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For a truncated family of distributions with a truncation parameter γ and a parameter θ as a nuisance parameter, we derive the stochastic expansions of bias-adjusted maximum likelihood estimators γ ̂ M L ∗ θ and γ ̂ M L ∗ of γ based on a sample of size n when θ is known and when θ is unknown, respectively. The asymptotic loss of γ ̂ M L ∗ relative to γ ̂ M L ∗ θ is obtained up to the second order, that is the order n−1. The results are a generalization of those for a one-sided truncated exponential family of distributions. Its application to truncated t-distributions is also given. [ABSTRACT FROM AUTHOR]

This paper elucidates the prerequisites for maximum likelihood estimation (MLE) of parameters within the exponential and scale parameter families. Estimation of these parameters is predicated on data derived from censored samples and seeks to adhere to stochastic ordering principles. The study establishes that for two independent normal distributions and a two-parameter exponential distribution discernible by the distinct parameter sets, the MLEs of the parameters evince a stochastically ordered relationship when evaluated using full datasets. Furthermore, this research is extended to corroborate the persistence of stochastic ordering in the MLEs of such parameters under conditions of fixed censoring of samples. [ABSTRACT FROM AUTHOR]

This study takes into account the efficient estimation of the probability density function (PDF) of the Weibull-Burr XII distribution and the cumulative distribution function (CDF), which allows for greater flexibility than many other generalized distributions in use today. Nine traditional estimators are used to produce the analytical formulations for the bias and the mean squared error (MSE). A simulated investigation is conducted to evaluate the finite sample performance of the suggested estimators in terms of MSE values. In addition to that a real data application is also demonstrated for illustration. [ABSTRACT FROM AUTHOR]

AbstractThis study focuses on parameter estimation in the presence of multicollinearity for the count response that follows the Waring distribution. The Waring regression model deals with over-dispersion. So, this study proposed the Waring ridge regression (WRR) model as a solution for multicollinearity with over-dispersion. We conducted a theoretical comparison between the ridge estimator and the maximum likelihood estimators using matrix and scalar mean squared error as a performance evaluation criterion. Several ridge parameters are considered for the WRR estimator. The performance of these parameters is numerically evaluated using a Monte Carlo simulation study and a real application. The results of the simulation and application demonstrate the superiority of the WRR model with different ridge parameters over the maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

One new factor driving the banking industry towards long-term, high-quality growth is digital marketing, which has arisen within the framework of the digital economy. The purpose of this research is to examine the effect of digital marketing on the financial results of MENA banks from 2010 to 2023. The research examines the impact of digital marketing techniques on the effectiveness of financial institutions through Tobit regression analysis, taking into account and controlling for sustainable practices (ESG), bank-specific characteristics (capital adequacy, bank size, liquidity, and cost efficiency), and macroeconomic variables (GDP and inflation). This empirical paper managed to collect the data for eleven countries in the MENA from the Refinitiv Eikon platform, world bank database, and the annual reports of relevant banks in the different stock markets. The final sample included 78 banks out of 120 listed banks. The results show that there is a clear association between the presence of digital marketing campaigns and improved profitability and market share growth for banks. Aligning digital initiatives with ESG principles is crucial for long-term value development, and sustainable practices increase these beneficial benefits even more. The research also shows that macroeconomic factors and bank-specific characteristics affect how effective digital marketing campaigns are. The significance of digital transformation and ESG integration in promoting competitive advantages and long-term growth in the MENA banking sector is highlighted by these findings, which have important implications for policy, investors, and bank executives. [ABSTRACT FROM AUTHOR]

In this article we, proposed a new two parameter distribution called inverted power modified Lindley distribution. The main objective is to introduce an extension to inverted modified Lindley distribution as an alternative to the inverted exponential, inverted gamma and inverted modified Lindley distributions, respectively. The proposed distribution is more flexible than the above mentioned distributions in terms of its hazard rate function. In the part of estimation of the proposed model, we first utilize the maximum likelihood (ML) estimator and parametric bootstrap confidence intervals, viz., standard bootstrap, percentile bootstrap, bias-corrected percentile (BCPB), bias-corrected accelerated bootstrap (BCAB) from the classical point of view as well the Bayesian estimation under different loss functions, squared error loss function, modified squared error loss function, and Bayes credible interval as to obtain the model parameter based on order statistics. A simulation study is carried out to check the efficiency of the classical and the Bayes estimators in terms of mean squared errors and posterior risks, respectively. Two real life data sets, have been analyzed for order statistics to demonstrate how the proposed methods may work in practice. [ABSTRACT FROM AUTHOR]

