Maximum Likelihood Parameter Estimation for the Quartic Exponential Distribution

The quartic exponential (QE) distribution defined by the probability density function of the type

is examined in detail.

The problem of obtaining maximum likelihood point estimates of the population parameters reduces to that of identifying the α as functions of the population moments μ _r ′, r = 1, 2.3.4.

The invalidity is explained of methods proposed by previous authors to deal with the nonlinear relationships involved, and a new algorithm is developed which overcomes these objections. The new algorithm is applied to practical data, and the resulting distributions fitted to observed frequencies are shown to compare favourably with those obtained by previous Methods.     

Exploiting Structure of Maximum Likelihood Estimators for Extreme Value Threshold Selection

To model the tail of a distribution, one has to define the threshold above or below which an extreme value model produces a suitable fit. Parameter stability plots, whereby one plots maximum likelihood estimates of supposedly threshold-independent parameters against threshold, form one of the main tools for threshold selection by practitioners, principally due to their simplicity. However, one repeated criticism of these plots is their lack of interpretability, with pointwise confidence intervals being strongly dependent across the range of thresholds. In this article, we exploit the independent-increments structure of maximum likelihood estimators to produce complementary plots with greater interpretability, and suggest a simple likelihood-based procedure that allows for automated threshold selection. Supplementary materials for this article are available online.     

半参数回归模型拟极大似然估计的弱相合性

本文考虑一类固定设计的半参数回归模型,其误差为一阶自回归时间序列。用权函数及拟极大似然估计方法得到了一些参数及非参数的拟极大似然估计量,在适当的条件下,研究了它们的弱相合性,从而丰富了该类半参数回归模型的估计理论与方法。     

Further Remarks on Maximum Likelihood Estimators for the Gamma Distribution

The maximum likelihood estimators , â, for the parameters ρ, a of the gamma density f(x) = k(x/a)^ρ–1 exp(?x/a) are solutions of the equations In – ψ() = ln(A/G), â = A, where Ψ is the logarithmic derivative of the gamma function, A and G being the sample (of size n) arithmetic and geometric means, respectively. The moments of and â are developed in descending powers of ρ. A comparison of the assessments of the moments by the present approach and a method involving series in descending powers of n is made.

Approximate expressions are also given for the first four moments of ? = l/â, which is the maximum likelihood estimator of c = l/a.     

Maximum likelihood estimation of change point from stationary to nonstationary in autoregressive models using dynamic linear model

Change point estimation is a useful concept in time series models that could be applied in several fields such as financing, quality control. It helps to decrease costs of decision making and production by monitoring stock market and production lines, respectively. In this paper, the maximum likelihood technique is developed to estimate change point at which the stationary AR(1) model changes to a nonstationary process. Filtering and smoothing of dynamic linear model are used to estimate unknown parameters after change point. We also assume that correlation exists between samples' statistics. Simulation results show the effectiveness of the proposed estimators to estimate the change point of stationary. In addition based on Shewhart control chart, filtering has a better accuracy in comparison to smoothing. A real example is provided to illustrate the application.     

Model Inadequacy and Residuals Control Charts for Autocorrelated Processes

As a result of time series parameter estimation based on previous data, the probability content of residuals control charts may vary when standard control limits are used. In this paper, we consider the AR(1) process with the autoregressive parameter being estimated from a sample of observations. The performance of the exponentially weighted moving average (EWMA) control chart for residuals is investigated. Modified control limits that account for the uncertainty in the parameter estimate are provided. Comparisons through simulation signify the importance of the modified control limits. Copyright © 2004 John Wiley & Sons, Ltd.     

