Finally, a real data set has been analyzed for illustrative purposes.

scholarly journals New Method for Generating New Families of Distributions

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 726
Author(s):  
Lamya A. Baharith ◽  
Wedad H. Aljuhani
Keyword(s):  
Exponential Distribution ◽  
Real Data ◽  
Rate Function ◽  
New Method ◽  
Likelihood Method ◽  
Alpha Power ◽  
Unknown Parameters ◽  
Failure Rate Function ◽  
New Approach ◽  
Numerical Studies

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this distribution can take asymmetric and near-symmetric shapes. Various asymmetric shapes, such as decreasing, increasing, L-shaped, near-symmetrical, and right-skewed shapes, are observed for the related failure rate function, making it more tractable for many modeling applications. Some significant mathematical features of the suggested distribution are determined. Estimates of the unknown parameters of the proposed distribution are obtained using the maximum likelihood method. Furthermore, some numerical studies were carried out, in order to evaluate the estimation performance. Three practical datasets are considered to analyze the usefulness and flexibility of the introduced distribution. The proposed alpha power Weibull–exponential distribution can outperform other well-known distributions, showing its great adaptability in the context of real data analysis.

An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications

2020 ◽  
Vol 70 (4) ◽  
pp. 953-978
Author(s):  
Mustafa Ç. Korkmaz ◽  
G. G. Hamedani
Keyword(s):  
Hazard Function ◽  
Mixture Distribution ◽  
Real Data ◽  
Quantile Function ◽  
Estimation Methods ◽  
Data Sets ◽  
Unknown Parameters ◽  
Lorenz Curves ◽  
Proposed Model ◽  
New Distribution

AbstractThis paper proposes a new extended Lindley distribution, which has a more flexible density and hazard rate shapes than the Lindley and Power Lindley distributions, based on the mixture distribution structure in order to model with new distribution characteristics real data phenomena. Its some distributional properties such as the shapes, moments, quantile function, Bonferonni and Lorenz curves, mean deviations and order statistics have been obtained. Characterizations based on two truncated moments, conditional expectation as well as in terms of the hazard function are presented. Different estimation procedures have been employed to estimate the unknown parameters and their performances are compared via Monte Carlo simulations. The flexibility and importance of the proposed model are illustrated by two real data sets.

scholarly journals Bayesian and E-Bayesian Estimations of Bathtub-Shaped Distribution under Generalized Type-I Hybrid Censoring

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 934
Author(s):  
Yuxuan Zhang ◽  
Kaiwei Liu ◽  
Wenhao Gui
Keyword(s):  
Bayesian Method ◽  
Real Data ◽  
Loss Functions ◽  
Type I ◽  
Statistical Efficiency ◽  
Unknown Parameters ◽  
Data Set ◽  
Hybrid Censoring ◽  
Bayesian Estimations ◽  
Generalized Type

For the purpose of improving the statistical efficiency of estimators in life-testing experiments, generalized Type-I hybrid censoring has lately been implemented by guaranteeing that experiments only terminate after a certain number of failures appear. With the wide applications of bathtub-shaped distribution in engineering areas and the recently introduced generalized Type-I hybrid censoring scheme, considering that there is no work coalescing this certain type of censoring model with a bathtub-shaped distribution, we consider the parameter inference under generalized Type-I hybrid censoring. First, estimations of the unknown scale parameter and the reliability function are obtained under the Bayesian method based on LINEX and squared error loss functions with a conjugate gamma prior. The comparison of estimations under the E-Bayesian method for different prior distributions and loss functions is analyzed. Additionally, Bayesian and E-Bayesian estimations with two unknown parameters are introduced. Furthermore, to verify the robustness of the estimations above, the Monte Carlo method is introduced for the simulation study. Finally, the application of the discussed inference in practice is illustrated by analyzing a real data set.

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