When the weight function depends on the lengths of the units of interest, the resulting distribution is called length biased. Length biased distribution is thus a special case of the more general form, known as weighted distribution. In this study, we introduce a novel probability distribution named the Length-Biased Sabur distribution (LBSD). This new distribution enhances the traditional Sabur distribution by incorporating a weighted transformation approach. The paper investigates the probability density function (pdf) and the cumulative distribution function (cdf) associated with the LBSD. A thorough examination of the distinctive structural properties of the proposed model is conducted, covering the survival function, conditional survival function, hazard function, cumulative hazard function, mean residual life, moments, moment generating function, characteristic function, likelihood ratio test, ordered statistics, entropy measures, and Bonferroni and Lorenz curve. [ABSTRACT FROM AUTHOR]

In many manufacturing industries, the lifetime performance index C L is utilized to assess the manufacturing process performance for products following some lifetime distributions and subjecting them to progressive type I interval censoring. This paper aims to explore the sampling design required to achieve a specified level of significance and test power for products with lifetimes following the Exponentiated Frech'et distribution. Since lifetime distribution is an asymmetrical probability distribution, this investigation is related to the topic of asymmetrical probability distributions and applications in various fields. When the termination time is fixed but the number of intervals is variable, the optimal number of inspection intervals and sample sizes yielding the minimized total experimental costs are determined and tabulated. When the termination time is varying, the optimal number of inspection intervals, sample sizes, and equal interval lengths achieving the minimum total experimental costs are determined and tabulated. Optimal parameter values are displayed in tabular form for feasible applications for users. Additionally, a practical example is provided to illustrate how this sampling design can be used to collect data by using the optimal setup of parameters, followed by a testing procedure to assess the capability of the production process. [ABSTRACT FROM AUTHOR]

Being an important probability distribution with moderate skewness, the two-parameter gamma distribution is widely used in statistics. However, the maximum likelihood estimators (MLEs) of its parameters do not have closed forms, making them difficult to be implemented in applications. Moreover, the MLE of its shape parameter has low estimation efficiency due to its considerable bias. Thus, many other estimators have been investigated in the literature. We propose an easy computation of the MLEs in this article, where the MLE of the shape parameter is modified to be highly efficient and significantly better than most of existing estimators. [ABSTRACT FROM AUTHOR]

We establish the exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator of unknown drift parameter in an ultraspherical Jacobi process using the change of measure method and precise asymptotic analysis techniques. As applications, the optimal uniform Berry–Esseen bound and optimal Cramér-type moderate deviation for the corresponding maximum likelihood estimator are obtained. [ABSTRACT FROM AUTHOR]

In analysis of survival data, the Accelerated Life Model (ALM) is one of the widely used semiparametric models, and we often encounter various types of censored survival data, such as right censored data, doubly censored data, interval censored data, partly interval-censored data, etc. For complicated types of censored data, the studies of statistical inferences on the ALM are very technical and challenging mathematically, thus up to now little work has been done. In this article, we extend the concept of weighted empirical likelihood (WEL) from univariate case to multivariate case, and we apply it to the ALM, which leads to an estimation approach, called weighted maximum likelihood estimator, as well as the WEL based confidence interval for the regression parameter. Our proposed procedures are applicable to various types of censored data under a unified framework, and some simulation results are presented. [ABSTRACT FROM AUTHOR]

This study takes into account the efficient estimation of the probability density function (PDF) of the Weibull-Burr XII distribution and the cumulative distribution function (CDF), which allows for greater flexibility than many other generalized distributions in use today. Nine traditional estimators are used to produce the analytical formulations for the bias and the mean squared error (MSE). A simulated investigation is conducted to evaluate the finite sample performance of the suggested estimators in terms of MSE values. In addition to that a real data application is also demonstrated for illustration.

