Estimation Of Distribution Research Articles

On the role of Volterra integral equations in self-consistent, product-limit, inverse probability of censoring weighted, and redistribution-to-the-right estimators for the survival function.

This paper reconsiders several results of historical and current importance to nonparametric estimation of the survival distribution for failure in the presence of right-censored observation times, demonstrating in particular how Volterra integral equations help inter-connect the resulting estimators. The paper begins by considering Efron's self-consistency equation, introduced in a seminal 1967 Berkeley symposium paper. Novel insights provided in the current work include the observations that (i) the self-consistency equation leads directly to an anticipating Volterra integral equation whose solution is given by a product-limit estimator for the censoring survival function; (ii) a definition used in this argument immediately establishes the familiar product-limit estimator for the failure survival function; (iii) the usual Volterra integral equation for the product-limit estimator of the failure survival function leads to an immediate and simple proof that it can be represented as an inverse probability of censoring weighted estimator; (iv) a simple identity characterizes the relationship between natural inverse probability of censoring weighted estimators for the survival and distribution functions of failure; (v) the resulting inverse probability of censoring weighted estimators, attributed to a highly influential 1992 paper of Robins and Rotnitzky, were implicitly introduced in Efron's 1967 paper in its development of the redistribution-to-the-right algorithm. All results developed herein allow for ties between failure and/or censored observations.

Weighted Jump in Random Walk graph sampling

Graph sampling is a challenging problem in network analysis due to the complex structures of networks. Currently, a series of graph sampling algorithms based on random walks achieve good results in graph sampling tasks. However, the existing methods often reduce the conductance of graphs, causing the sampler to stay in the same node for a long time. This results in undersampling. In this paper, we propose a novel Weighted Jump Random Walk (WJRW) algorithm to generate representative samples. We design a parameter in the WJRW algorithm that can adjust the proportions of random walk and random jump in every step. According to the issue of repeated sample nodes in the Generalized Maximum Degree (GMD) method and the problem of large deviations in the Simple Random Walk (SRW), WJRW addresses the weaknesses of the GMD method and enhances the diffusion of a random walker on graphs, leading to a more representative sample. Then, WJRW addresses the issue of large deviations in SRW and enhances the efficiency of the unbiased estimator. By generating smoother stationary distributions. Numerical experiments with extensive real-world networks verify that our method achieves higher accuracy than classical and state-of-the-art methods in estimating distribution estimation.

Healthy Lifespan Statistics Derived From Cross-Sectional Prevalence Data Using the Sullivan Method are Informative Summary Measures of Population Health

Health expectancy (HE), commonly derived from cross-sectional prevalence data using the Sullivan method, serves as the most frequently used summary measure of population health. Like lifespan distribution statistics, which are often discussed alongside life expectancy (LE) in demographic studies, analogous statistics on healthy lifespans can provide valuable information on population health. We examine whether healthy lifespan distribution statistics beyond HE can be estimated based on cross-sectional prevalence data and the life table, the data inputs of the Sullivan method. To do so, we treat the Sullivan method as an extension of the stationary population model to health and distinguish between health conditions with and without recovery from the state of decreased health. Our empirical demonstration is based on the prevalence of chronic diseases in selected European countries in 2017 from the Survey of Health, Ageing and Retirement in Europe (SHARE), as well as on life tables from EUROSTAT. We find that the Sullivan method, when considered as an extension of the stationary population model to health, allows for the estimation of a healthy survival distribution and its statistics, beyond HE, for health characteristics with no recovery from the state of decreased health. We show that for such health conditions, the method requires that the number of persons in full health in a stationary population does not increase with age. We argue that for such health dimensions, HE conditional on being in good health at the life table radix age is of relevance for health policy interventions. In our empirical application, we show that the conditional and unconditional measures of HE can give substantially different pictures of population health. Furthermore, we show that across European countries, in contrast to the negative relationship between LE and lifespan inequality, higher HE is associated with greater inequality in healthy years lived when conditional on being healthy at age 50. Overall, the Sullivan method, when considered as an extension of the stationary population model, proves to be a valuable tool for deriving summary statistics of population health beyond HE, which are highly relevant to public policy. * This article belongs to a special issue on “Levels and Trends of Health Expectancy: Understanding its Measurement and Estimation Sensitivity”.

