Likelihood Function Of Sample Research Articles

A digital teaching model for jurisprudence courses based on a linear regression model

Abstract To cultivate highly qualified legal talents, this paper designs a digital teaching model for jurisprudence courses based on a linear regression model to enhance the teaching effectiveness of jurisprudence courses. Regression modeling deals with the correlation between teaching variables and establishes a linear regression model in the design process. The sample likelihood function is constructed to output the teaching regression coefficients and the metrics of the model fitting effect, the raw data are dimensionless, and the dimensionless model covariance is calculated according to the nature of teaching expectations. On this basis, the digital teaching model of the jurisprudence course was explored to optimize digital teaching resources, adjust the structure, and establish a student evaluation mechanism. To verify the feasibility of this teaching model, its teaching effect was tested. The results showed that the digital teaching model of jurisprudence based on linear regression led to a teaching level of 2.6594%, and students’ mastery of knowledge reached 2.7952%. And the number of students with failing grades decreased by 480%, and the number of students receiving excellent grades increased by 3.53 times, with a teaching quality rating range of [3.95-4.39] points. It can be seen that the linear regression model promoted the jurisprudence course to break through the traditional teaching model and further improve the teaching quality of the jurisprudence course.

ANALISIS METODE BAYESIAN PADA SISTEM ANTREAN RAWAT JALAN DI RSUP Dr. KARIADI DENGAN DISTRIBUSI SAMPEL POISSON DAN GEOMETRIK

Hospital service facilities cannot be separated from queuing events. Queues are an unavoidable part of life, but they can be minimized with a good system. The purpose of this study was to find out how the queuing system at Dr. Kariadi. Bayesian method is used to combine previous research and this research in order to obtain new information. The sample distribution and prior distribution obtained from previous studies are combined with the sample likelihood function to obtain a posterior distribution. After calculating the posterior distribution, it was found that the queuing model in the outpatient installation at Dr. Kariadi Semarang is (G/G/c): (GD/∞/∞) where each polyclinic has met steady state conditions and the level of busyness is greater than the unemployment rate so that the queuing system at Dr. Kariadi is categorized as good, except in internal medicine poly.

A Note on Parameters Estimation for Nonlinear Wiener Processes With Measurement Errors

The nonlinear Wiener process has been widely used as a model for the degradation process. This note concerns parameters estimation of nonlinear Wiener processes with measurement errors (WPME) by the maximum likelihood estimation method. Firstly, we prove a rule that the estimated results based on the sample likelihood function developed through observations at each point are equal to the results from the first differences of the observations. This rule indicates that for reducing computation complexity the first differences of the observations may develop the sample likelihood function. Then we present a simple method to calculate the determinant and the inverse matrix of the covariance matrix of the WPME. This simple method could avoid the overflow error when calculating the determinant of the covariance matrix and the case that the inverse matrix is close to be singular, which could result in wrong estimation results. Secondly, we highlight the unit-specific assumption, which has a significant impact on parameters estimation but has been neglected in many papers. Then, we propose a modified expectation maximization algorithm for parameters estimation with random effects. Finally, to demonstrate the application and superiority of the proposed method, we provide a numerical example and a case study with comparison to several representative methods in the literature.

Statistical inference using generalized linear mixed models under informative cluster sampling

AbstractWhen a sample is obtained from a two‐stage cluster sampling scheme with unequal selection probabilities the sample distribution can differ from that of the population and the sampling design can be informative. In this case making valid inference under generalized linear mixed models can be quite challenging. We propose a novel approach for parameter estimation using an EM algorithm based on the approximate predictive distribution of the random effect. In the approximate predictive distribution instead of using the intractable sample likelihood function we use a normal approximation of the sampling distribution of the profile pseudo maximum likelihood estimator of the random effects in the level‐one model. Two limited simulation studies show that the proposed method using the normal approximation performs well for modest cluster sizes. The proposed method is applied to the real data arising from 2011 Private Education Expenditures Survey (PEES) in Korea. The Canadian Journal of Statistics 45: 479–497; 2017 © 2017 Statistical Society of Canada