In analysis of survival data, the Accelerated Life Model (ALM) is one of the widely used semiparametric models, and we often encounter various types of censored survival data, such as right censored data, doubly censored data, interval censored data, partly interval-censored data, etc. For complicated types of censored data, the studies of statistical inferences on the ALM are very technical and challenging mathematically, thus up to now little work has been done. In this article, we extend the concept of weighted empirical likelihood (WEL) from univariate case to multivariate case, and we apply it to the ALM, which leads to an estimation approach, called weighted maximum likelihood estimator, as well as the WEL based confidence interval for the regression parameter. Our proposed procedures are applicable to various types of censored data under a unified framework, and some simulation results are presented.

In this paper, we used the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength $ R = P(Y < X) $ based on independent adaptive progressive censored samples that were taken from the Chen distribution. An approximate confidence interval of $ R $ was constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. Additionally, the asymptotic distribution theory and delta approach were used to generate the approximate confidence interval. Further, the Bayesian estimation of $ R $ was obtained based on the balanced loss function, which came in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating $ R $ using the Bayesian approach, all the unknown parameters of the Chen distribution were assumed to be independently distributed and to have informative gamma priors. Additionally, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm was used to compute the Bayes estimate of $ R $ and the associated highest posterior density credible interval. In the end, simulation research was used to assess the general overall performance of the proposed estimators and a real dataset was provided to exemplify the theoretical results.

Up to now, almost all existing methods for joint modeling survival data and longitudinal data rely on parametric/semiparametric assumptions on longitudinal covariate process, and the resulting inferences critically depend on the validity of these assumptions that are difficult to verify in practice. The kernel method-based procedures rely on choices of kernel function and bandwidth, and none of the existing methods provides estimate for the baseline distribution in proportional hazards model. This article proposes a proportional hazards model for joint modeling right censored survival data and intensive longitudinal data taking into account of within-subject historic change patterns. Without any parametric/semiparametric assumptions or use of kernel method, we derive empirical likelihood-based maximum likelihood estimators and partial likelihood estimators for the regression parameter and the baseline distribution function. We develop stable computing algorithms and present some simulation results. Analyses of real dataset are conducted for smoking cessation data and liver disease data. [ABSTRACT FROM AUTHOR]

The lifetime performance index is commonly used in the manufacturing industry to evaluate the performance of the capabilities of the production process. For products with multiple components, the comprehensive lifetime performance index, which is a monotonically increasing function of the overall process yield, is used to relate to each individual lifetime performance index. For products where the lifetime of the ith component follows an exponentiated Fréchet lifetime distribution, we examine the maximum likelihood estimators for both the comprehensive and individual lifetime performance indices based on the progressive type I interval-censored samples, deriving their asymptotic distributions. By specifying the target level for the comprehensive lifetime performance index, we can set the desired level for individual indices. A testing procedure, using the maximum likelihood estimator as the test statistic, was developed to determine if the comprehensive lifetime performance index meets the target. Given that the lifetime distribution is asymmetric, this study pertains to asymmetrical probability distributions and their applications across diverse fields. We illustrate the power analysis of this testing procedure with figures and summarize key findings. Finally, we demonstrate the application of this testing algorithm with a practical example involving two components to verify if the overall production process achieves the assigned target level. [ABSTRACT FROM AUTHOR]

In this paper, we used the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength R = P(Y < X) based on independent adaptive progressive censored samples that were taken from the Chen distribution. An approximate confidence interval of R was constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. Additionally, the asymptotic distribution theory and delta approach were used to generate the approximate confidence interval. Further, the Bayesian estimation of R was obtained based on the balanced loss function, which came in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating R using the Bayesian approach, all the unknown parameters of the Chen distribution were assumed to be independently distributed and to have informative gamma priors. Additionally, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm was used to compute the Bayes estimate of R and the associated highest posterior density credible interval. In the end, simulation research was used to assess the general overall performance of the proposed estimators and a real dataset was provided to exemplify the theoretical results. [ABSTRACT FROM AUTHOR]