Analysis of human ABO Blood Group as a risk factor with knee osteoarthritis at tertiary care hospital in Pakistan.

To investigate the association and risk estimation of ABO blood group distribution and clinical attributes in patients with Knee Osteoarthritis. This was a hospital-based study conducted at, Liaquat University Hospital Hyderabad from December, 2019 to December, 2022 to investigate this least researched area of highly prevalent musculoskeletal disease in Pakistan. Non-Probability Convenience Sampling was used for selecting 190 cases of confirm Knee Osteoarthritis patients diagnosed by Orthopedic surgeon based on standard clinical and radiographic criteria. Data were analyzed using IBM-SPSS version 23.0. Percentages and frequencies were counted for categorical data. Pearson Chi Square test and fisher's exact test were used to check the association and Multinomial Logistic Regression was used to estimate the risk for moderate and severe kellgren grading Knee Osteoarthritis (KOA) cases with ABO blood grouping in comparison of mild Kellgren grading. A total of 190 cases of Knee Osteoarthritis (KOA) were included in the study. Females (61.6%) and patients with age 50 and above were 40.5 % were found in greater proportion. Majority (41.6%) were classified radiologically as mild cases with O group (39.5%) and positive Rh antigen (95.8%). Strong association (p = <0.01) was found between gender, age group and ABO blood group with KOA radiological Kellgren and Lawrence score. There is strong relation in between radiological grading of knee osteoarthritis severity and A blood group, gender and age.

Aerosol deposition and emission from a United States Pharmacopeia‐induction port when using a pressurized metered‐dose inhaler with and without a valved holding chamber

AbstractThe testing of pharmaceutical aerosols includes measuring the aerodynamic particle size distribution, which is usually performed on cascade impactors. In the next generation impactor (NGI), the aerosol dose is introduced through an induction port (IP), being separated into different aerodynamic diameter ranges by seven stages and a micro‐orifice collector. The IP plays an important role in estimating the oropharyngeal deposition fraction. While the IP retains mainly large particles, it also tends to retain particles in the respirable range. In this work, the deposition of particles in the IP of an NGI is studied, both experimentally and through computational fluid dynamic (CFD) simulations. Experimental tests are conducted both with the pressurized metered‐dose inhaler (pMDI) alone and in conjunction with a cylindrical valved holding chamber (VHC). For each case, the total mass deposition that occurs in the VHC, IP, and NGI stages is measured, as well as the mass median aerodynamic diameter of the aerosol leaving the IP. The CFD simulations show minimal variations in results regardless of the injection method due to the relatively low velocity and spray angle of the pMDI‐generated aerosol. If the flap‐valve is removed from the VHC, significant particle deposition occurs upstream or within the valve region, with downstream air recirculation contributing to small particle deposition. Based on the CFD results, a correlation is proposed to accurately predict the particle escape fraction of the IP and VHC, which allows estimation of the tannin distribution of particles collected in the NGI stages, especially those corresponding to the range of respirable particles.

Unveiling dynamics changes: Singular spectrum analysis-based method for detecting concept drift in industrial data streams