非理想信道估计条件下OFDM系统的最大似然检测算法

基于最大似然的准则,研究了理想信道估计条件下和非理想信道估条件下OFDM系统的最优检测算法。研究结果表明,当发送信号为PSK调制方式时,无论是理想信道估计还是非理想信道估计,最大似然检测算法与传统的迫零检测算法等价。但当信道估计非理想且发送信号的调制方式为16QAM或高阶QAM时,采用最大似然检测算法才能够获得更好的性能。     

Regression Estimators: A Comparative Study

We develop diagnostics for fixed-effects generalized linear models (GLMs) with functional predictors. As a first step, we extend the GLM to allow a functional predictor to depend on a vector of parameters. A power transformation and a cubic spline transformation are the two primary models considered. We discuss methods for fitting the models, then derive the score test and a graphical tool for assessing the need for a transformation in models with functional predictors. These methods are natural extensions of the Box–Tidwell score test and constructed residual plots used to examine nonlinearities in classical GLMs. We consider both penalized and nonpenalized maximum likelihood methods and use two data sets to illustrate the methods.     

The Efficiencies of Maximum Likelihood and Minimum Variance Unbiased Estimators of Fraction Defective in the Normal Case

This paper compares two point estimators of fraction defective of a normal distribution when both population parameters are unknown; the minimum variance unbiased estimator, (x), and the maximum likelihood estimator, (x). Using minimum mean squared error as a criterion, it is shown that the choice of estimator depends upon the true value of F(x), and the sample size. In the domain .0005 ≤ F(x) ≤ .50, the maximum likelihood estimator is generally superior even for small sample sizes, except for F(x) less than about 0.01, or greater than 0.25. Furthermore, the bias in the m.l.e. is slight over much of the domain where this estimator has smaller mean squared error.

As a practical solution to the estimation problem. it is suggested that the m.v.u.e. be calculated, and if this estimate is between 0.01 and 0.25, it should be replaced with the m.l.e. This combined estimator is shown to be nearly as efficient as the better of the m.v.u.e. and m.l.e. throughout the domain of F(x).     

基于残差的趋势性分析检验线性回归关系

通过对回归模型的拟合残差的趋势性分析,分别在重复测量和非重复测量下提出了检验线性回归关系的方法。实例分析与模拟试验表明所提出的检验方法具有满意的功效。     

一般增长曲线模型中随机回归系数线性估计的可容许性

本文在矩阵损失下研究了一般增长曲线模型中随机回归系数线性估计的可容许性。分别在齐次线性估计类和非齐次线性估计类中得到了随机回归系数的一个线性估计是可容许的充要条件。     

Behavior of the Residuals of a Regression Least-Squares Model Which Is Linear in Its Parameters When the Number of Parameters Is Increased. Part 1. State of the Question. Correlation Coefficient between the Residuals

A brief critical review is given of methods and recommendations utilized when analyzing the residuals of a regression least-squares model which is linear in its parameters. The absence of the necessary rigorous solutions and the explicit contradictions present in the recommendations considerably reduce the introduction and utilization efficiency of the information contained in the residuals. A formula is proposed for calculating the correlation coefficient between the residuals together with a compact program created in the Maple V R5 mathematical software applied program package, and examples of calculations are presented.     

The Generalized Likelihood Ratio Chart for Monitoring a Proportion with Autocorrelation

When monitoring a proportion p, it is usually assumed that the binary observations are independent. This paper investigates the problem of monitoring p when the binary observations follow a two‐state Markov chain model with first‐order dependence. A Markov binary generalized likelihood ratio (MBGLR) chart based on a likelihood ratio statistic with an upper bound on the estimate of p is proposed. The MBGLR chart is used to monitor a continuous stream of autocorrelated binary observation. The MBGLR chart with a relatively large upper bound has good overall performance over a wide range of shifts. The extra number of defectives is defined to measure the loss when using control charts for monitoring p. The MBGLR chart is optimized over a range of upper bounds for the MLE of p. The numerical results show that the optimized MBGLR chart has a smaller extra number of defectives than the optimized Markov binary cumulative sum chart that can detect a shift in p much faster than a Shewhart‐type chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Maximum Likelihood Estimation for Stochastic Differential Equations Using Sequential Gaussian-Process-Based Optimization

