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The mathematical modeling of the coronavirus disease-19 (COVID-19) pandemic has been attempted by a large number of researchers from the very beginning of cases worldwide. The purpose of this research work is to find and classify the modelling of COVID-19 data by determining the optimal statistical modelling to evaluate the regular count of new COVID-19 fatalities, thus requiring discrete distributions. Some discrete models are checked and reviewed, such as Binomial, Poisson, Hypergeometric, discrete negative binomial, beta-binomial, Skellam, beta negative binomial, Burr, discrete Lindley, discrete alpha power inverse Lomax, discrete generalized exponential, discrete Marshall-Olkin Generalized exponential, discrete Gompertz-G-exponential, discrete Weibull, discrete inverse Weibull, exponentiated discrete Weibull, discrete Rayleigh, and new discrete Lindley. The probability mass function and the hazard rate function are addressed. Discrete models are discussed based on the maximum likelihood estimates for the parameters. A numerical analysis uses the regular count of new casualties in the countries of Angola,Ethiopia, French Guiana, El Salvador, Estonia, and Greece. The empirical findings are interpreted in-depth. [ABSTRACT FROM AUTHOR]

Introduction: Analyzing long term survivors such as diabetic patients can't be done using the usual survival models. One approach to analyze it is using defective distribution that doesn't force a pre-assumption of cure fraction to the model. To study more than one random variable interacting together, multivariate distributions may be used. However, most of multivariate distributions have complicated forms, which make the computations difficult. Besides, it may be hard to find a multivariate distribution that fits the data properly, especially in health care field. To get over this problem, one can use copula approach. In literature, to the best of our knowledge, only one paper handled copula defective models and didn't consider the effect of covariates. In this paper, we take into consideration not only existed covariates but also unobserved ones by including frailty term. Methods: Two new models are introduced. The first model, used Gumbel copula to take the dependence into consideration together with the observed covariates. The second one take into consideration not only the dependence but also the unobserved covariates by integrating frailty term in to the model. Results: A diabetic retinopathy data is analyzed. The two models indicated the existence of long-term survivals through negative parameters without the need of pre-assuming the existence of it. Including frailty term to the model helped in capturing more dependence between the variables. We compared the results using goodness of fit methods, and the results suggested that the model with frailty term is the best to be used. Conclusion: The two introduced models correctly detected the existence of cure fraction with less estimated parameters than that in mixture cure fraction models. Also, it has the advantage of not pre-assuming the existence of cure fraction to the model. comparing both models, the model with frailty term fitted the data better.

In this paper, we consider the problem of two sided hypothesis testing for the parameter of coefficient of variation of an inverse Gaussian population. An approach used here is the modified signed log-likelihood ratio (MSLR) method which is the modification of traditional signed log-likelihood ratio test. Previous works show that this proposed method has third-order accuracy whereas the traditional approach has first-order one. Indeed, these methods are based on likelihood with a higher order of accuracy. For this reason, we are interested in using this method for inference about the parameter of coefficient of variation of an inverse Gaussian distribution. All necessary formulas for obtaining MSLR statistic are provided. Numerically, the performances of this method are compared with classical approaches, in terms of empirical type-I error rate and empirical test power. Simulation results show that the empirical type-I error rates of MSLR are close to nominal type-I error rate, even for small sample sizes whereas the traditional approaches are reliable only for large sample sizes. Comparing the empirical power sizes shows that the power of MSLR method is superior to other considered methods in some settings, by regarding that the competing approaches cannot perform well in controlling the type-I error probability because their empirical type-I error rates are far from the nominal type-I error rate. Finally, we illustrate the proposed methods using a real data set and then we conclude the paper.

To extract valid results from time series analysis of tides observations, noise reduction is vital. This study aimed to use a precise statistical model to investigate noise types. Noise component amplitude of the proposed model was studied by Least Square Estimation (LS-VCE) through different statistical models: (1) white noise and auto-regressive noise, (2) white noise and Flicker noise, (3) white noise and random walk noise, (4) white noise and Flicker noise and random walk, and (5) auto-regressive noise and Flicker noise. Based on the values obtained for the Likelihood Function, it was concluded that the noise model that can be considered for observations of the Buoy time series includes two white and Flicker noises. In addition, tide forecasting for all stations was done by extracting important frequency calculated in two cases: (1) the first case in which matrix of observation weight matrix was considered as the unit matrix or the noise model was just a white noise (2) the case in which matrix of observation weight matrix was considered as a combination of white and Flicker noises. The results show that use of precise observation weight matrix resulted in 11 millimeter difference compared to the case in which observation with unit weight matrix was used.