In some real-life situations, we will face restrictions of time and sample size which cause a researcher to not have access to all of the data. Therefore, it is valuable to study the estimation of parameters based on information of available data. In such situations, using appropriate sampling schemes, to more efficient estimators are important. The aim of the present paper is to study the Bayes estimators of parameters of the Pareto type-I model under different loss functions and compare among them as well as with the classical estimator named maximum likelihood estimator based on upper record ranked set sampling scheme. Here the informative Gamma prior is used as the conjugate prior distribution for finding the Bayes estimator. We also used symmetric loss functions such as squared error loss function and asymmetric loss functions such as linear-exponential loss function. We present the analysis of a Monte Carlo simulation to compare the performance of the estimators with respect to their risks (average loss over sample space) based on upper record ranked set sampling. Finally, one real data set is analyzed to illustrate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

The article deals with numerical estimation of the drift parameter in the continuous-time linear model with two independent fractional Brownian motions. The main focus is given to the computational difficulties of the maximum likelihood approach, in particular, to the construction of the approximate solution to the Fredholm integral equation of the second kind with a singular kernel. We also introduce two alternative estimators and investigate their asymptotic properties. The performance of all estimators is studied numerically. [ABSTRACT FROM AUTHOR]

This paper proposes a bootstrap testing approach based on an approximate maximum likelihood method to discern whether a diffusion process is linear or whether there are threshold effects in the drift, the diffusion term or in both. It complements an alternative method based on the least-squares estimator which focuses on threshold effects in the drift. Monte Carlo simulations illustrate that the proposed testing approach is able to detect the source of the non-linearity. Two empirical applications show the importance of modeling threshold effects in the diffusion instead of the drift. [ABSTRACT FROM AUTHOR]

The Poisson-Inverse Gaussian regression model (PIGRM) is commonly used to analyze count datasets with over-dispersion. While the maximum likelihood estimator (MLE) is a standard choice for estimating PIGRM parameters, its performance may be suboptimal in the presence of correlated explanatory variables. To overcome this limitation, we introduce a novel Liu-type estimator for PIGRM. Our analysis includes an examination of the matrix mean square error (MMSE) and scalar mean square error (SMSE) properties of the proposed estimator, comparing them with those of the MLE, ridge, and Liu estimators. We also present several parameters of the Liu-type estimator for PIGRM. We evaluated the performance of the proposed estimator through a simulation study and application to real-life data, using SMSE as the primary evaluation criterion. Our results demonstrate that the proposed estimators outperform the MLE, ridge, and Liu estimators in both simulated and real-world scenarios. [ABSTRACT FROM AUTHOR]

Based on a progressive first-failure censoring (PFFC) sample, we discuss the statistical inferences of the entropy of a Rayleigh distribution. In particular, the Maximum likelihood and the different Bayes estimates for entropy are derived and compared via a Monte Carlo simulation study. Bayes estimators are developed using both symmetric and asymmetric loss functions. Approximate confidence intervals (CIs) and credible intervals (CrIs) of the entropy of the model are also performed. Numerical examples and a real data set are given to illustrate the proposed estimators. [ABSTRACT FROM AUTHOR]

The article offers a method for estimating the volatility covariance matrix of vectors of financial time series data using a change point approach. The proposed method supersedes general varying-coefficient parametric models, such as GARCH, whose coefficients may vary with time, by a change point model. In this study, an adaptive pointwise selection of homogeneous segments with a given right-end point by a local change point analysis is introduced. Sufficient conditions are obtained under which the maximum likelihood process is adaptive against the covariance estimate to yield an optimal rate of convergence with respect to the change size. This rate is preserved while allowing the jump size to diminish. Under these circumstances, argmax results of a two-sided negative Brownian motion or a two-sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, provide inference for the change point parameter. Theoretical results are supported by the Monte–Carlo simulation study. A bivariate data on daily log returns of two US stock market indices as well as tri-variate data on daily log returns of three banks are analyzed by constructing confidence interval estimates for multiple change points that have been identified previously for each of the two data sets. [ABSTRACT FROM AUTHOR]

AbstractThe two-parameter standard two-sided power family of distributions on (0, 1) is considered in this article. We propose bootstrap standard errors for the maximum likelihood estimators, as well as bootstrap confidence intervals for its parameters, once these important statistical measures of accuracy cannot be computed based on first-order asymptotic theory. We consider Monte Carlo simulation experiments to verify the performance of the bootstrap methods, and the numerical results are quite promising. Applications to real data are also considered to illustrate the proposed methodology in practice. [ABSTRACT FROM AUTHOR]

