Usual Maximum Likelihood Estimator Research Articles

Bootstrapping State-Space Models: Distribution-Free Estimation in View of Prediction and Forecasting

Linear models, seasonal autoregressive integrated moving average (SARIMA) models, and state-space models have been widely adopted to model and forecast economic data. While modeling using linear models and SARIMA models is well established in the literature, modeling using state-space models has been extended with the proposal of alternative estimation methods to the maximum likelihood. However, maximum likelihood estimation assumes, as a rule, that the errors are normal. This paper suggests implementing the bootstrap methodology, utilizing the model’s innovation representation, to derive distribution-free estimates—both point and interval—of the parameters in the time-varying state-space model. Additionally, it aims to estimate the standard errors of these parameters through the bootstrap methodology. The simulation study demonstrated that the distribution-free estimation, coupled with the bootstrap methodology, yields point forecasts with a lower mean-squared error, particularly for small time series or when dealing with smaller values of the autoregressive parameter in the state equation of state-space models. In this context, distribution-free estimation with the bootstrap methodology serves as an alternative to maximum likelihood estimation, eliminating the need for distributional assumptions. The application of this methodology to real data showed that it performed well when compared to the usual maximum likelihood estimation and even produced prediction intervals with a similar amplitude for the same level of confidence without any distributional assumptions about the errors.

A Note on Bias Correction for the Standard Two‐Sided Power Distribution

AbstractIt is verified that the maximum likelihood (ML) estimators of the standard two‐sided power distribution (STSP) parameters introduced by van Dorp and Kotz on the interval (0,1) can be severely biased. Since the usual analytical methods of bias correction cannot be applied in such a parametric distribution, bootstrap bias correction methods are proposed. The numerical study favors a particular bootstrap estimator based on parametric resampling. Real data applications are also considered to illustrate the impact of bias correction of the usual maximum likelihood estimators of the standard two‐sided power distribution parameters when the sample size is small.

On three-parameter generalized exponential distribution

In this article, we consider the estimation of the three-parameter generalized exponential distribution. In presence of the unknown location parameter the usual maximum likelihood estimators do not exist for The maximum product of spacings method serves as good alternative since it always exists and yields consistent estimators in the entire parametric space. We develop the asymptotic distribution of the proposed estimators along-with a detailed discussion about the computational intricacies involved in implementing the product of spacing method. Extensive simulations have been performed to demonstrate the effectiveness of the proposed method for α in the range (0, 1), in addition to a comparative study with some standard techniques known to provide consistent estimators, even for Furthermore, two real data sets have been analyzed to demonstrate the applicability of the proposed method.

A nonparametric mixture approach to density and null proportion estimation in large‐scale multiple comparison problems

SummaryA new method for estimating the proportion of null effects is proposed for solving large‐scale multiple comparison problems. It utilises maximum likelihood estimation of nonparametric mixtures, which also provides a density estimate of the test statistics. It overcomes the problem of the usual nonparametric maximum likelihood estimator that cannot produce a positive probability at the location of null effects in the process of estimating nonparametrically a mixing distribution. The profile likelihood is further used to help produce a range of null proportion values, corresponding to which the density estimates are all consistent. With a proper choice of a threshold function on the profile likelihood ratio, the upper endpoint of this range can be shown to be a consistent estimator of the null proportion. Numerical studies show that the proposed method has an apparently convergent trend in all cases studied and performs favourably when compared with existing methods in the literature.

A Simultaneous Estimation of the Baseline Intensity and Parameters for Modulated Renewal Processes

This paper proposes a semiparametric solution to estimate the intensity (hazard) function of modulated renewal processes: a nonparametric estimate for the baseline intensity function together with a parametric estimate of the model parameters of the covariate processes. Based on the martingale property associated with the conditional intensity, we construct a statistic from a residual analysis to estimate the baseline renewal intensity function, when the model parameters of the covariate processes are known. In addition, when the baseline intensity is obtained, the model parameters can be estimated using the usual maximum likelihood estimation. In practice, both the baseline intensity and model parameters are suggested to be estimated simultaneously via an expectation–maximization (E–M)-type iterative algorithm. A more important feature of the newly proposed algorithm is that, given n events in the observation dataset, its computation time is of order O(n2), while the Nelson–Aalen–Breslow estimator takes a computation time of order O(n3). For illustration, we apply the proposed estimation procedure to a set of data simulated from a modulated gamma renewal process and the aftershock sequence following the Ms8 Wenchuan earthquake, which occurred in Sichuan Province, China on 12 May 2008.