This paper presents a nonlinear state space model with considering a first-order autoregressive model for measurement noises. A recursive method using Taylor series based approximations for filtering, prediction and smoothing problem of hidden states from the noisy observations is designed. Also, an expectation-maximization algorithm for calculating the maximum likelihood estimators of parameters is presented. The closed form solutions are obtained for estimating of the hidden states and the unknown parameters. Finally, the performance of the designed methods are verified in a simulation study. [ABSTRACT FROM AUTHOR]

Recently, Mahmoudi and Mahmoodian [7] introduced a new class of distributions which contains univariate normal–geometric distribution as a special case. This class of distributions are very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the normal–geometric distribution marginals. Different properties of this new bivariate distribution have been studied. This distribution has five unknown parameters. The EM algorithm is used to determine the maximum likelihood estimates of the parameters. We analyze one series of real data set for illustrative purposes.

This paper introduces the four parameter transmuted generalized Gompertz distribution which includes the transmuted Gompertz, transmuted generalized exponential, transmuted exponential, Gompertz, generalized exponential and exponential distributions as special cases and studies its statistical properties. Explicit expressions are derived for the quantile, moments, moment generating function and entropies. Maximum likelihood estimation is used to estimate the model parameters. Finally, two applications of the new distribution is illustrated using reliability data sets.

The Danish fire insurance data have recently been modeled by composite distributions, i.e., distributions made up by piecing together two or more distributions. Here, we introduce a new non composite distribution that performs well with respect to the Danish fire insurance data. It fits better than almost all of the commonly known heavy-tailed distributions and some of the composite distributions. [ABSTRACT FROM AUTHOR]

It is demonstrated that the confidence intervals (CIs) for the probability of eventual extinction and other parameters of a Galton–Watson branching process based upon the maximum likelihood estimators can often have substantially lower coverage when compared to the desired nominal confidence coefficient, especially in small, more realistic sample sizes. The same conclusion holds for the traditional bootstrap CIs. We propose several adjustments to these CIs, which greatly improves coverage in most cases. We also make a correction in an asymptotic variance formula given in Stigler (1971). The focus here is on implementation of the CIs which have good coverage, in a wide variety of cases. We also consider expected CI lengths. Some recommendations are made. [ABSTRACT FROM PUBLISHER]

In this paper, a new four parameter continuous distribution, called the Kumaraswamy Rayleigh (KW-ER) distribution is proposed and studied. Some mathematical properties are presented and discussed, these properties involve expansions for the cumulative and density functions, skewness and kurtosis based on the quantile function and explicit expressions for the moments and generating function. The density function of the order statistics and obtain their moments are provided. An explicit expression for Renyi entropy is obtained. The method of maximum likelihood is used for estimating the distribution parameters and the observed information matrix is derived. A data set is used to illustrate the application of the new distribution.

This research introduces the three parameter transmuted Rayleigh distribution with an application to fatigue fracture data. Using the quadratic rank transmutation map method proposed by Shaw et al. [25] we develop the three parameter transmuted Rayleigh distribution. This research also introduces the new class of weighted Rayleigh distribution by using Azzalini [2] method. Some structural properties of the new distribution are derived such as moments, incomplete moments, probability weighted moments, moment generating function, entropies, mean deviation and the kth moment of order statistics. The parameters of the proposed model are estimated using the maximum likelihood estimation and obtain the observed information matrix. The potentiality of the proposed model is illustrated using fatigue fracture data.

In this paper, we present a methodology for estimating the parame-ters of a model for an electrical arc furnace, by using maximum likelihood estimation. Maximum likelihood estimation is one of the most employed methods for parameter estimation in practical settings. The model for the electrical arc furnace that we consider, takes into account the non-periodic and non-linear variations in the voltage-current characteristic. We use NETLAB, an open source MATLAB® toolbox, for solving a set of non-linear algebraic equations that relate all the parameters to be estimated. Results obtained through simulation of the model in PSCADTM, are contrasted against real measurements taken during the furnance's most critical operating point. We show how the model for the electrical arc furnace, with appropriate parameter tuning, captures with great detail the real voltage and current waveforms generated by the system. Results obtained show a maximum error of 5% for the current's root mean square error.

