Medical overpayment estimation: A Bayesian approach

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

The paper presents a novel statistical approach for analyzing the daily coronavirus case and fatality statistics. The survival discretization method was used to generate a two-parameter discrete distribution. The resulting distribution is referred to as the "Discrete Marshall–Olkin Length Biased Exponential (DMOLBE) distribution". Because of the varied forms of its probability mass and failure rate functions, the DMOLBE distribution is adaptable. We calculated the mean and variance, skewness, kurtosis, dispersion index, hazard and survival functions, and second failure rate function for the suggested distribution. The DI index demonstrates that the proposed model can represent both over-dispersed and under-dispersed data sets. We estimated the parameters of the DMOLBE distribution. The behavior of ML estimates is checked via a comprehensive simulation study. The behavior of Bayesian estimates is checked by generating 10,000 iterations of Markov chain Monte Carlo techniques, plotting the trace, and checking the proposed distribution. From simulation studies, it was observed that the bias and mean square error decreased with an increase in sample size. To show the importance and flexibility of DMOLBE distribution using two data sets about deaths due to coronavirus in China and Pakistan are analyzed. The DMOLBE distribution provides a better fit than some important discrete models namely the discrete Burr-XII, discrete Bilal, discrete Burr-Hatke, discrete Rayleigh distribution, and Poisson distributions. We conclude that the new proposed distribution works well in analyzing these data sets. The data sets used in the paper was collected from 2020 year.

Central and local limit theorems for the coefficients of polynomials of binomial type

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with Dmax first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

The law of large number (LLN) and central limit theorem (CLT) have been important in probability, statistics, finance, and other fields for several centuries. They are also core theorems in probability. Through the work of Laplace, Chebyshev, Markov, Lyapunov, Bernoulli, and other famous mathematicians, the proof and forms of LLN and CLT have been greatly improved and expanded. They enable people to extract useful information from a large amount of data for statistical inference and prediction. Nevertheless, more scholars nowadays propose slightly improved proof for them, which may inspire people to use them better and understand their core ideas inside their elegant form. This study unravels certain proofs and applications of these two theorems to try to inspire scholars some further study of these theorems. This article uses a combination of proof of LLN and CLT and their application in different fields. Using different lemmas, such as Markov’s inequality and Chebyshev’s inequality, it discusses the process of different proofs of LLN and CLT. Based on many scholars’ research, it summarizes the application of LLN and CLT in many different fields. This article can provide learners a fundamental idea of LLN and CTL. These can help readers quickly catch the important points of LLN and CLT, which can help many different investigations in different fields, including Math, Physics, Economics, and various fields with data analysis.

The central limit theorem (CLT) is one of the most important theorems in statistics, and it is often introduced to social sciences researchers in an introductory statistics course. However, the recent replication crisis in the social sciences prompts us to investigate just how common certain misconceptions of statistical concepts are. The main purposes of this article are to investigate the misconceptions of the CLT among social sciences researchers and to address these misconceptions by clarifying the definition and properties of the CLT in a manner that is approachable to social science researchers. As part of our article, we conducted a survey to examine the misconceptions of the CLT among graduate students and researchers in the social sciences. We found that the most common misconception of the CLT is that researchers think the CLT is about the convergence of sample data to the normal distribution. We also found that most researchers did not realize that the CLT applies to both sample means and sample sums, and that the CLT has implications for many common statistical concepts and techniques. Our article addresses these misconceptions of the CLT by explaining the preliminaries needed to understand the CLT, introducing the formal definition of the CLT, and elaborating on the implications of the CLT. We hope that through this article, researchers can obtain a more accurate and nuanced understanding of how the CLT operates as well as its role in a variety of statistical concepts and techniques. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Background and Objective: Currently, no published studies are available that compare central limit theorem model with traditionally used growth models in predictive food microbiology to describe bacterial growth behaviors of Pseudomonas spp. in beefs. The major objectives of the present study were to develop a novel growth model based on the central limit theorem and compare the prediction capability of the model with those of various growth models (modified Gompertz, logistic, Baranyi and Huang models) commonly used in predictive food microbiology. Material and Methods: Bacterial growth data for Pseudomonas spp. were collected from previously published studies on beefs stored at isothermal storage temperatures (0, 4, 7, 10, 15 and 20 °C). Temperature dependent kinetic parameters (maximum specific growth rate ‘µmax’ and lag phase duration ‘λ’) collected from various primary models were described as functions of storage temperatures using Ratkowsky model. Fitting capability of the novel growth model based on the central limit theorem was compared with other growth models using mean square error and coefficient of determination. Results and Conclusion: The novel growth model developed in this study provided mean square errors less than 0.104 and coefficients of determination greater than 0.962. No significant differences (p>0.05) were seen between the statistical indices of this developed model and traditionally used growth models. Results have shown that the novel growth model based on the central limit theorem can be used to describe the growth behaviors of microorganisms as alternative to traditionally used growth models of modified Gompertz, logistic, Baranyi and Huang models in predictive food microbiology. Furthermore, this novel model can be used for the prediction of shelf-life of beefs as a function of temperature since spoilage of beefs is directly linked to the load of Pseudomonas spp.