A closed-form solution for the stochastic volatility model with applications on international stock markets

This paper proposes a closed-form estimator for the stochastic volatility (SV) model. Compared to the usual maximum likelihood estimation (MLE), which is difficult to perform without appropriate approximations, the proposed method can be easily implemented and does not require the use of any numerical optimizer or starting values for iterations. Moreover, closed-form estimates can be supplied as initial values to MLE, for instance, conducted with a novel Laplace approximation. Denoted by MLE-C, this method consistently outperforms other estimators including the Markov chain Monte Carlo (MCMC). This is confirmed with simulation studies consisting of various combinations of true parameters and sample sizes. Our empirical data include daily returns of S&P 500, Nikkei 225 and DAX 100 over 2011–2020. The SV model estimated by MLE-C almost uniformly beats the popular GARCH counterparty, based on both the in-sample fit and out-of-sample forecasting criteria. Value-at-Risk analyses further demonstrate the capability of the SV model to accurately describe the tail behaviors of negative returns.

BACKWARD CUSUM FOR TESTING AND MONITORING STRUCTURAL CHANGE WITH AN APPLICATION TO COVID-19 PANDEMIC DATA

It is well known that the conventional cumulative sum (CUSUM) test suffers from low power and large detection delay. In order to improve the power of the test, we propose two alternative statistics. The backward CUSUM detector considers the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector sequentially cumulates a triangular array of backwardly cumulated residuals. A multivariate invariance principle for partial sums of recursive residuals is given, and the limiting distributions of the test statistics are derived under local alternatives. In the retrospective context, the local power of the tests is shown to be substantially higher than that of the conventional CUSUM test if a break occurs in the middle or at the end of the sample. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is found to be much shorter than that of the conventional monitoring CUSUM procedure. Furthermore, we propose an estimator of the break date based on the backward CUSUM detector and show that in monitoring exercises this estimator tends to outperform the usual maximum likelihood estimator. Finally, an application of the methodology to COVID-19 data is presented.

On the performance of some biased estimators in the gamma regression model: simulation and applications

The gamma regression model is widely applied when the response variable is continuous and positively skewed. In the multicollinearity problem, the usual maximum likelihood estimator is inadequate due to its inflated variance. To reduce this effect, well-known ridge and Liu estimators are generally used. In this study, we propose some shrinkage parameters for the new estimator and compared with some best ridge parameters indicated by Amin et al. [Performance of some ridge estimators for the gamma regression model. Stat Pap. 2020;61:997–1026] and two proposed ridge parameters for the GRM specified by Lukman et al. [A new ridge-type estimator for the gamma regression model. Scientifica. 2021;2021:5545356]. A Monte Carlo simulation study and an empirical application are conducted to assess the effectiveness of the proposed and other estimators. Based on the findings of simulation results and applications, we found that one of our proposed estimators performed the best for small dispersion levels.

A closed-form estimator for the Markov switching in mean model

This paper revisits the Markov switching in mean model which is commonly fitted by maximizing its log-likelihood. To effectively resolve the computational complexity caused by the nolinear nature and iterative components in the log-likelihood, we propose a closed-form solution inspired by moment-based and Yule–Walker methods. Associated asymptotics are discussed with numerical evidence. For practical considerations, we demonstrate the usefulness of the proposed estimates when supplied as initial values to obtain the usual maximum likelihood estimates for reliable statistical inferences.

Estimating logit models with small samples

AbstractIn small samples, maximum likelihood (ML) estimates of logit model coefficients have substantial bias away from zero. As a solution, we remind political scientists of Firth's (1993,Biometrika,80, 27–38) penalized maximum likelihood (PML) estimator. Prior research has described and used PML, especially in the context of separation, but its small sample properties remain under-appreciated. The PML estimator eliminates most of the bias and, perhaps more importantly, greatly reduces the variance of the usual ML estimator. Thus, researchers do not face a bias-variance tradeoff when choosing between the ML and PML estimators—the PML estimator has a smaller biasanda smaller variance. We use Monte Carlo simulations and a re-analysis of George and Epstein (1992,American Political Science Review,86, 323–337) to show that the PML estimator offers a substantial improvement in small samples (e.g., 50 observations) and noticeable improvement even in larger samples (e.g., 1000 observations).

On the stochastic restricted Liu estimator in logistic regression model

In this paper, we study the effects of near-singularity which is known as multicollinearity in the binary logistic regression. Furthermore, we also assume the presence of stochastic non-sample linear restrictions. The well-known logistic Liu estimator is combined with the stochastic linear restrictions in order to propose a new method, namely, the stochastic restricted Liu estimation. Theoretical comparisons between the usual maximum likelihood estimator, Liu estimator, stochastic restricted maximum-likelihood estimator and the new stochastic restricted Liu estimator are derived using matrix mean-squared errors of the estimators. A Monte Carlo simulation experiment is designed to evaluate the performances of the listed estimators in terms of mean-squared error and mean absolute error criteria. Artificial data are used to show how to interpret the theorems. According to the results of the simulation, the new method beats the other estimators when the data matrix has the problem of collinearity along with the stochastic restrictions.