In this paper we develop a signal processing algorithm for the detection, ranging and Doppler localization of moving target using Golay code as the phase coding signal. We exploit the side-lobes suppression property of Golay code, that occurs in complementary code pair, for side-lobe free target detection, range estimation and Doppler localization. Golay code based phase coding is achieved using QPSK (Quadrature Phase Shift Keying) scheme. When the transmitted signal is reflected from a moving target, its complex demodulation is carried out at the receiver. This is followed by correlation of each component of demodulated signal with one member of the complementary Golay code pair separately. The correlation results of the two channels are added together, leading to effective side-lobes suppression when phase and frequency of the demodulating carriers match with that of the received signal. The shift in the time index of the correlation peak gives the range to the target. Frequency of the demodulating carriers at which side-lobe free correlation peak occurs at the output indicates Doppler frequency corresponding to the target speed. The developed technique exhibits excellent Doppler localization, target detection and range estimation performance by overcoming the side-lobe limitations of the conventional phase coding signals in ranging applications.

The Birnbaum-Saunders model is widely applied to model fatigue failures caused by cyclic stresses both in the case of standard and accelerated life tests. This latter kind of tests are adopted when the product of interest is very reliable, in order to obtain failure data in a reasonably short amount of time. Estimates of the product's reliability or the long-term performances at normal use condition are then obtained, from accelerated failure data, adopting functional relationships accounting for the effect of the accelerating variables on the product's lifetime distribution. Customarily these models are formulated assuming that the accelerating variables affect the values of the lifetime distribution parameters, and not its form . In particular, in literature the Birnbaum-Saunders distribution is usually applied to accelerated data under the hypothesis that the scale parameter only depends on the stress conditions, while the shape parameter doesn’t depend on them. In this paper, an applicative example is presented in which this standard model does not work satisfactorily. In fact, it is shown as, in the case of the considered real set of accelerated fatigue failure data, the Birnbaum-Saunders distribution, in which both scale and shape parameters depend on the stress conditions, fits the data significantly better than the abovementioned standard option. Difference among lifetime distribution estimates provided by the two different considered models are highlighted and discussed. [ABSTRACT FROM AUTHOR]

The mathematical modeling of the coronavirus disease-19 (COVID-19) pandemic has been attempted by a large number of researchers from the very beginning of cases worldwide. The purpose of this research work is to find and classify the modelling of COVID-19 data by determining the optimal statistical modelling to evaluate the regular count of new COVID-19 fatalities, thus requiring discrete distributions. Some discrete models are checked and reviewed, such as Binomial, Poisson, Hypergeometric, discrete negative binomial, beta-binomial, Skellam, beta negative binomial, Burr, discrete Lindley, discrete alpha power inverse Lomax, discrete generalized exponential, discrete Marshall-Olkin Generalized exponential, discrete Gompertz-G-exponential, discrete Weibull, discrete inverse Weibull, exponentiated discrete Weibull, discrete Rayleigh, and new discrete Lindley. The probability mass function and the hazard rate function are addressed. Discrete models are discussed based on the maximum likelihood estimates for the parameters. A numerical analysis uses the regular count of new casualties in the countries of Angola,Ethiopia, French Guiana, El Salvador, Estonia, and Greece. The empirical findings are interpreted in-depth., Competing Interests: Conflict of InterestThe authors declare no conflicts of interest, (© Indian Statistical Institute 2022, Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)

This paper introduces a transmuted modified Weibull distribution as an important competitive model which contains eleven life time distributions as special cases. We generalized the three parameter modified Weibull distribution using the quadratic rank transmutation map studied by Shaw et al. [12] to develop a transmuted modified Weibull distribution. The properties of the transmuted modified Weibull distribution are discussed. Least square estimation is used to evaluate the parameters. Explicit expressions are derived for the quantiles. We derive the moments and examine the order statistics. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. This model is capable of modeling of various shapes of aging and failure criteria. [ABSTRACT FROM AUTHOR]

Cognitive radio employs spectrum sensing to facilitate coexistence of different communication systems over a same frequency band. A peculiar feature of this technology is the possible presence of interference within the signal bandwidth, which considerably complicates the synchronization task. This paper investigates the problem of carrier frequency estimation in an orthogonal frequency-division multiplexing (OFDM)-based cognitive radio system that operates in the presence of narrowband interference (NBI). Synchronization algorithms devised for conventional OFDM transmissions are expected to suffer from significant performance degradation when the received signal is plagued by NBI. To overcome this difficulty, we propose a novel scheme in which the carrier frequency offset (CFO) and interference power on each subcarrier are jointly estimated through maximum likelihood (ML) methods. In doing so we exploit two pilot blocks. The first one is composed of several repeated parts in the time-domain and provides a CFO estimate which may be affected by a certain residual ambiguity. The second block conveys a known pseudo-noise sequence in the frequency-domain and is used to resolve the ambiguity. The performance of the proposed algorithm is assessed by simulation in a scenario inspired by the IEEE 802.11g WLAN system in the presence of a Bluetooth interferer. [ABSTRACT FROM PUBLISHER]

