Comparison of Five Methods to Estimate the Parameters for the Three-Parameter Lindley Distribution with Application to Life Data

We have studied one of the most common distributions, namely, Lindley distribution, which is an important continuous mixed distribution with great ability to represent different systems. We studied this distribution with three parameters because of its high flexibility in modelling life data. The parameters were estimated by five different methods, namely, maximum likelihood estimation, ordinary least squares, weighted least squares, maximum product of spacing, and Cramér-von Mises. Simulation experiments were performed with different sample sizes and different parameter values. The different methods were compared on the generated data by mean square error and mean absolute error. In addition, we compared the methods for real data, which represent COVID-19 data in Iraq/Anbar Province.

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Weighted generalized Quasi Lindley distribution: Different methods of estimation, applications for Covid-19 and engineering data

AIMS Mathematics ◽

10.3934/math.2021688 ◽

2021 ◽

Vol 6 (11) ◽

pp. 11850-11878

Author(s):

SidAhmed Benchiha ◽

◽

Amer Ibrahim Al-Omari ◽

Naif Alotaibi ◽

Mansour Shrahili ◽

...

Keyword(s):

Least Squares ◽

Mean Squared Error ◽

Weighted Least Squares ◽

Stochastic Ordering ◽

Real Data ◽

Ordinary Least Squares ◽

Lifetime Distribution ◽

Model Parameters ◽

Lindley Distribution ◽

Von Mises

<abstract>Recently, a new lifetime distribution known as a generalized Quasi Lindley distribution (GQLD) is suggested. In this paper, we modified the GQLD and suggested a two parameters lifetime distribution called as a weighted generalized Quasi Lindley distribution (WGQLD). The main mathematical properties of the WGQLD including the moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis, stochastic ordering, median deviation, harmonic mean, and reliability functions are derived. The model parameters are estimated by using the ordinary least squares, weighted least squares, maximum likelihood, maximum product of spacing's, Anderson-Darling and Cramer-von-Mises methods. The performances of the proposed estimators are compared based on numerical calculations for various values of the distribution parameters and sample sizes in terms of the mean squared error (MSE) and estimated values (Es). To demonstrate the applicability of the new model, four applications of various real data sets consist of the infected cases in Covid-19 in Algeria and Saudi Arabia, carbon fibers and rain fall are analyzed for illustration. It turns out that the WGQLD is empirically better than the other competing distributions considered in this study.</abstract>

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Improving hydrological forecasts through temporal hierarchal reconciliation

10.5194/egusphere-egu21-13303 ◽

2021 ◽

Author(s):

Mohammad Sina Jahangir ◽

John Quilty

Keyword(s):