Theoretically, the maximum likelihood estimator has the sandwich-type asymptotic variance-covariance matrix under model misspecification. Its empirical estimator, that is called the robust variance estimator, is consistent. Thus, the estimator is asymptotically valid even under model misspecification. In practice, the robust variance estimator is used for computation of standard errors in longitudinal data analysis. Recently, Golden et al. (2019 Econometrics, 7, 1-27) showed that the maximum likelihood estimator retains a sandwich-type asymptotic variance-covariance matrix in the presence of missing data even when the missing-data mechanism is missing not at random. Although they revealed the asymptotic validity of the robust variance estimator in the simultaneous presence of both model misspecification and missing data, its finite-sample performance did not be investigated. In this article, we evaluated the finite-sample performance via simulation studies and clarify its small-sample problems. In addition, we illustrated the robust variance estimator using longitudinal CD4 count data from a randomized double-blind study. [ABSTRACT FROM AUTHOR]

We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order 1/4. We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to −1 or 1. [ABSTRACT FROM AUTHOR]

A shock model with two types of shocks functioning in the presence of an additional risk is proposed. Survival probability and mean residual life times of the proposed models are derived and assessed through the data of life testing experiment. Model validation and estimation of survival probability and mean residual life times is done through simulation studies. Comparison of survival probabilities and mean residual life times of models functioning without and with additional risk is made. [ABSTRACT FROM AUTHOR]

In this study, we explore a new probability distribution termed as the Length-Biased Transmuted Mukherjee-Islam (LBTMI) distribution. This exploration enhances the conventional Transmuted Mukherjee-Islam distribution by integrating a weighted transformation approach. The paper examines the probability density function and the corresponding cumulative distribution function associated with the LBTMI distribution. A comprehensive examination of the unique structural properties of the proposed model is carried out, including the survival function, conditional survival function, hazard function, cumulative hazard function, mean residual life, moments, moment generating function (MGF), characteristic function (CF), cumulant generating function (CGF), likelihood ratio test, ordered statistics, entropy measures, and Bonferroni and Lorenz curves. To ensure precise estimation of model parameters, the study employs the maximum likelihood estimation method, contributing significantly to the advancement of statistical modelling in this domain. [ABSTRACT FROM AUTHOR]

In this study, we employ a weighted transformation approach to introduce a novel model that generalises the Transmuted Mukherjee-Islam distribution. The resulting generalized distribution is referred to as the Weighted Transmuted Mukherjee-Islam (WTMI) distribution The paper thoroughly explores the probability density function (PDF) and the corresponding cumulative distribution function (CDF) associated with the WTMI distribution. A thorough investigation of the distinctive structural properties of the proposed model is conducted, including survival function, conditional survival function, hazard function, cumulative hazard function, mean residual life, moments, moment generating function (MGF), characteristics function (CF), cumulant generating function (CGF), likelihood ratio test, ordered statistics, entropy measures, and Bonferroni and Lorenz curves. The maximum likelihood estimation method is employed for the precise estimation of model parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the maximum likelihood estimator (MLE) for the parameter θ in the probability density function f (x ; θ) . We specifically focus on the application of moving extremes ranked set sampling (MERSS) and analyze its properties in large samples. We establish the existence and uniqueness of the MLE for two common distributions when utilizing MERSS. Our theoretical analysis demonstrates that the MLE obtained through MERSS is, at the very least, as efficient as the MLE obtained through simple random sampling with an equivalent sample size. To substantiate these theoretical findings, we conduct numerical experiments. Furthermore, we explore the implications of imperfect ranking and provide a practical illustration by applying our approach to a real dataset. [ABSTRACT FROM AUTHOR]

In this article, the explicit expression for the moments of record values is derived from the inverted Topp-Leone distribution. We have also derived the recurrence relation for single and product moments of the inverted Topp-Leone distribution based on record values. These results were utilized to obtain the best linear unbiased estimator for the location and scale parameter of the inverted Topp-Leone distribution. The best linear unbiased predictor of future record is also computed. Further, based on records, the maximum likelihood estimator for the scale and shape parameters of the inverted Topp-Leone distribution is also derived. Also, the exact confidence intervals for scale and shape parameters of the inverted Topp-Leone distribution are constructed in terms of upper records. We have also conducted a simulation study to show the performances of derived point and interval estimators. In addition to that, we have also presented a real data study to discuss the significance of derived results in real-life scenarios. This study is useful when the data are heavily right-tailed, follow an inverted Topp-Leone distribution, and are in the form of a record sequence. [ABSTRACT FROM AUTHOR]