Industrial data streams frequently experience concept drifts. Current drift detection methods, focusing on prediction performance or data distribution, often neglect temporal dependencies and require prior distribution assumptions. These limitations have catalyzed the development of dynamic theory-based approaches that identify drifts by discerning variations in data dynamics. However, the majority of these exhibit high complexity in dynamics characterization and still underperform in industrial drift detection. To overcome these challenges, we propose a singular spectrum analysis-based drift detection method for industry data streams, comprising three modules: common dynamics extraction, noise-robust drift detection, and threshold adaptation. Our approach uses singular spectrum analysis (SSA) as the dynamics representation method instead of traditional high-dimensional phase space embedding and Fourier transform. SSA allows for the efficient description of principal dynamics by intelligently mixing the Fourier basis functions according to given data. The first two modules are specifically engineered to address the common dynamics and stochastic noise inherent in industrial data streams through a well-designed online common dynamics extraction algorithm and a noise-robust detection index respectively. Threshold adaptation module provides a more reasonable way of updating the drift threshold based on the extreme value theory, ensuring global consistency for parameter estimation of extreme value distribution. Our proposed method was tested on simulated examples and a real-world sintering process, exhibiting enhanced drift detection performance.

Measuring The Spatio-Temporal Distribution of Sulfur Dioxide (SO2) with Copernicus Sentinel-5P Near Real Time in Medan City

This study aims to monitor the impact of sulfur dioxide (SO2), a significant contributor to atmospheric corrosion and associated with air pollution, in urban and industrial areas through empirical models of spatial distribution and temporal estimation in Medan. Datasets of Sentinel 5P NRTI are worth using to monitor the formation of sulfate aerosols or SO2 emissions from natural sources and anthropogenic origin in near real-time from 2019 until 2023 on the local scale, Medan. The source dataset covers 1,11 km of spatial image resolution on orbital in daily revisit time with a spectral bound between the ultraviolet and the shortwave infrared. The measurement of atmospheric SO2 concentrations is generated by enumerating the SO2 vertical column density technique at the soil surface with the cloud-based platform The Earth Engine (EE) Code Editor to handle large data sizes and produce interactive mapping. Based on the availability of auxiliary results, the number of spreading SO2 concentrations fell from 0,92 mmol/m2 in 2019 to 0,41 mmol/m2 in 2023, a reduction of 49.98 percent. The average of SO2 concentrations has decreased substantially over the years. Consequently, these pollutants have recorded long-term health and meteorological parameters impact that substances can rise or reduce acid rain.Keywords: Air pollution, Meteorological parameters, Sentinel 5P NRTI, SO2

Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

AbstractRecently, a growing interest is evident in modelling dependent competing risks in lifetime prognosis problems. In this work, we propose to model the dependent competing risks by Marshal‐Olkin bivariate exponential distribution. The observable data consists of a number of failures due to different causes across different time intervals. The failure count data is common in instances like one‐shot devices where the state of the subjects is inspected at different inspection times rather than the exact failure times. The point estimation of the lifetime distribution in the presence of competing risk has been studied through a divergence‐based robust estimation method called minimum density power divergence estimation (MDPDE) with and without constraint. The optimal value of the tuning parameter has been obtained. The testing of the hypothesis is performed based on a Wald‐type test statistic. The influence function is derived for the point estimator and the test statistic, reflecting the degree of robustness. Another key contribution of this work is determining the optimal inspection times based on predefined objectives. This article presents the determination of multi‐criteria‐based optimal design. Population‐based heuristic algorithm nondominated sorting‐based multiobjective Genetic algorithm is exploited to solve this optimization problem.

Using dispersion models at microscale to assess long-term air pollution in urban hot spots: A FAIRMODE joint intercomparison exercise for a case study in Antwerp