Stochastic differential equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and maximize the likelihood function. While importance sampling methods have allowed for the accurate evaluation of likelihoods at fixed parameter values, there is still a question of how to find the maximum likelihood estimate. In this article, we propose an efficient Gaussian-process-based method for exploring the parameter space using estimates of the likelihood from an importance sampler. Our technique accounts for the inherent Monte Carlo variability of the estimated likelihood, and does not require knowledge of gradients. The procedure adds potential parameter values by maximizing the so-called expected improvement, leveraging the fact that the likelihood function is assumed to be smooth. Our simulations demonstrate that our method has significant computational and efficiency gains over existing grid- and gradient-based techniques. Our method is applied to the estimation of ocean circulation from Lagrangian drift data in the South Atlantic ocean.     

Behavior of the Residuals of a Regression Least-Squares Model Which Is Linear in Its Parameters when the Number of Parameters Is Increased. Part 2. Interval for Series of Residuals. Problem of Gross Errors

The distribution of residuals in one series of measurements is obtained. It is shown that the normally used t interval for residuals is always wider than the correct one. The question of the detection of gross errors in the graphs of residuals is discussed. Reliable detection of gross errors is possible only for extremely small ratios of the number of parameters of the model to the number of nodes (0.1).     

A Combined Sensitive Matrix Method and Maximum Likelihood Method for Uncertainty Inverse Problems

The uncertainty inverse problems with insufficiency and imprecision in the input and/or output parameters are widely existing and unsolved in the practical engineering. The insufficiency refers to the partly known parameters in the input and/or output, and the imprecision refers to the measurement errors of these ones. In this paper, a combined method is proposed to deal with such problems. In this method, the imprecision of these known parameters can be described by probability distribution with a certain mean value and variance. Sensitive matrix method is first used to transform the insufficient formulation in the input and/or output to a resolvable one, and then the mean values of these unknown parameters can be identified by maximizing the likelihood of the measurements. Finally, to quantify the uncertainty propagation, confidence intervals of the obtained solutions are calculated based on linearization and Monte Carlo methods. Two numerical examples are presented to demonstrate the effectiveness of the present method.     

纵向数据半参数回归模型估计的强相合性

本文考虑如下纵向数据半参数回归模型:yij=x'ijβ g(tij) eij。基于最小二乘法和一般的非参数权函数方法给出了模型中参数β,回归函数g(·)和误差方差σ2的估计,并在适当条件下证明了估计量的强相合性。     

Change Point Estimation in the Mean of Multivariate Linear Profiles with No Change Type Assumption via Dynamic Linear Model

Change point estimation is a useful concept that helps quality engineers to effectively search for assignable causes and improve quality of the process or product. In this paper, the maximum likelihood approach is developed to estimate change point in the mean of multivariate linear profiles in Phase II. After the change point, parameters are estimated through filtering and smoothing approaches in dynamic linear model. The proposed change point estimator can be applied without any prior knowledge about the change type against existing estimators which assume change type is known in advance. Besides, sporadic change point can be identified as well. Simulation results show the effectiveness of the proposed estimators to estimate step, drift and monotonic, as well as sporadic changes in small to large shifts. In addition, effect of different values of the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart smoothing coefficient on the performance of the proposed estimator is investigated presenting that the smoothing estimator has more uniform performance. Copyright © 2015 John Wiley & Sons, Ltd.     

A common conjugate prior structure for several randomized response models

In this paper we present a common Bayesian approach to four randomized response models, including Warner's (1965) and other modification for it that appeared thereafter in the literature. Suitable truncated beta distributions are used throughout in a common conjugate prior structure to obtain the Bayes estimates for the proportion of a “sensitive” attribute in the population of interest. The results of this common conjugate prior approach are contrasted with those of Winkler and Franklin's (1979), in which non-conjugate priors have been used in the context of Warner's model. The results are illustrated numerically in several cases and exemplified further with data reported in Liu and Chow (1976) concerning incidents of induced abortions.     

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.