Censoring plays an important role in the reliability and life testing trials due to its cost optimality and time reduction properties. The unified hybrid censoring scheme is the combination of the generalized type-I and type-II hybrid censoring schemes. In this paper, our objective is to study the classical and Bayesian estimation methods of the parameter and reliability characteristics from the inverse Pareto lifetime model under the unified hybrid censoring scheme. In the classical estimation methods, the maximum likelihood and associated asymptotic confidence interval estimators are derived. In Bayesian estimation, the Bayes estimators under squared error loss function and the highest posterior density (HPD) credible intervals based on the informative and non-informative priors are developed. For the Bayesian computations, the Markov chain Monte Carlo techniques are used to compute Bayes and HPD credible interval estimates. A quantitative outcome of the objectives has been shown by a Monte Carlo simulation and with the help of a real-life application. [ABSTRACT FROM AUTHOR]

This research focuses on estimating the stress-strength reliability in a system characterized by the influence of two random stresses on its strength, employing both frequentist and Bayesian approaches. The reliability of such systems is represented by the function δ = P (U < V < W) , where V denotes the system's strength, and U and W represent the stresses. The analysis is performed under a progressive Type-II censoring scheme, considering the random variables U, V, and W as independent and following the Burr-XII distribution. In a frequentist setup, both the maximum likelihood estimator and the maximum product spacings estimator of δ have been obtained. In the Bayesian paradigm, the Bayes estimator of δ under the squared error loss function is derived utilizing the Markov chain Monte Carlo method, considering independent gamma priors for the unknown parameters. In addition, asymptotic confidence intervals and highest probability density credible intervals for δ are also formulated. An extensive simulation experiment is carried out to compare the performances of the different developed estimators. Finally, a real-life application is presented to demonstrate the practical applicability of the proposed theory. [ABSTRACT FROM AUTHOR]

ABSTRACT Measuring the abundance of microbes in a sample is a common procedure with a long history, but best practices are not well-conserved across microbiological fields. Serial dilution methods are commonly used to dilute bacterial cultures to produce countable numbers of colonies, and from these counts, to infer bacterial concentrations measured in colony-forming units (CFUs). The most common methods to generate data for CFU point estimates involve plating bacteria on (or in) a solid growth medium and counting their resulting colonies or counting the number of tubes at a given dilution that have growth. Traditionally, these types of data have been analyzed separately using different analytic methods. Here, we build a direct correspondence between these approaches, which allows one to extend the use of the most probable number method from the liquid tubes experiments, for which it was developed, to the growth plates by viewing colony-sized patches of a plate as equivalent to individual tubes. We also discuss how to combine measurements taken at different dilutions, and we review several ways of analyzing colony counts, including the Poisson and truncated Poisson methods. We test all point estimate methods computationally using simulated data. For all methods, we discuss their relevant error bounds, assumptions, strengths, and weaknesses. We provide an online calculator for these estimators.Estimation of the number of microbes in a sample is an important problem with a long history. Yet common practices, such as combining results from different measurements, remain sub-optimal. We provide a comparison of methods for estimating abundance of microbes and detail a mapping between different methods, which allows to extend their range of applicability. This mapping enables higher precision estimates of colony-forming units (CFUs) using the same data already collected for traditional CFU estimation methods. Furthermore, we provide recommendations for how to combine measurements of colony counts taken across dilutions, correcting several misconceptions in the literature.