In the framework of the Forum for Air Quality Modelling in Europe (FAIRMODE), a modelling intercomparison exercise for computing NO2 long-term average concentrations in urban districts with a very high spatial resolution was carried out. This exercise was undertaken for a district of Antwerp (Belgium). Air quality data includes data recorded in air quality monitoring stations and 73 passive samplers deployed during one-month period in 2016. The modelling domain was 800 × 800 m2. Nine modelling teams participated in this exercise providing results from fifteen different modelling applications based on different kinds of model approaches (CFD – Computational Fluid Dynamics-, Lagrangian, Gaussian, and Artificial Intelligence). Some approaches consisted of models running the complete one-month period on an hourly basis, but most others used a scenario approach, which relies on simulations of scenarios representative of wind conditions combined with post-processing to retrieve a one-month average of NO2 concentrations.The objective of this study is to evaluate what type of modelling system is better suited to get a good estimate of long-term averages in complex urban districts. This is very important for air quality assessment under the European ambient air quality directives. The time evolution of NO2 hourly concentrations during a day of relative high pollution was rather well estimated by all models. Relative to high resolution spatial distribution of one-month NO2 averaged concentrations, Gaussian models were not able to give detailed information, unless they include building data and street-canyon parameterizations. The models that account for complex urban geometries (i.e. CFD, Lagrangian, and AI models) appear to provide better estimates of the spatial distribution of one-month NO2 averages concentrations in the urban canopy. Approaches based on steady CFD-RANS (Reynolds Averaged Navier Stokes) model simulations of meteorological scenarios seem to provide good results with similar quality to those obtained with an unsteady one-month period CFD-RANS simulations.

A roundtrip probability estimation method for mechanical equipment fault detection under imbalanced samples

Aiming at high misdetection of mechanical faults under imbalanced samples, a roundtrip probability-based method is proposed. By roundtrip mapping between latent variables and real fault data, biased estimation of the probability distribution of real fault data is obtained. Further, virtual fault data are sampled according to such distribution to increase sample amount. For recognition of real and virtual data, loss function based on binary cross-entropy is designed. For reconstruction between fault data and its roundtrip mapped results, objective function based on mean square error is designed. Thus, it preserves boundary data and avoids too many virtual data in central area. Meanwhile, a strategy for eliminating abnormal samples is designed to reduce boundary deviation. For supporting the advantage of roundtrip, in-depth reasons for misdetection are analyzed from empirical risk and structural risk. Experiments on 30 benchmark imbalanced test sets show that fault detection rate increases after amount enhancement. Additionally, it is verified on blade cracking and bearing fault detection. Results show that F1 score increases from 0.485 to 0.51 and 0.725 to 0.775 for such two cases.

Analysis of the turbostratic structures in PAN-based carbon fibers with wide-angle x-ray diffraction

Carbon fiber (CF) is a versatile material renowned for its excellent mechanical, thermal, and electrical properties. Polyacrylonitrile (PAN)-based CFs dominate the market in part due to their high tensile strength, rendering them suitable for structural applications in a wide variety of applications ranging from sporting goods to aerospace. Over five decades of commercial development of PAN-based CF has resulted in a range of varieties with different tensile moduli and tensile strengths. The microstructures, nanostructures, and crystal structures of PAN-based CF all play pivotal roles in the macroscale properties of this material. In particular, the crystal structure and crystallite orientation in CF is closely related to the mechanical properties. The crystal structure of PAN-based CF generally consists of turbostratic carbon, a disordered form of graphite, the characteristics of which can be effectively characterized in a bulk format through wide-angle x-ray diffraction (WAXD). In this work, we employed a three-part approach to the analysis of WAXD patterns collected from four intermediate modulus PAN-based CFs. The approach incorporates a Scherrer analysis, a Debye analysis, and an orientational analysis to provide precise estimates of crystallite sizes, crystallite distributions, and crystallite orientations with the fiber axis. The results presented here suggest that intermediate modulus PAN-based CF mostly consists of small turbostratic crystallites (<4 nm), with larger crystallites having increased orientation with the fiber axis. The results here imply the presence of curvature and/or wrinkling of the turbostratic layers within the CF structure.