Performances of different methods of estimating the diameter distribution based on simple stand structure variables in monospecific regular temperate European forests

Using the three characteristic points of a forest stand, dg (mean quadratic diameter), dmin (diameter of the smallest tree) and dmax (diameter of the largest tree), appears informative enough to determine the parameters of the whole diameter distribution and, hence, the standing volume, with an accuracy of 2–3%. This is related principally with a particular feature of the Weibull distribution function, and the empirical dependency of the main scale parameter α + β from the mean quadratic diameter (dg): This allows the prediction of the parameter β with an unexpectedly high likelihood. This feature could be used for growth modelling as well as inventory purposes, at least for monospecific and even-aged stands and, maybe more, because this feature is proper to the function itself. One of the most appealing applications of diameter distribution functions is to predict compliant stand diameters without needing to tally all stems, but in determining the function parameters only on the base of simple stand characteristics. This can be applied for yield model construction or inventory purposes. The aim of this paper is to compare different methods of estimating the Weibull distribution parameters, partly based on parameter recovery method (PRM). They use a remarkable, empiric property of the Weibull function. Their performances are assessed in comparison to real distributions from a wide database of permanent Swiss yield plots repeatedly measured (time series) for Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.). The Weibull distribution offers the advantage of simple but reliable estimation procedures. One of these is the main (scale) parameter β being given at a remarkable point of the function free, i.e. independent from the shape parameter γ. Because dg lies very close to this point, it correlates empirically very tightly with this parameter and thus allows for a trustful simple estimation. We compare, with appropriate statistic tests, real distribution with such obtained with the usual maximum likelihood estimation (MLE) of the Weibull parameters and those obtained with these new procedures. The results obtained from a set of 800 yield plots of regular spruce stands and 596 of beech in Switzerland illustrate the good performances of the two much simpler procedures. The accuracy of estimating the standing volume is about 1.4% for beech and 2.8% for spruce when the site index (SI) is known. The three considered characteristic stand attributes (dg, dmin and dmax) appear robust enough for determining the diameter distribution with a respectable accuracy. This is enough reason for a revival of the old, but very ingenious, method of angle count sampling of Walter Bitterlich (1947).

DFT-based low-complexity optimal cell ID estimation in NB-IoT

This paper deals with cell identifier (ID) estimation for narrowband-Internet of things (NB-IoT) system. It is suggested to transform the usual maximum likelihood (ML) estimator expression in order to highlight a discrete Fourier transform (DFT), which can be computed with fast algorithms. Therefore, the proposed method is a DFT-based low-complexity cell ID estimator that can be qualified as optimal in the ML sense. The principle is extended to the practical case where the channel is unknown and must be estimated. In this scenario, the concentrated likelihood function needs to be maximized, in which the ML channel estimate is a function of the unknown cell ID parameter. This operation only involves a few additional multiplications. Simulation results reveal that the performance of the proposed method actually matches the optimal one of the ML cell ID estimator. Furthermore, the technique is robust to residual frequency offset up to several hundreds of Hertz. We also show that the mean square error of channel estimation reaches its Cramér-Rao bound (CRB).

Lomax exponential distribution with an application to real-life data.

In this paper, a new modification of the Lomax distribution is considered named as Lomax exponential distribution (LE). The proposed distribution is quite flexible in modeling the lifetime data with both decreasing and increasing shapes (non-monotonic). We derive the explicit expressions for the incomplete moments, quantile function, the density function for the order statistics etc. The Renyi entropy for the proposed distribution is also obtained. Moreover, the paper discusses the estimates of the parameters by the usual maximum likelihood estimation method along with determining the information matrix. In addition, the potentiality of the proposed distribution is illustrated using two real data sets. To judge the performance of the model, the goodness of fit measures, AIC, CAIC, BIC, and HQIC are used. Form the results it is concluded that the proposed model performs better than the Lomax distribution, Weibull Lomax distribution, and exponential Lomax distribution.