Abstract Time-delay and Doppler estimation is crucial in various engineering fields, as estimating these parameters constitutes one of the key initial steps in the receiver’s operational sequence. Due to its importance, several expressions of the Cramér–Rao Bound (CRB) and Maximum Likelihood Estimation (MLE) have been derived over the years. Previous contributions started from the assumption that the transmission process introduces an unknown phase, which hindered the explicit consideration of the time-delay parameter in the carrier-phase component in theoretical derivations. However, this contribution takes into account this additional term under the assumption that such an unknown phase is inferred and compensated for. This new condition leads to the derivation of a novel MLE. Subsequently, a closed-form expression of the achievable Mean Squared Error (MSE) for the time-delay and Doppler parameters is provided for the asymptotic region, assuming the signal is band-limited. Both expressions are validated via Monte Carlo simulations. This analysis reveals five distinct regions of operation of the MLE, refining existing knowledge and providing valuable insights into time-delay estimation

One of the flexible count data models for dealing with over and under-dispersion with extra zeroes is the zero-inflated Conway–Maxwell Poisson (ZICOMP). The ZICOMP regression coefficients are generally estimated using the maximum likelihood estimator (MLE). In the ZICOMP regression model, when the explanatory variables are correlated, the MLE does not give efficient results. To overcome the effect of multicollinearitymode in the ZICOPM regression, we proposed the ridge regression estimator. To evaluate the performance of the estimator, we use mean squared error (MSE) as the performance evaluation criteria. A theoretical comparison of the ridge estimator with MLE is made to show the superiority of the estimator. The proposed estimator is evaluated with the help of a simulation study and a real application. The results of the simulation study and real application show the superiority of the proposed estimator because it produces a smaller MSE as compared to the MLE. [ABSTRACT FROM AUTHOR]

In this article, we proposed a new probability distribution called length-biased Chris-Jerry distribution in which length-biased distribution is a specific case of weighted distribution. Also, we determined the structural properties of length-biased Chris-Jerry distribution and the unknown parameters were estimated by maximum likelihood estimator and tested by likelihood ratio test. Finally, we emphasized a real lifetime data set to reveal how the proposed distribution worked in it. [ABSTRACT FROM AUTHOR]

Non-homogeneous Poisson process with power law intensity function has often been used as a model for describing the failure pattern of repairable systems. Maximum likelihood and Bayesian estimation are used to estimate model parameters. Simulation and realistic application are used and represented by shutting down the gas power plant in Mosul. Stops in hours are designed with the power law random process model in order to obtain a model that represents the average stop time of the units throughout the study period in the best way. The results of the application on the data of the three concerned stations show that the Bayes estimate is better than the maximum likelihood estimate. This proves that the Bayes methods are very accurate and effective in estimating the rate of occurrence parameters. [ABSTRACT FROM AUTHOR]

Estimation of parameters of the generalized inverse Lindley (GIL) distribution is considered under a hybrid censoring scheme. The point estimators, such as the maximum likelihood estimators using the Expectation-Maximization (E-M) algorithm, have been derived. The two approximate Bayes estimators using Tierney and Kadane's method and Gibbs sampling procedure, using the gamma prior and the general entropy loss (GEL) function, have been obtained. Several confidence intervals are proposed, such as the asymptotic confidence intervals (ACIs), bootstrap confidence intervals, and the highest posterior density (HPD) credible intervals. The prediction for future observations has been considered under one and two-sample Bayesian prediction methods using the type-i hybrid censoring scheme. An extensive simulation study has been conducted to numerically evaluate all the estimators' performances. The point estimators are compared through their biases and mean squared errors (MSEs). The performances of confidence intervals are evaluated using coverage probability (CP), average length (AL), and probability coverage density (PCD). Two real-life datasets have been considered for illustrative purposes. [ABSTRACT FROM AUTHOR]

Time-delay and Doppler estimation is crucial in various engineering fields, as estimating these parameters constitutes one of the key initial steps in the receiver's operational sequence. Due to its importance, several expressions of the Cramér–Rao Bound (CRB) and Maximum Likelihood Estimation (MLE) have been derived over the years. Previous contributions started from the assumption that the transmission process introduces an unknown phase, which hindered the explicit consideration of the time-delay parameter in the carrier-phase component in theoretical derivations. However, this contribution takes into account this additional term under the assumption that such an unknown phase is inferred and compensated for. This new condition leads to the derivation of a novel MLE. Subsequently, a closed-form expression of the achievable Mean Squared Error (MSE) for the time-delay and Doppler parameters is provided for the asymptotic region, assuming the signal is band-limited. Both expressions are validated via Monte Carlo simulations. This analysis reveals five distinct regions of operation of the MLE, refining existing knowledge and providing valuable insights into time-delay estimation [ABSTRACT FROM AUTHOR]