Assessment of the intake and sources of total sugar among Filipino children and adolescents based on national consumption data

Diabetes is a public health issue in the Philippines, commonly resulting from high sugar intake. This study examined the consumption and sources of total sugar among Filipinos aged 7 months to 18 years old using nationally representative data. Total sugar consumption was estimated using two non-consecutive 24-h recalls and food composition tables. Mean intakes were converted into usual intakes to reduce measurement errors using the Software for intake distribution estimation software. The percentile of usual intakes, energy from sugar consumption, and the amount of sugar in minimally processed and processed foods for different age groups were calculated using STATA. Changes in consumption between 2008 and 2013 were also examined. Children aged 7 &amp;ndash; 11 months old consumed 26% of energy from total sugar, higher than that of older age groups. Those in the richest income quintile and living in urban areas had the highest percentage intakes. The median intake of total sugar exceeded 25 g/day, with children aged 1 &amp;ndash; 2 years old in the highest income quintile consuming the greatest amount (54.7 g/day). Total sugar comprised 5.3 &amp;ndash; 30.9% of energy intake. Processed foods contributed 64 &amp;ndash; 75% of total sugar. Food categories that contributed the highest amounts (&amp;ge;10%) of total sugar to different age groups were as follows: (i) 7 &amp;ndash; 11 months old: milk formula, milk products, baked products, sugar, and sweets; (ii) 1 &amp;ndash; 2 years old: Sugar, sweets, milk formula, milk products, baked products, and non-alcoholic beverages; and (iii) 3 &amp;ndash; 18 years old: Sugar, sweets, baked products, and non-alcoholic beverages. From 2008 to 2013, the consumption of sweetened beverages and baked products increased significantly, while the consumption of fresh and minimally processed fruits and vegetables decreased significantly. Therefore, it is suggested to reduce added sugar in processed foods, provide unsweetened alternatives, reduce the intake of table sugar, baked goods, and confections, and increase the intake of fresh fruits and vegetables.

Comparative study between poorly differentiated thyroid cancer and anaplastic thyroid cancer: real-world pathological distribution, death attribution, and prognostic factor estimation.

The clinic-pathological boundary between poorly differentiated thyroid cancer (PDTC) and anaplastic thyroid cancer (ATC) is unclear due to a wide spectrum of histopathological features and the rarity of the disease. In addition to that, with the highest mortality rate and non-standard treatment modality, the PDTC/ATC population has not been subjected to comprehensive description and comparison with the extent of histological characteristics, therapeutic response, prognostic factors, and death attribution analysis. A total of 4,947 PDTC/ATC patients from 2000 to 2018 were identified from the Surveillance, Epidemiology, and End Results (SEER) database. Kaplan-Meier survival curve estimation and Cox proportional hazard regression were applied. Overall, the 5- and 10-year DSS for PDTC were 71.9% and 68.0%, respectively, whereas the 5- and 10-year OS are 59.3% and 51.2%, respectively. The median survival time for ATC patients was 3 months with 1-year OS being 26.9% and 1-year DSS being 31.2%. During the follow-up period, 68.1% of the PDTC/ATC cohort were dead, 51.6% of which were attributed to thyroid malignancies and 16.5% to non-thyroid causes. The top three common non-thyroid causes of death were miscellaneous cancers, lower respiratory system disease, and heart disease. The histological feature of papillary thyroid cancer (PTC) was the leading pathological category for PDTC patients (51.7%), whereas 76.7% of ATC patients' pathological feature was characterized as unidentifiable. Sarcoma histological characteristics found in ATC cases suffer the highest overall mortality (vs. PTC, HR = 2.61, 95% CI 1.68-4.06, P < 0.001). Older age unidentifiable histology feature, more advanced AJCC N1b, AJCC M1, and SEER stage, tumor size larger than 5cm, and more invasive tumor extension were independent bad outcome predictors. The populational analysis of the PDTC/ATC cohort has provided reliable support for better understanding of the difference between PDTC and ATC cases and the guidance of clinical practice and further studies.