The Alpha-Beta Skew Generalized t Distribution: Properties and Applications

n this paper, we propose a new distribution, namely alpha-beta-skew generalized tdistribution. The proposed distribution is really flxible and includes as special models some important distributions like Normal, t-student, Cauchy and etc as its marginal component distributions. It features a probability density function with up to three modes. The moment generating function as well as the main moments are provided. Inference is based on a usual maximum-likelihood estimation approach and a small Monte Carlo simulation is conducted for studying the asymptotic properties of the maximum-likelihood estimate. The usefulness of the new model is illustrated in a real data

Bayesian estimation of the survival characteristics for Hjorth distribution under progressive type-II censoring

In this paper, the Bayes estimation procedure for the parameters and survival characteristics (survival and hazard functions) of the Hjorth distribution has been proposed with progressively type-II censored data. The Bayes estimators are derived with gamma prior and evaluated under squared error loss function. It is known that the censored observations create the complexity in Bayes estimation procedures. Therefore, two approximation techniques, namely Tierney–Kadane approximation method and Markov Chain Monte Carlo method have been used to compute the approximate Bayes estimators. The proposed estimates are compared with the usual maximum likelihood estimators through Monte Carlo simulations. Lastly, a medical data set has been considered to show the applicability of the proposed model as well study in real life scenario.

Does the generalized mean have the potential to control outliers?

The efficacy of the generalized mean in controlling outliers is explored in this paper. We found that in the presence of outliers in the data, the generalized mean estimates the mean of the dominating population more accurately compared to the usual maximum likelihood estimator. Thus the generalized mean allows stable estimation of the target mean parameter without invoking the complications of sophisticated robust techniques. For example, while doing experimentation on the growth of species, the data on the size or growth rate of a particular species are often contaminated with those from other species, where the behavior of the latter component is similar to that of a bunch of outlying observations. To carry out realistic growth related inference on the mean growth of the primary component in this situation, the generalized mean is recommended as a useful tool to the experimental biologists. There are innumerable other real-life scenarios where a suitably chosen generalized mean can provide better input in doing inference with real data compared to the arithmetic and other standard means.

Stein-type shrinkage estimators in gamma regression model with application to prostate cancer data.

Gamma regression is applied in several areas such as life testing, forecasting cancer incidences, genomics, rainfall prediction, experimental designs, and quality control. Gamma regression models allow for a monotone and no constant hazard in survival models. Owing to the broad applicability of gamma regression, we propose some novel and improved methods to estimate the coefficients of gamma regression model. We combine the unrestricted maximum likelihood (ML) estimators and the estimators that are restricted by linear hypothesis, and we present Stein-type shrinkage estimators (SEs). We then develop an asymptotic theory for SEs and obtain their asymptotic quadratic risks. In addition, we conduct Monte Carlo simulations to study the performance of the estimators in terms of their simulated relative efficiencies. It is evident from our studies that the proposed SEs outperform the usual ML estimators. Furthermore, some tabular and graphical representations are given as proofs of our assertions. This study is finally ended by appraising the performance of our estimators for a real prostate cancer data.

A new decision theoretic sampling plan for exponential distribution under Type-I censoring

In this paper a new decision theoretic sampling plan (DSP) is proposed for Type-I censored exponential distribution. The proposed DSP is based on a new estimator of the expected lifetime of an exponential distribution which always exists, unlike the usual maximum likelihood estimator. The DSP is a modification of the Bayesian variable sampling plan of Lam. An optimum DSP is derived in the sense that it minimizes the Bayes risk. In terms of the Bayes risks, it performs better than Lam’s sampling plan and its performance is as good as the Bayesian sampling plan of Lin, Liang and Haung, although implementation of the DSP is very simple. Analytically it is more tractable than the Bayesian sampling plan of Lin, Liang and Haung, and it can be easily generalized for any other loss functions also. A finite algorithm is provided to obtain the optimal plan and the corresponding minimum Bayes risk is calculated. Extensive numerical comparisons with the optimal Bayesian sampling plan proposed by Lin, Liang and Haung are made. The results have been extended for three degree polynomial loss function and for Type-I hybrid censoring scheme.

The Kumaraswamy Exponentiated U-Quadratic Distribution: Properties and Application

In this paper, a new lifetime model called Kumaraswamy exponentiated U-quadratic (KwEUq) distribution is proposed. Several mathematical and statistical properties are derived and studied such as the explicit form of the quantile function, moments, moment generating function, order statistics, probability weighted moments, Shannon entropy and Renyi entropy. We also found that the usual maximum likelihood estimates (MLEs) fail to hold for the KwEUq distribution. Two alternative methods are suggested for the parameter estimation of the KwEUq, the alternative maximum likelihood estimation (AMLE) and modified maximum likelihood estimation (MMLE). Simulation studies were conducted to assess the finite sample behavior of the AMLEs and MMLEs. Finally, we provide application of the KwEUq for illustration purposes.