Least Squares ◽

Performance Metrics ◽

Moving Average ◽

Weighted Least Squares ◽

Absolute Error ◽

Efficiency Index ◽

Ordinary Least Squares ◽

Data Driven ◽

Wide Range ◽

Different Levels

Hydrological forecasts at different horizons are often made using different models. These forecasts are usually temporally inconsistent (e.g., monthly forecasts may not sum to yearly forecasts), which may lead to misaligned or conflicting decisions. Temporal hierarchal reconciliation (or simply, hierarchical reconciliation) methods can be used for obtaining consistent forecasts at different horizons. However, their effectiveness in the field of hydrology has not yet been investigated. Thus, this research assesses hierarchal reconciliation for precipitation forecasting due to its high importance in hydrological applications (e.g., reservoir operations, irrigation, drought and flood forecasting). Original precipitation forecasts (ORF) were produced using three different models, including &#8216;automatic&#8217; Exponential Time-Series Smoothing (ETS), Artificial Neural Networks (ANN), and Seasonal Auto-Regressive Integrated Moving Average (SARIMA). The forecasts were produced at six timescales, namely, monthly, 2-monthly, quarterly, 4-monthly, bi-annual, and annual, for 84 basins selected from the Canadian model parameter experiment (CANOPEX) dataset. Hierarchical reconciliation methods including Hierarchical Least Squares (HLS), Weighted Least Squares (WLS), and Ordinary Least Squares (OLS) along with the Bottom-Up (BU) method were applied to obtain consistent forecasts at all timescales.Generally, ETS and ANN showed the best and worst performance, respectively, according to a wide range of performance metrics (root mean square error (RMSE), normalized RMSE (nRMSE), mean absolute error (MAE), normalized MAE (nMAE), and Nash-Sutcliffe Efficiency index (NSE)). The results indicated that hierarchal reconciliation has a dissimilar impact on the ORFs&#8217; accuracy in different basins and timescales, improving the RMSE in some cases while decreasing it in others. Also, it was highlighted that for different forecast models, hierarchical reconciliation methods showed different levels of performance. According to the RMSE and MAE, the BU method outperformed the hierarchical methods for ETS forecasts, while for ANN and SARIMA forecasts, HLS and OLS improved the forecasts more substantially, respectively. The sensitivity of ORF to hierarchical reconciliation was assessed using the RMSE. It was shown that both accurate and inaccurate ORF could be improved through hierarchical reconciliation; in particular, the effectiveness of hierarchical reconciliation appears to be more dependent on the ORF accuracy than it is on the type of hierarchical reconciliation method.While in the present work, the effectiveness of hierarchical reconciliation for hydrological forecasting was assessed via data-driven models, the methodology can easily be extended to process-based or hybrid (process-based data-driven) models. Further, since hydrological forecasts at different timescales may have different levels of importance to water resources managers and/or policymakers, hierarchical reconciliation can be used to weight the different timescales according to the user&#8217;s preference/desired goals.

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Cubic Transmuted-G Family of Distributions and Its Properties

Stochastics and Quality Control ◽

10.1515/eqc-2017-0027 ◽

2018 ◽

Vol 33 (2) ◽

pp. 103-112 ◽

Cited By ~ 2

Author(s):

Muhammad Aslam ◽

Zawar Hussain ◽

Zahid Asghar

Keyword(s):

Maximum Likelihood ◽

Least Squares ◽

Weighted Least Squares ◽

Real Life ◽

Likelihood Estimation ◽

Real Data ◽

Information Criteria ◽

Data Sets ◽

New Family ◽

Family Of Distributions

Abstract In this article, a new family of distributions is introduced by using transmutation maps. The proposed family of distributions is expected to be useful in modeling real data sets. The genesis of the proposed family, including several statistical and reliability properties, is presented. Methods of estimation like maximum likelihood, least squares, weighted least squares, and maximum product spacing are discussed. Maximum likelihood estimation under censoring schemes is also considered. Further, we explore some special models of the proposed family of distributions and examined different properties of these special models. We compare three particular models of the proposed family with several existing distributions using different information criteria. It is observed that the proposed particular models perform better than different competing models. Applications of the particular models of the proposed family of distributions are finally presented to establish the applicability in real life situations.

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Evaluation for estimating of the PDF and the CDF of Generalized Inverted Exponential Distribution with Application in Industry

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.39 ◽

2020 ◽

pp. 507-522

Author(s):

Parisa Torkaman

Keyword(s):

Least Squares ◽

Exponential Distribution ◽

Mean Squared Error ◽

Weighted Least Squares ◽

Real Data ◽

Minimum Variance ◽

Cumulative Distribution ◽

Estimation Methods ◽

Data Set ◽

Better Than

The generalized inverted exponential distribution is introduced as a lifetime model with good statistical properties. This paper, the estimation of the probability density function and the cumulative distribution function of with five different estimation methods: uniformly minimum variance unbiased(UMVU), maximum likelihood(ML), least squares(LS), weighted least squares (WLS) and percentile(PC) estimators are considered. The performance of these estimation procedures, based on the mean squared error (MSE) by numerical simulations are compared. Simulation studies express that the UMVU estimator performs better than others and when the sample size is large enough the ML and UMVU estimators are almost equivalent and efficient than LS, WLS and PC. Finally, the result using a real data set are analyzed.