Compositional kriging analysis: A spatial interpolation method for distributions

Effective development of subsurface petroleum resources relies on the estimation of spatially distributed parameters at undrilled locations. Established geostatistical algorithms based on kriging exist for the estimation of scalar spatial variables such as porosity and permeability. This may not be suitable for the estimation of distributions of properties, such as grain size, whose whole distribution varies spatially. The current conventional approach is to fit a normal or log-normal distribution to the available data, and then to estimate the parameters of the distribution, the mean and standard deviation, spatially using kriging, taking care to consider any dependence between the mean and standard deviation. The assumption that all the grain size distributions can be approximated by a single distribution type is unsatisfactory, since datasets have very different-looking distributions, with variations in skewness, kurtosis, and modality. This paper presents an alternative approach that can handle significant variability in the distribution shape by separating the distribution into bins, like a histogram, and treating these bins as percentages in a composition. Compositional data needs to be mapped onto a simplex to overcome spurious correlations between those components, in addition, spatial estimation methods for compositional data have already been developed. However, the contribution to this field is the mapping of continuous data from distributions into a composition that enables the compositional kriging method to predict distributions at new locations. Moreover, the results showed the prediction distributions in the presence of varying modality, skewness, and kurtosis. Therefore, the grain size datasets in this paper have been working with the confidentiality restrictions so it explains the technique with a dataset of population ages from the US census 2010 for the state of Texas, which shows similar variability in distribution.

Prescriptive analytics models for vessel inspection planning in maritime transportation

Port state control (PSC) inspections are crucial for maritime safety and pollution reduction. The inspection process involves identifying high-risk vessels, allocating surveyors, and conducting onboard checks. This study aims to optimize the selection and assignment process through a two-stage framework, balancing the benefits of identifying deficiencies against the costs of inspection delays. Initially, we employ a predict-then-optimize approach, predicting the number of vessel deficiencies using a k-nearest neighbor (kNN) model, which informs the inspection decisions. However, due to the nonlinear nature of the optimization in relation to predicted values, we also explore an estimate-then-optimize framework that estimates distributions of potential deficiencies. We enhance two prescriptive analytics models and introduce an advanced global model with a pre-processing algorithm for better distribution estimation. A case study using data from the Hong Kong port demonstrates that the estimate-then-optimize models surpass the predict-then-optimize approach, offering solutions closer to the optimal policy. Furthermore, our improved model outperforms existing methods, proving more effective in practical applications.

Joint-mode diffusion analysis of discontinuous Galerkin methods: Towards superior dissipation estimates for nonlinear problems and implicit LES

We present a new linear eigensolution analysis technique that provides superior estimates of dissipation distribution in wavenumber space for the discontinuous Galerkin (DG) method. The technique builds upon traditional dispersion-diffusion analyses that have been applied to spectral/hp element methods, but in particular is an improvement upon the non-modal eigenanalysis approach proposed by Fernandez et al. in [1]. The present technique takes into account the indirect effects that dispersion may have on dissipation, as recently discussed by Moura et al. in [2], in order to better represent dissipation itself. Also, a concept often used with dynamic mode decomposition (DMD) techniques is invoked to weight the relative contribution of the multiple diffusion curves that stem from temporal eigenanalysis. This allows for obtaining a single dissipation profile in wavenumber space, so that the proposed technique is named joint-mode analysis. Although the non-modal approach also provides a single diffusion curve, the joint-mode dissipation curve is shown to correlate significantly better with the energy spectrum of Burgers' turbulence at large and intermediate scales, which is particularly relevant for implicit large-eddy simulation (LES). The proposed technique is readily extensible to other spectral/hp element methods.

Estimation of daylight spectral power distribution from uncalibrated hyperspectral radiance images.

This paper introduces a novel framework for estimating the spectral power distribution of daylight illuminants in uncalibrated hyperspectral images, particularly beneficial for drone-based applications in agriculture and forestry. The proposed method uniquely combines image-dependent plausible spectra with a database of physically possible spectra, utilizing an image-independent principal component space (PCS) for estimations. This approach effectively narrows the search space in the spectral domain and employs a random walk methodology to generate spectral candidates, which are then intersected with a pre-trained PCS to predict the illuminant. We demonstrate superior performance compared to existing statistics-based methods across various metrics, validating the framework's efficacy in accurately estimating illuminants and recovering reflectance values from radiance data. The method is validated within the spectral range of 382-1002nm and shows potential for extension to broader spectral ranges.