ABSTRACTFrequentists’ inference often delivers point estimators associated with confidence intervals or sets for parameters of interest. Constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analyses of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal effects, and for obtaining the asymptotic repeated sampling distributions of the causal effect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, cluster-randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of finite population central limit theorems that have not been established before. Our new theorems fill this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference. Supplementary materials for this article are available online.

The central limit theorem is a kind of important theorem in the probability theory that the distribution of the sequence of random variables is asymptotically normal. In this paper, we fi rst give some common central limit theorems from the content of the central limit theorem, and then prove the central limit theorem in the supply of electricity, device prices, shopping malls management, cigarette manufacturing, social life, military problem and so on. Finally, the advantages and disadvantages of the central limit theorem are analyzed.

In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\\mathscr{F}$. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded $\\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in $C\\lbrack 0, 1\\rbrack$ and $D\\lbrack 0, 1\\rbrack$.

Due to the increasing complexity of urban morphology, there is a growing interest for a reliable calibration of vertical wind profiles for the assessment of wind loading on structures. It is found that use of anemometric methods for the assessment of roughness parameters provides reasonably reliable results. However, there are issues related to such methods inducing uncertainty in the calibrated parameters. In this field, the paper proposes a probabilistic framework as an alternative to the classic deterministic anemometric methods for the assessment of wind profile parameters. Based on the Bayesian approach, the framework also allows quantification of uncertainty in the parameters and its propagation to the wind loading on structures. The proposed framework is first applied to a database of pseudo-experimental wind profiles to show its use and assess its robustness, as well as to highlight its advantages with respect to the deterministic approach. Then, it is applied to field measurements in a complex urban environment obtained with a LiDAR wind profiler to show its practical application. It is found that the proposed framework is able to estimate reliable roughness parameters, with a bias mainly depending on the minimum measurement height and uncertainty decreasing with increasing sample size. The use of field measurements is shown to introduce further uncertainty in calibrated parameters. • Calibration of wind profiles in urban environments represents a still opened issue. • Deterministic anemometric methods introduce uncertainty in the calibrated parameters. • A probabilistic framework is proposed for the uncertainty assessment. • The main sources of uncertainty in assessing wind action and its effects on buildings are identified.

1. Probability Models. 1.1 Discrete Probability Models. 1.2 Conditional Probability and Independence. 1.3 Random Variables. 1.4 Expectation. 1.5 The Variance. 1.6 Covariance and Correlation. 2. Special Discrete Distributions. 2.1 The Binomial Distribution. 2.2 The Hypergeometric Distribution. 2.3 The Geometric and Negative Binomial Distributions. 2.4 The Poisson Distribution. 3. Continuous Random Variables. 4.1 Continuous RV's and Their Distributions. 4.2 Expected Values and Variances. 4.3 Transformations of Random Variables. 4.4Joint Densities. 4 Special Continuous Distributions. 4.1 The Normal Distribution. 4.2 The Gamma Distribution. 5. Conditional Distributions. 5.1 The Discrete Case. 5.2 Conditional Expectations for the Discrete Case. 5.3 Conditional Densities and Expectations for Continuous RV's. 6. Limit Laws. 6.1 Moment Generating Functions. 6.2 Convergence in Probability and in Distribution. 6.3 The Central Limit Theorem. 6.4 The Delta-Method. 7. Estimation. 7.1 Point Estimation. 7.2 The Method of Moments. 7.3 Maximum Likelihood. 7.4 Consistency. 7.5 The OMEGA-Method. 7.6 Confidence Intervals. 7.7 Fisher Information, The Cramer-Rao Bound, and Asymptotic Normality of MLE's. 7.8 Sufficiency. 8. Testing Hypotheses. 8.1 Introduction. 8.2 The Neyman-Pearson Lemma. 8.3 The Likelihood Ratio Test. 8.4 The p-Value and the Relationship Between Tests of Hypotheses and Confidence Intervals. 9. The Multivariate Normal, Chi-square, t, and F-Distributions. 9.1 The Multivariate Normal Distribution. 9.2 The Central and Noncentral Chi-Square Distributions. 9.3 Student's t-Distribution. 9.4 The F-Distribution. 10.3 Nonparametric Statistics. 10.1 The Wilcoxon Test and Estimator. 10.2 One Sample Methods. 10.3 The Kolmogorov-Smirnov Tests. 11. Linear Models. 11.1 The Principle of Least Squares. 11.2 Linear Models. 11.3 F-Tests for H0. 11.4 Two-Way Analysis of Variance. 12. Frequency Data. 12.1 Logistic Regression. 12.2 Two-Way Frequency Tables. 12.3 Chi-Square Goodness of Fit Tests. 13. Miscellaneous Topics. 13.1 Survival Analysis. 13.2 Bootstrapping. 13.3 Bayesian Statistics. 13.4 Sampling.