In this paper, we describe a 4π Compton imager composed of a single 15 mm × 15 mm × 10 mm CdZnTe detector. Full 4π images are reconstructed via list-mode maximum likelihood (ML). A new weighting method for ML reconstruction is proposed in which the contributions of small-uncertainty sequences are enhanced relative to sequences with large uncertainties. The new reconstruction method is compared with traditional ML techniques for measured imaging data. The 4π Compton imager has a measured intrinsic imaging efficiency of nearly 2% and an imaging resolution using the weighted ML reconstruction method of 17° at 662 keV after 10 iterations. [ABSTRACT FROM AUTHOR]

It is shown how to implement an EM algorithm for maximum likelihood estimation of hierarchical nonlinear models for data sets consisting of more than two levels of nesting. This upward–downward algorithm makes use of the conditional independence assumptions implied by the hierarchical model. It cannot only be used for the estimation of models with a parametric specification of the random effects, but also to extend the two-level nonparametric approach – sometimes referred to as latent class regression – to three or more levels. The proposed approach is illustrated with an empirical application. [ABSTRACT FROM AUTHOR]

Summary. Estimation of breastmilk infectivity in HIV-1 infected mothers is difficult because transmission can occur while the fetus is in utero, during delivery, or through breastfeeding. Since transmission can only be detected through periodic testing, however, it may be impossible to determine the actual mode of transmission in any individual child. In this article we develop a model to estimate breastmilk infectivity, along with the probabilities of in-utero and intrapartum transmission. In addition, the model allows separate estimation of early and late breastmilk infectivity, and individual variation in maternal infectivity. Methods for hypothesis testing of binary risk factors and a method for assessing goodness of fit are also described. Data from a randomized trial of breastfeeding versus formula feeding among HIV-1 infected mothers in Nairobi, Kenya, are used to illustrate the methods. [ABSTRACT FROM AUTHOR]

This study is motivated by a problem concerning comparison of atmospheric chemistry data obtained from two different measurement networks. The data describe pollutant concentrations in air. One network analyzes gases and particles captured on filters daily, the other weekly. The networks use similar but not identical instrumentation and measurement methods. Chemical concentrations form usually non-stationary time series. The goal of the paper is to present a multivariate linear Gaussian model with time dependent correlation matrix for description of such series, a method for parameter estimation and fitting of the model and the procedure used for inference about the mentioned data. [ABSTRACT FROM AUTHOR]

This paper develops an asymptotic theory for time series binary choice models with nonstationary explanatory variables generated as integrated processes. Both logit and probit models are covered. The maximum likelihood (ML) estimator is consistent but a new phenomenon arises in its limit distribution theory. The estimator consists of a mixture of two components, one of which is parallel to and the other orthogonal to the direction of the true parameter vector, with the latter being the principal component. The ML estimator is shown to converge at a rate of n3/4 along its principal component but has the slower rate of n1/4 convergence in all other directions. This is the first instance known to the authors of multiple convergence rates in models where the regressors have the same (full rank) stochastic order and where the parameters appear in linear forms of these regressors. It is a consequence of the fact that the estimating equations involve nonlinear integrable transformations of linear forms of integrated processes as well as polynomials in these processes, and the asymptotic behavior of these elements is quite different. The limit distribution of the ML estimator is derived and is shown to be a mixture of two mixed normal distributions with mixing variates that are dependent upon Brownian local time as well as Brownian motion. It is further shown that the sample proportion of binary choices follows an arc sine law and therefore spends most of its time in the neighborhood of zero or unity. The result has implications for policy decision making that involves binary choices and where the decisions depend on economic fundamentals that involve stochastic trends. Our limit theory shows that, in such conditions, policy is likely to manifest streams of little intervention or intensive intervention. [ABSTRACT FROM AUTHOR]

We develop an improved empirical Bayes estimation methodology for the analysis of two-state Markov chains observed from heterogeneous individuals. First, the two transition probabilities corresponding to each chain are assumed to be drawn from a common, bivariate distribution that has beta marginals. Second, randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations. A likelihood ratio test is used to test for dependence between the transition probabilities. Posterior distributions for the transition probabilities are also derived, as is an approximation for the equilibrium probabilities. The proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data. [ABSTRACT FROM AUTHOR]

