Errors Of Maximum Likelihood Estimates Research Articles

Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications

In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median correlation coefficient, and conditional expectation, are derived. After proposing the general class, one special model of the new bivariate family is discussed in detail. The maximum likelihood approach is utilized to estimate the family parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, the importance of the new bivariate family is explained by means of two distinctive real data sets in various fields.

A New Family of Odd Nakagami Exponential (NE-G) Distributions

In this study, a new family of odd nakagami exponential (NE-G) distributions is introduced and investigated as a new generator of continuous distributions. Quantile, hazard rate function, moments, incomplete moments, order statistics, and entropies are only a few of the statistical features that are investigated. A unique model is presented and thoroughly examined. To estimate model parameters based on describing real-life data sets, the maximum likelihood method is applied. The bias and mean square error of maximum likelihood estimators are investigated using a comprehensive simulation exercise. Finally, the new family adaptability is demonstrated via application to real-world data sets.

The odd flexible Weibull-H family of distributions: Properties and estimation with applications to complete and upper record data

In this paper, a new generator of continuous distributions called the odd flexible Weibull-H family is proposed and studied. Some of its statistical properties including quantile, skewness, kurtosis, hazard rate function, moments, incomplete moments, mean deviations, coefficient of variation, Bonferroni and Lorenz curves, moments of the residual (past) lifetimes and entropies are studied. Two special models are introduced and discussed in-detail. The maximum likelihood method is used to estimate the model parameters based on complete and upper record data. Adetailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, three applications to real data sets show the flexibility of the new family.

Maximum likelihood estimation error and operational value-at-risk stability

The challenge of using small sample sizes for operational risk capital models fitted via maximum likelihood estimation is well recognized, yet the literature generally provides warning examples rather than a systematic approach. We present a general framework for analyzing maximum likelihood estimation error on operational value-at-risk as a function of sample size for five severity distributions commonly used in operational risk capital models. More specifically, we study the estimation error along three dimensions: the choice of severity distribution, the sample size and the heaviness of the underlying losses. We apply these results to model selection and explore implications for operational risk modeling.

Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data

This paper studies the application of extreme value statistics (EVS) theory on analysis for stock data, based on interior penalty function algorithm and Bootstrap methods. The generalized Pareto distribution (GPD) models are considered in analyzing the closing price data of Shanghai stock market. The maximum likelihood estimates (MLEs) are obtained by using the interior penalty function algorithm. Correspondingly, the bias and standard errors of MLEs, and the hypothesis test on the shape parameter are concerned through Bootstrap methods. Some simulations are performed to demonstrate the efficacy of parameter estimation and the power of the test. The estimates of the tail index in this paper are compared with those obtained via classical methods. At last, the model is diagnosed by numerical and graphical methods and the Value-at-Risk (VaR) is estimated.

Bias of Standard Errors in Latent Class Model Applications Using Newton-Raphson and EM Algorithms

The EM algorithm is a popular method for computing maximum likelihood estimates. It tends to be numerically stable, reduces execution time compared to other estimation procedures and is easy to implement in latent class models. However, the EM algorithm fails to provide a consistent estimator of the standard errors of maximum likelihood estimates in incomplete data applications. Correct standard errors can be obtained by numerical differentiation. The technique requires computation of a complete-data gradient vector and Hessian matrix, but not those associated with the incomplete data likelihood. Obtaining first and second derivatives numerically is computationally very intensive and execution time may become very expensive when fitting Latent class models using a Newton-type algorithm. When the execution time is too high one is motivated to use the EM algorithm solution to initialize the Newton Raphson algorithm. We also investigate the effect on the execution time when a final Newton-Raphson step follows the EM algorithm after convergence. In this paper we compare the standard errors provided by the EM and Newton-Raphson algorithms for two models and analyze how this bias is affected by the number of parameters in the model fit.

03.1.1. Deriving the Observed Information Matrix in Ordered Probit and Logit Models Using the Complete-Data Likelihood Function

Louis (1982) presents a method for computing the observed information matrix and standard errors of maximum likelihood estimates obtained via the EM algorithm based on the complete-data log likelihood function. The problem illustrates the well-known method of Louis (1982) for a widely used qualitative response model in econometrics. The observed-data log likelihood function for the following model can, of course, be easily differentiated to obtain the observed information matrix; our objective is to illustrate the method and not to recommend its use for this model.

On Estimating Standard Errors in Multilevel Analysis

Standard errors of maximum likelihood estimators are commonly estimated from the inverse of the information matrix. When the information matrix is a function of parameters, some of which are estimated with little precision, the standard error may be estimated very poorly. This problem is discussed in the context of two-level (random-coefficient) models, and some remedies are proposed.

MUSIC and maximum likelihood techniques on two-dimensional DOA estimation with uniform circular array

The authors describe the application of multiple signal classification (MUSIC) and maximum likelihood (ML) techniques to the joint azimuthal and elevational directions-of-arrival (AEDOA) estimation with a uniform circular array. Both deterministic and random source signal models are considered. The asymptotic statistical properties of MUSIC and ML estimation error vectors for AEDOA parameters are investigated. In particular, explicit analytical expressions are derived for asymptotic MUSIC and ML covariance matrices as well as Cramer-Rao lower bounds (CRLBs). These analytical formulae are employed in the theoretical performance study. Computer simulation results are presented to validate theoretical predictions and compare the performance of MUSIC and ML methods. It is shown that the performance of unconditional ML is superior to that of deterministic ML, which is, in turn, better than that of MUSIC.

The standard error and bias of normal density estimators

The standard error of maximum likelihood estimators is derived for the following three cases: only m is unknown, only σ is unknown, both parameters of the normal distribution are unknown. Explicit analytical expressions are obtained for the bias of the maximum likelihood estimators in these cases.

The downward bias of asymptotic standard errors of maximum likelihood estimates of non-linear systems

A simulation study indicates that the asymptotic standard errors understate the actual inaccuracy of ML system estimates even when the number of coefficients estimated is small. The understatement increases in size when the system is larger.

AN INVESTIGATION OF METHODS FOR REDUCING SAMPLING ERROR IN CERTAIN IRT PROCEDURES*

ABSTRACTThe sampling errors of maximum likelihood estimates of item‐response theory parameters are studied in the case where both people and item parameters are estimated simultaneously. A check on the validity of the standard error formulas is carried out. The effect of varying sample size, test length, and the shape of the ability distribution is investigated. Finally, the effect of anchor‐test length on the standard error of item parameters is studied numerically for the situation, common in equating studies, where two groups of examinees each take a different test form together with the same anchor test. The results encourage the use of rectangular or bimodal ability distributions, also the use of very short anchor tests.