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A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries

Axioms ◽

10.3390/axioms10010025 ◽

2021 ◽

Vol 10 (1) ◽

pp. 25 ◽

Cited By ~ 2

Author(s):

Ehab Almetwally ◽

Randa Alharbi ◽

Dalia Alnagar ◽

Eslam Hafez

Keyword(s):

Statistical Model ◽

Least Squares ◽

Weighted Least Squares ◽

Likelihood Estimation ◽

Linear Representation ◽

Rate Function ◽

Estimation Methods ◽

Unknown Parameters ◽

Least Squares Estimators ◽

New Distribution

This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new lifetime distribution with two-parameter is introduced by a combination of inverted Topp-Leone distribution and modified Kies family to produce the modified Kies inverted Topp-Leone (MKITL) distribution, which covers a lot of application that both the traditional inverted Topp-Leone and the modified Kies provide poor fitting for them. This new distribution has many valuable properties as simple linear representation, hazard rate function, and moment function. We made several methods of estimations as maximum likelihood estimation, least squares estimators, weighted least-squares estimators, maximum product spacing, Crame´r-von Mises estimators, and Anderson-Darling estimators methods are applied to estimate the unknown parameters of MKITL distribution. A numerical result of the Monte Carlo simulation is obtained to assess the use of estimation methods. also, we applied different data sets to the new distribution to assess its performance in modeling data.

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WAVELET ESTIMATORS FOR LONG MEMORY IN STOCK MARKETS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024909005233 ◽

2009 ◽

Vol 12 (03) ◽

pp. 297-317 ◽

Cited By ~ 4

Author(s):

ANOUAR BEN MABROUK ◽

HEDI KORTAS ◽

SAMIR BEN AMMOU

Keyword(s):

Maximum Likelihood ◽

Least Squares ◽

Stock Returns ◽

Long Memory ◽

Weighted Least Squares ◽

Ordinary Least Squares ◽

Estimation Accuracy ◽

Approximate Maximum Likelihood ◽

Approximate Maximum Likelihood Estimator ◽

Wavelet Estimators

In this paper, fractional integrating dynamics in the return and the volatility series of stock market indices are investigated. The investigation is conducted using wavelet ordinary least squares, wavelet weighted least squares and the approximate Maximum Likelihood estimator. It is shown that the long memory property in stock returns is approximately associated with emerging markets rather than developed ones while strong evidence of long range dependence is found for all volatility series. The relevance of the wavelet-based estimators, especially, the approximate Maximum Likelihood and the weighted least squares techniques is proved in terms of stability and estimation accuracy.

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Tolerance Intervals in a Heteroscedastic Linear Regression Context with Applications to Aerospace Equipment Surveillance

International Journal of Quality Statistics and Reliability ◽

10.1155/2009/126283 ◽

2009 ◽

Vol 2009 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

Janet Myhre ◽

Daniel R. Jeske ◽

Michael Rennie ◽

Yingtao Bi

Keyword(s):

Linear Regression ◽

Least Squares ◽

Performance Metrics ◽

Weighted Least Squares ◽

Ordinary Least Squares ◽

Automated System ◽

Least Squares Regression ◽

Data Set ◽

Tolerance Intervals ◽

Power Of The Test

A heteroscedastic linear regression model is developed from plausible assumptions that describe the time evolution of performance metrics for equipment. The inherited motivation for the related weighted least squares analysis of the model is an essential and attractive selling point to engineers with interest in equipment surveillance methodologies. A simple test for the significance of the heteroscedasticity suggested by a data set is derived and a simulation study is used to evaluate the power of the test and compare it with several other applicable tests that were designed under different contexts. Tolerance intervals within the context of the model are derived, thus generalizing well-known tolerance intervals for ordinary least squares regression. Use of the model and its associated analyses is illustrated with an aerospace application where hundreds of electronic components are continuously monitored by an automated system that flags components that are suspected of unusual degradation patterns.