Mechanisms for Robust Local Differential Privacy.

We consider privacy mechanisms for releasing data X=(S,U), where S is sensitive and U is non-sensitive. We introduce the robust local differential privacy (RLDP) framework, which provides strong privacy guarantees, while preserving utility. This is achieved by providing robust privacy: our mechanisms do not only provide privacy with respect to a publicly available estimate of the unknown true distribution, but also with respect to similar distributions. Such robustness mitigates the potential privacy leaks that might arise from the difference between the true distribution and the estimated one. At the same time, we mitigate the utility penalties that come with ordinary differential privacy, which involves making worst-case assumptions and dealing with extreme cases. We achieve robustness in privacy by constructing an uncertainty set based on a Rényi divergence. By analyzing the structure of this set and approximating it with a polytope, we can use robust optimization to find mechanisms with high utility. However, this relies on vertex enumeration and becomes computationally inaccessible for large input spaces. Therefore, we also introduce two low-complexity algorithms that build on existing LDP mechanisms. We evaluate the utility and robustness of the mechanisms using numerical experiments and demonstrate that our mechanisms provide robust privacy, while achieving a utility that is close to optimal.

Bayesian inference for amplitude distribution with application to radar clutter

The performance of telecommunication systems is significantly subject to scattered signals superposed at the receivers. Notably, if the superposed scattered signals are impulsive in nature, this leads to burst effect on the communicated symbols. Thus, attaining an accurate estimate of the parameters of the underlying statistical model of the aggregate scattered signals becomes more important. More specifically, in the case of radar applications, finding efficient estimators for the amplitude distribution has been found to be of highly importance. In this article, the Bayesian approach has been adopted for parameter estimation of the amplitude distribution. To this end, within a Gibbs sampling framework, this would be done by the sampling from four full conditionals which can be performed in a straightforward manner. Depending on the size of sample, two types of priors, i.e. Jeffreys (small size) and conjugate (non-small size), are considered for the scale parameter of the amplitude distribution. While using the conjugate prior, hyperparameters can be found through the empirical Bayes. For the purpose of validation of the proposed approach, performance of the adopted Bayesian paradigm will be demonstrated through the simulation study. Furthermore, analysis through real datasets of radar signals reveals that the proposed Bayesian approach outperforms the known moment and log-moment estimators of the amplitude distribution and works better than the maximum likelihood estimator when sample size is small.

Polynomial approaches in improving accuracy of probability distribution estimation using the method of moments

AbstractBACKGROUNDDetermination of a probability density function (PDF) is an area of active research in engineering sciences as it can improve process systems. A previously developed polynomial method‐of‐moments‐based PDF estimation model has been applied in the research to produce accurate approximations to both standard and more complex PDF. A model with a different polynomial basis than a monomial is still to be developed and evaluated. This is the work that is undertaken in this study.RESULTSA set of standard PDF (Normal, Weibull, Log Normal and Bimodal) and more complex distributions (solutions to the Smoluchowski coagulation equation and Population Balance equation) were approximated by the method‐of‐moments using Chebyshev, Hermite and Lagrange polynomial‐based density functions. Results show that Lagrange polynomial‐based models improve the fit compared to monomial based‐modeling in terms of RMSE and Kolmogorov–Smirnov test statistic estimates. The Kolmogorov–Smirnov test‐statistics decreased by 19% and the RMSE values were improved by around 85% compared to the standard monomial basis when using Lagrange polynomial basis.CONCLUSIONThis study indicates that the procedure using Lagrange polynomials with method‐of‐moments is a more reliable reconstruction procedure that calculates the approximate distribution using lesser number of moments, which is desirable. © 2024 The Authors. Journal of Chemical Technology and Biotechnology published by John Wiley &amp; Sons Ltd on behalf of Society of Chemical Industry (SCI).