The standard contagious distributions (see Douglas 1980) have been used in such varied fields as biology and automobile insurance, often to model various physical phenomena as well as provide a good fit to count data when other models are inadequate. Unfortunately, the parameterizations often used when working with these distributions normally lead to extremely high correlations of the maximum likelihood estimators (MLE's). This tends to lead to mathematical complexities, and causes difficulty or even errors in their interpretation. Furthermore, numerical difficulties may arise when using numerical procedures to locate the estimates. Some of these difficulties were discussed by Douglas (1980, pp. 171, 204-205), who suggested that a reparameterization to reduce or even eliminate such correlation is desirable. If the MLE's are asymptotically uncorrelated, the parameterization is orthogonal. Philpot (1964) derived an orthogonal parameterization for the Neyman Type A distribution; Stein, Zucchini, and Juritz (1987) derived for the Poisson mixture by the inverse Gaussian distribution. Parameter orthogonality has several attractive features in the present context. Since there is no correlation asymptotically, the estimates (with their standard errors) provide a simpler summary of the data than in the absence of such orthogonality. The use of a parameterization where the MLEs are highly correlated can lead to a misleading analysis, or at best a more complicated analysis that would be necessary if an orthogonal parameterization had been used. To the extent that a high correlation exists, the parameters involved tend to measure similar quantities, and orthogonality separates information about the parameters from each other. This article gives an... [ABSTRACT FROM AUTHOR]

When maximum likelihood estimates of the coefficients in a nonlinear model such as the probit model are obtained there are a number of asymptotically equivalent covariance matrix estimators that can be used. These covariance matrix estimators are typically associated with different computer algorithms. For example, with the Newton-Raphson algorithm the inverse of the negative of the Hessian matrix from the log-likelihood function is used; with the method of scoring the inverse of the information matrix is used; and with a procedure proposed by Berndt, Hall, Hall, and Hausman (1974), the inverse of the outer product of the first derivatives of the log-likelihood function is used. Although these three estimators are asymptotically equivalent, their performance can vary in finite samples. The main objective of this article is to use a Monte Carlo experiment to investigate the finite sample properties of the three covariance matrix estimators, in the context of maximum likelihood estimation of the probit model. Related questions concerning the empirical distributions of test statistics and the properties of a preliminary test estimator are also examined, under varying degrees of multicollinearity. We find that, on average, the Hessian matrix and the information matrix give almost identical results and lead to more accurate estimates of the asymptotic covariance matrix than does the estimator based on first derivatives. The finite sample mean squared error of the maximum likelihood estimator, however, is considerably greater than the asymptotic covariance matrix, and the estimator based on first derivatives provides a better estimate of finite sample mean squared error. All three estimators lead to empirical distributions that can be approximated by an asymptotic normal distribution. The pretest estimator formed by testing for the... [ABSTRACT FROM AUTHOR]

We examine the forecasting problem for a finite series of observations from either a nonseasonal or a seasonal autoregressive-moving average process. Four sources of bias in the usual estimator of forecast mean squared error are identified and analyzed for particular models using maximum likelihood and least squares parameter estimates. The usual estimator is found to be biased downwards, especially near the boundary of a stationary or invertible region, and the bias is severe for least squares estimators. An alternative estimator is proposed for the maximum likelihood case, which is shown generally to have reduced bias. [ABSTRACT FROM AUTHOR]

A model with dichotomous indicators of latent variables is developed. The latent variables are related to each other and to a set of exogenous variables in a system of structural relations. Identification and maximum likelihood estimation of the model are treated. A sociological application is presented in which a theoretical construct (an attitude) is related to a set of background variables. The construct is not measured directly, but is indicated by the answers to a pair of questionnaire statements. [ABSTRACT FROM AUTHOR]

We develop estimation methods for continuous-time Markov and semi-Markov non-homogeneous man- power systems using the notion of calendar time divided into 'time windows' by change points. The model parameters may only change at these change points but remain constant between them. Our estimation methods employ a competing risks approach and allow for left truncated and right censored data. Maximum likelihood estimators are given for the hazard and survivor functions describing length of stay in any grade of the manpower system. The models are fitted to data from the Northern Ireland nursing service. [ABSTRACT FROM AUTHOR]

A bivariate failure model is proposed in which the residual lifetime of one component is dependent on the working status of the other. General properties of the model are discussed, and the maximum likelihood estimates of the parameters are found in a bivariate exponential-life special case. [ABSTRACT FROM AUTHOR]

An efficient method of estimating the linear expenditure system (LES) by maximum likelihood is introduced. A test for insuring that parameter estimates obtained by numerical maximization satisfy the first-order conditions is given. In some experiments the new method of estimating the LES is compared to the more conventional method of estimation and is found to use significantly less computer time than the conventional method. [ABSTRACT FROM AUTHOR]