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Improved Dividend Estimation from Intraday Quotes

Entropy ◽

10.3390/e24010095 ◽

2022 ◽

Vol 24 (1) ◽

pp. 95

Author(s):

Pontus Söderbäck ◽

Jörgen Blomvall ◽

Martin Singull

Keyword(s):

Least Squares ◽

Weighted Least Squares ◽

Statistical Significance ◽

Ordinary Least Squares ◽

Options Market ◽

Market Data ◽

Out Of Sample ◽

Intraday Data ◽

Least Squares Estimates ◽

Estimation Quality

Liquid financial markets, such as the options market of the S&P 500 index, create vast amounts of data every day, i.e., so-called intraday data. However, this highly granular data is often reduced to single-time when used to estimate financial quantities. This under-utilization of the data may reduce the quality of the estimates. In this paper, we study the impacts on estimation quality when using intraday data to estimate dividends. The methodology is based on earlier linear regression (ordinary least squares) estimates, which have been adapted to intraday data. Further, the method is also generalized in two aspects. First, the dividends are expressed as present values of future dividends rather than dividend yields. Second, to account for heteroscedasticity, the estimation methodology was formulated as a weighted least squares, where the weights are determined from the market data. This method is compared with a traditional method on out-of-sample S&P 500 European options market data. The results show that estimations based on intraday data have, with statistical significance, a higher quality than the corresponding single-times estimates. Additionally, the two generalizations of the methodology are shown to improve the estimation quality further.

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Jaya algorithm in estimation of P[X > Y] for two parameter Weibull distribution

AIMS Mathematics ◽

10.3934/math.2022156 ◽

2022 ◽

Vol 7 (2) ◽

pp. 2820-2839

Author(s):

Saurabh L. Raikar ◽

◽

Dr. Rajesh S. Prabhu Gaonkar ◽

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Weibull Distribution ◽

Least Squares ◽

Weighted Least Squares ◽

Real Life ◽

Likelihood Estimation ◽

Estimation Methods ◽

Jaya Algorithm ◽

Two Parameter

<abstract> Jaya algorithm is a highly effective recent metaheuristic technique. This article presents a simple, precise, and faster method to estimate stress strength reliability for a two-parameter, Weibull distribution with common scale parameters but different shape parameters. The three most widely used estimation methods, namely the maximum likelihood estimation, least squares, and weighted least squares have been used, and their comparative analysis in estimating reliability has been presented. The simulation studies are carried out with different parameters and sample sizes to validate the proposed methodology. The technique is also applied to real-life data to demonstrate its implementation. The results show that the proposed methodology's reliability estimates are close to the actual values and proceeds closer as the sample size increases for all estimation methods. Jaya algorithm with maximum likelihood estimation outperforms the other methods regarding the bias and mean squared error. </abstract>

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Robust Weighted Least Squares Estimation of Regression Parameter in the Presence of Outliers and Heteroscedastic Errors

Jurnal Teknologi ◽

10.11113/jt.v71.3609 ◽

2014 ◽

Vol 71 (1) ◽

Cited By ~ 1

Author(s):

Bello Abdulkadir Rasheed ◽

Robiah Adnan ◽

Seyed Ehsan Saffari ◽

Kafi Dano Pati

Keyword(s):

Linear Regression ◽

Least Squares ◽

Robust Regression ◽

Weighted Least Squares ◽

Ordinary Least Squares ◽

Least Square ◽

Model Parameters ◽

Constant Variance ◽

Regression Parameters ◽

M Estimation

In a linear regression model, the ordinary least squares (OLS) method is considered the best method to estimate the regression parameters if the assumptions are met. However, if the data does not satisfy the underlying assumptions, the results will be misleading. The violation for the assumption of constant variance in the least squares regression is caused by the presence of outliers and heteroscedasticity in the data. This assumption of constant variance (homoscedasticity) is very important in linear regression in which the least squares estimators enjoy the property of minimum variance. Therefor e robust regression method is required to handle the problem of outlier in the data. However, this research will use the weighted least square techniques to estimate the parameter of regression coefficients when the assumption of error variance is violated in the data. Estimation of WLS is the same as carrying out the OLS in a transformed variables procedure. The WLS can easily be affected by outliers. To remedy this, We have suggested a strong technique for the estimation of regression parameters in the existence of heteroscedasticity and outliers. Here we apply the robust regression of M-estimation using iterative reweighted least squares (IRWLS) of Huber and Tukey Bisquare function and resistance regression estimator of least trimmed squares to estimating the model parameters of state-wide crime of united states in 1993. The outcomes from the study indicate the estimators obtained from the M-estimation techniques and the least trimmed method are more effective compared with those obtained from the OLS.

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