On Bayesian procedure for maximum earthquake magnitude estimation

This work is focused on the Bayesian procedure for the estimation of the regional maximum possible earthquake magnitude <em>m</em><sub>max</sub>. The paper briefly discusses the currently used Bayesian procedure for m<sub>max</sub>, as developed by Cornell, and a statistically justifiable alternative approach is suggested. The fundamental problem in the application of the current Bayesian formalism for <em>m</em><sub>max</sub> estimation is that one of the components of the posterior distribution is the sample likelihood function, for which the range of observations (earthquake magnitudes) depends on the unknown parameter <em>m</em><sub>max</sub>. This dependence violates the property of regularity of the maximum likelihood function. The resulting likelihood function, therefore, reaches its maximum at the maximum observed earthquake magnitude <em>m</em><sup>obs</sup><sub>max</sub> and not at the required maximum <em>possible</em> magnitude <em>m</em><sub>max</sub>. Since the sample likelihood function is a key component of the posterior distribution, the posterior estimate of <em>m^</em><sub>max</sub> is biased. The degree of the bias and its sign depend on the applied Bayesian estimator, the quantity of information provided by the prior distribution, and the sample likelihood function. It has been shown that if the maximum posterior estimate is used, the bias is negative and the resulting underestimation of <em>m</em><sub>max</sub> can be as big as 0.5 units of magnitude. This study explores only the maximum posterior estimate of <em>m</em><sub>max</sub>, which is conceptionally close to the classic maximum likelihood estimation. However, conclusions regarding the shortfall of the current Bayesian procedure are applicable to all Bayesian estimators, <em>e.g.</em> posterior mean and posterior median. A simple, <em>ad hoc</em> solution of this non-regular maximum likelihood problem is also presented.

Competing Risks Models using Mortgage Duration Data under the ProportionalHazards Assumption

This paper demonstrates two important results related to the estimation of a competing risks model under the proportional hazards assumption with grouped duration data, a model that has become the canonical model for the termination of mortgages with prepayment and default as two competing risks. First, the model with non-parametric baseline hazards is unidentifiable with only grouped mortgage duration data. Therefore, assumption on the functional form of the baseline hazard is necessary for any meaningful inference. Second, under some parametric assumptions such as piece-wise constant baseline hazards, the sample likelihood function has an explicit analytical form. Therefore, there is no need for the approximation formula widely adopted in the previous literature. Both Monte Carlo simulations and actual mortgage data are used to demonstrate the adverse impact of the approximation.

Two-stage informative cluster sampling—estimation and prediction with applications for small-area models

This paper considers the effects of informative two-stage cluster sampling on estimation and prediction. The aims of this article are twofold: first to estimate the parameters of the superpopulation model for two-stage cluster sampling from a finite population, when the sampling design for both stages is informative, using maximum likelihood estimation methods based on the sample-likelihood function; secondly to predict the finite population total and to predict the cluster-specific effects and the cluster totals for clusters in the sample and for clusters not in the sample. To achieve this we derive the sample and sample-complement distributions and the moments of the first and second stage measurements. Also we derive the conditional sample and conditional sample-complement distributions and the moments of the cluster-specific effects given the cluster measurements. It should be noted that classical design-based inference that consists of weighting the sample observations by the inverse of sample selection probabilities cannot be applied for the prediction of the cluster-specific effects for clusters not in the sample. Also we give an alternative justification of the Royall [1976. The linear least squares prediction approach to two-stage sampling. Journal of the American Statistical Association 71, 657–664] predictor of the finite population total under two-stage cluster population. Furthermore, small-area models are studied under informative sampling.

A more informative estimation procedure for the parameters of a diffusion process

The estimation procedures for the parameters of a diffusion process with constant coefficients have mainly focused on volatility. Nevertheless, even if the knowledge of the volatility alone suffices to compute the Black and Scholes option prices, other financial application models assume that the price dynamics follows a log-normal process and requires the knowledge of both parameters. On the other hand, while the usual ML estimator of volatility gives satisfactory results, the estimation of drift is much less accurate; moreover, the drift-estimated value highly depends on the phases of the business cycle included in the sample data. This contribution explicitly imposes a risk aversion or risk neutral assumption into the ML estimation procedure and makes a constrained maximization of the sample likelihood function. The aim is twofold: to obtain estimated values which are consistent with a widely accepted assumption and use the risk aversion constraint in order to improve the accuracy of the estimates.