This work utilizes generalized order statistics (GOSs) to study the q-Weibull distribution from several statistical perspectives. First, we explain how to obtain the maximum likelihood estimates (MLEs) and utilize Bayesian techniques to estimate the parameters of the model. The Fisher information matrix (FIM) required for asymptotic confidence intervals (CIs) is generated by obtaining explicit expressions. A Monte Carlo simulation study is conducted to compare the performances of these estimates based on type II censored samples. Two well-established measures of information are presented, namely extropy and weighted extropy. In this context, the order statistics (OSs) and sequential OSs (SOSs) for these two measures are studied based on this distribution. A bivariate q-Weibull distribution based on the Farlie-Gumbel-Morgenstern (FGM) family and its relevant concomitants are studied. Finally, two concrete instances of medical real data are ultimately provided. [ABSTRACT FROM AUTHOR]

This study introduces a new distribution, a Lehmann-type exponentiated distribution, which is built upon the unit exponentiated half-logistic distribution. The analytical characteristics of the proposed distribution, like moments, moment-generating function, quantiles, and stress-strength reliability, are explored in detail. The renowned maximum likelihood estimation method is employed for the statistical inference of the distribution’s parameters. A computer experiment is run to explore the performance of the maximum likelihood estimates of the distribution parameters under diverse scenarios. Additionally, the practicality and efficacy of the distribution are illustrated through a numerical example using a real-world dataset. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of efficient estimation for the drift parameter θ ∈ R d in the linear model Z t : = θ t + σ 1 B H 1 (t) + σ 2 B H 2 (t) , t ∈ [ 0 , T ]. Where B H 1 and B H 2 are two independent d-dimensional fractional Brownian motions with Hurst indices H1 and H2 such that 1 2 ≤ H 1 < H 2 < 1. The main goal is firstly to define the maximum likelihood estimator (MLE) of the drift θ, and secondly to provide a sufficient condition for the James-Stein type estimators which dominate, under the usual quadratic risk, the usual estimator (MLE). [ABSTRACT FROM AUTHOR]

In recent years, various forms of progressive hybrid censoring schemes (PHCS) have gained significant traction in survival and reliability analysis studies due to their versatility. However, these PHCS variants are often characterized by complexity stemming from the multitude of parameters involved in their specification. Consequently, the primary objective of this paper is to propose a unified approach termed combined type II progressive hybrid censoring scheme ( ComT 2 PHCS) capable of encompassing several existing PHCS variations. Our analysis focuses specifically on the exponential distribution (ExDist). Bayesian inference techniques are employed to estimate the parameters of the ExDist under the ComT 2 PHCS. Additionally, we conduct fundamental distributional analyses and likelihood inference procedures. We derive the conditional moment-generating function (CondMGF) of maximum likelihood estimator (MLE) for parameters of the ExDist under ComT 2 PHCS. Further, we use CondMGF for the distribution of MLE for parameters of ExDist under ComT 2 PHCS. Finally, we provide an illustrative example to elucidate the inference methods derived in this paper. [ABSTRACT FROM AUTHOR]

This paper establishes an Edgeworth expansion for the t-statistic of the Whittle Maximum Likelihood Estimator (WMLE) of a linear regression model whose residual component is stationary, Gaussian, and strongly dependent time series. Under the widely used set of assumptions and two more mild additional conditions on the spectral density function and the parametric values, an Edgeworh expansion of the t-statistic of arbitrarily large order of the process is proved to have an error of o (n 1 − s / 2) where s is a positive integer. [ABSTRACT FROM AUTHOR]

New parameters can be added to expand families of distribution for greater flexibility or to construct covariate models in several ways. In this study, a trigonometric-type distribution called Sine-Weibull distribution was developed by adopting the Weibull distribution as the baseline distribution and Sine-G Family as the generator to generate a flexible probability distribution without the need for extra parameters. The moment, moment generating function, entropy, and order statistics are some of the mathematical aspects of this distribution that were derived. The Maximum Likelihood approach was used to estimate the new distribution's parameters. Using actual datasets, the Sine-Weibull distribution's applicability was demonstrated. [ABSTRACT FROM AUTHOR]