An effective and economic estimation of population mean in stratified random sampling using a linear cost function.

Improvement in the estimation of population mean has been an area of interest in sampling theory. So many estimators have been suggested for elevated estimation of the population mean in stratified random sampling, but there is still a gap for more closely estimating the population mean. In this paper, the authors propose a ratio-product-cum-exponential-cum-logarithmic type estimator for the enhanced estimation of population mean by implying one auxiliary variable in stratified random sampling using conventional ratio, exponential ratio, and logarithmic ratio type estimators. The suggested estimator is a generalization of ratio, exponential ratio, and logarithmic ratio type estimators, and therefore these are special cases of the proposed estimator. The proposed estimator's bias and MSE are determined and compared with those of influential estimators, with the linear cost function being used to investigate and compare alternatives. Use Cramer's rule to determine the optimal value of the proposed estimator. The proposed estimator is more effective than other existing estimators, according to theoretical observations. For various applications, we suggest using a proposed estimator with the minimal MSE, which is verified by a numerical example, to have practical applicability of theoretical conclusions in real life.

Exact analytical soliton solutions of the M-fractional Akbota equation.

In this paper we explore the new analytical soliton solutions of the truncated M-fractional nonlinear ( 1 + 1 ) -dimensional Akbota equation by applying the exp a function technique, Sardar sub-equation and generalized kudryashov techniques. Akbota is an integrable equation which is Heisenberg ferromagnetic type equation and have much importance for the analysis of curve as well as surface geometry, in optics and in magnets. The obtained results are in the form of dark, bright, periodic and other soliton solutions. The gained results are verified as well as represented by two-dimensional, three-dimensional and contour graphs. The gained results are newer than the existing results in the literature due to the use of fractional derivative. The obtained results are very helpful in optical fibers, optics, telecommunications and other fields. Hence, the gained solutions are fruitful in the future study for these models. The used techniques provide the different variety of solutions. At the end, the applied techniques are simple, fruitful and reliable to solve the other models in mathematical physics.

An innovative fractional-order evolutionary game theoretical study of personal protection, quarantine, and isolation policies for combating epidemic diseases.

This study uses imposed control techniques and vaccination game theory to study disease dynamics with transitory or diminishing immunity. Our model uses the ABC fractional-order derivative mechanism to show the effect of non-pharmaceutical interventions such as personal protection or awareness, quarantine, and isolation to simulate the essential control strategies against an infectious disease spread in an infinite and uniformly distributed population. A comprehensive evolutionary game theory study quantified the significant influence of people's vaccination choices, with government forces participating in vaccination programs to improve obligatory control measures to reduce epidemic spread. This model uses the intervention options described above as a control strategy to reduce disease prevalence in human societies. Again, our simulated results show that a combined control strategy works exquisitely when the disease spreads even faster. A sluggish dissemination rate slows an epidemic outbreak, but modest control techniques can reestablish a disease-free equilibrium. Preventive vaccination regulates the border between the three phases, while personal protection, quarantine, and isolation methods reduce disease transmission in existing places. Thus, successfully combining these three intervention measures reduces epidemic or pandemic size, as represented by line graphs and 3D surface diagrams. For the first time, we use a fractional-order derivate to display the phase-portrayed trajectory graph to show the model's dynamics if immunity wanes at a specific pace, considering various vaccination cost and effectiveness settings.

Parameter free AEWMA control chart for dispersion in semiconductor manufacturing.

The study presents a new parameter free adaptive exponentially weighted moving average (AEWMA) control chart tailored for monitoring process dispersion, utilizing an adaptive approach for determining the smoothing constant. This chart is crafted to adeptly detect shifts within anticipated ranges in process dispersion by dynamically computing the smoothing constant. To assess its effectiveness, the chart's performance is measured through concise run-length profiles generated from Monte Carlo simulations. A notable aspect is the incorporation of an unbiased estimator in computing the smoothing constant through the suggested function, thereby improving the chart's capability to identify different levels of increasing and decreasing shifts in process dispersion. The comparison with an established adaptive EWMA-S2 dispersion chart highlights the considerable efficiency of the proposed chart in addressing diverse magnitudes of process dispersion shifts. Additionally, the study includes an application to a real-life dataset, showcasing the practicality and user-friendly nature of the proposed chart in real-world situations.

Estimation of population median using bivariate auxiliary information in simple random sampling.

To estimate the unknown population median, several researchers have developed efficient estimators but these estimators are unable to provide efficient results in the existence of outliers. Keeping this point in view, the present work suggests enhanced class of robust estimators to estimate population median under simple random sampling in case of outliers/extreme observations. The suggested estimators are a mixture of bivariate auxiliary information and robust measures with the linear combination of deciles mean, tri-mean and Hodges Lehmann estimator. Mathematical properties associated with the improved class of robust estimators are evaluated in terms of bias and mean squared error. Moreover, the potentiality of our suggested estimators as compared to already available estimators is checked by considering two real-life data sets with outlier(s). In addition, a simulation study is also added in this regard. From theoretical and numerical findings, it is observed that our newly suggested estimators outperforms as compared to its competitors.

An adaptive Bayesian approach for improved sensitivity in joint monitoring of mean and variance using Max-EWMA control chart.

This article introduces an adaptive approach within the Bayesian Max-EWMA control chart framework. Various Bayesian loss functions were used to jointly monitor process deviations from the mean and variance of normally distributed processes. Our study proposes the mechanism of using a function-based adaptive method that picks self-adjusting weights incorporated in Bayesian Max-EWMA for the estimation of mean and variance. This adaptive mechanism significantly enhances the effectiveness and sensitivity of the Max-EWMA chart in detecting process shifts in both the mean and dispersion. The Monte Carlo simulation technique was used to calculate the run-length profiles of different combinations. A comparative performance analysis with an existing chart demonstrates its effectiveness. A practical example from the hard-bake process in semiconductor manufacturing is presented for practical context and illustration of the chart settings and performance. The empirical results showcase the superior performance of the Adaptive Bayesian Max-EWMA control chart in identifying out-of-control signals. The chart's ability to jointly monitor the mean and variance of a process, its adaptive nature, and its Bayesian framework make it a useful and effective control chart.

Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable.

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.