Modified Inverse Weibull Distribution: Using Python Approach to Evaluate Customer Transaction Data

In this paper, we have derived and studied new probability distributions by extending the 2‐dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3‐dimensional Rayleigh distribution (3‐DRD) and k‐dimensional Rayleigh distribution (k‐DRD), respectively. For both 3‐DRD and k‐DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi‐square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3‐DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3‐DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh‐based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

：A new distribution called the exponentiated Burr XII Weibull(EBW) distributions is proposed and presented. This distribution contains several new and known distributions such as exponentiated log-logistic Weibull, exponentiated log-logistic Rayleigh, exponentiated log-logistic exponential, exponentiated Lomax Weibull, exponentiated Lomax Rayleigh, exponentiated Lomax Exponential, Lomax Weibull, Lomax Rayleigh Lomax exponential, Weibull, Rayleigh, exponential and log-logistic distributions as special cases. A comprehensive investigation of the properties of this generalized distribution including series expansion of probability density function and cumulative distribution function, hazard and reverse hazard functions, quantile function, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, R´enyi entropy and distribution of order statistics are presented. Parameters of the model are estimated using maximum likelihood estimation technique and real data sets are used to illustrate the usefulness and applicability of the new generalized distribution compared with other distributions.

The error estimate of Borgonovo's moment-independent index δi is considered, and it shows that the possible computational complexity of δi is mainly due to the probability density function (PDF) estimate because the PDF estimate is an ill-posed problem and its convergence rate is quite slow. So it reminds us to compute Borgonovo's index using other methods. To avoid the PDF estimate, δi, which is based on the PDF, is first approximatively represented by the cumulative distribution function (CDF). The CDF estimate is well posed and its convergence rate is always faster than that of the PDF estimate. From the representation, a stable approach is proposed to compute δi with an adaptive procedure. Since the small probability multidimensional integral needs to be computed in this procedure, a computational strategy named asymptotic space integration is introduced to reduce a high-dimensional integral to a one-dimensional integral. Then we can compute the small probability multidimensional integral by adaptive numerical integration in one dimension with an improved convergence rate. From the comparison of numerical error analysis of some examples, it can be shown that the proposed method is an effective approach to uncertainty importance measure computation.

This study presents the Kumaraswamy Marshal–Olkin Flexible Weibull Extension (KMO-FWE) distribution, a newly developed probability model designed to extend existing lifetime distributions. This extension enhances the ability to model various hazard rate patterns with greater flexibility. The study explores the fundamental properties of this distribution, including the cumulative distribution function (CDF), probability density function (PDF), survival function, hazard function, moments, moment generating function, quantile function, and order statistics. To estimate the distribution’s parameters, several statistical methods are assessed, with a primary focus on Maximum Likelihood Estimation (MLE). Additionally, alternative estimation techniques such as Maximum Product Spacing (MPS), Least Squares (LS), Weighted Least Squares (WLS), and Percentile Estimation (PE) are considered. A Monte Carlo simulation study is conducted to evaluate the accuracy and efficiency of these estimation approaches under both complete data and Type-II censoring conditions. The results indicate that MLE is highly effective for large sample sizes but may introduce bias in smaller samples. In contrast, MPS provides stable parameter estimates under more complex conditions. While LS and WLS are computationally straightforward, their performance diminishes in the presence of censoring. Meanwhile, PE proves beneficial in reliability analysis based on percentile estimation. Overall, the findings highlight the KMO-FWE distribution as a versatile and reliable model, suitable for applications in reliability engineering, survival analysis, and biomedical research.

This study introduces a new probability distribution, the Uniform–Exponential Distribution, constructed by combining the Uniform and Exponential distributions to capture phenomena characterized by both constant and exponentially decaying behavior. The distribution is developed within the T-X family framework, and its fundamental properties are derived, including the probability density function (PDF), cumulative distribution function (CDF), hazard function, reverse hazard function, and moment-generating function (MGF). Analytical expressions for key moments—mean, variance, skewness, and kurtosis—are presented to describe the distribution’s shape and behavior. Parameter estimation is conducted using maximum likelihood estimation (MLE), ensuring statistical rigor in model fitting. Theoretical examples are provided to illustrate the distribution’s practical relevance. Potential applications are identified in reliability analysis, survival modeling, and environmental science, underscoring the distribution’s flexibility and utility in modeling diverse real-world processes.

In this exploratory study of integrated marketing communications (IMC), we bring together relational and transactional data, and sales and marketing views, to investigate whether firms can enhance their position in the market in terms of both marketing-oriented and sales-based performance benchmarks. This article examines, in a business-to-business services context, the relationships between collecting various types of customer transaction and relational data, customer data quality, and two types of business unit performance. The first type of performance is a more traditional sales-oriented metric of sales-oriented business growth, growth in sales, and net income. The second type is a less traditional marketing-oriented metric, called marketing-oriented customer performance, a summed measure of retention rate, share of wallet, lifetime customer value, and return on investment (ROl). An extensive review of different types of transaction and relational data produced two different categories of transaction and relationship information.

Harmonic reducers exhibit non-stationary and phase-dependent degradation behavior during long-term service, challenging the ability of classical stochastic degradation models to accurately assess reliability. To address phase-dependent differences in degradation behavior, this paper proposes a reliability assessment model based on a two-phase hybrid stochastic degradation process. In the proposed framework, the Wiener process is employed to characterize early-phase gradual degradation dominated by stochastic fluctuations, while the Inverse Gaussian process is used to describe later-phase monotonically accelerated degradation driven by cumulative damage. The framework allows for sample-level variability in transition times to more realistically capture individual degradation behavior. The Schwarz Information Criterion is also adopted to detect change points. Maximum likelihood estimation is performed for model parameter inference, and analytical expressions for the reliability function, cumulative distribution function, and probability density function are derived. Numerical results indicate that a change point exists for each tested product and that the proposed model achieves the best goodness of fit among the considered candidates, demonstrating its superiority in capturing phase-dependent characteristics of harmonic reducer degradation. In terms of reliability assessment bias, the proposed model (0.06%) significantly outperforms the Wiener degradation model (32%) and the IG degradation model (9.9%). These results further confirm that, under an identical failure threshold, the proposed approach yields more accurate and realistic reliability assessment outcomes.

In many applications such as transaction data analysis, the classification of long chains of sequences is required. For example, brand purchase history in customer transaction data is in a form like AABCABAA, where A, B, and C are brands of a consumer product. The decision tree-based package mbonsai is designed to handle sequence data of varying lengths using one or multiple variables of interest as predictor variables. This software package uses tree growing and pruning strategies adopted from C4.5 and CART algorithms, and includes new features for handling sequence data and indexing for classification purpose. The software uses a simple command line program for learning and predicting processes, and has the ability to generate user-friendly graphics depicting decision trees. The underlying C++ codes are designed to efficiently process large data sets in ASCII files. Two examples from transaction data sets are used to illustrate the application of mbonsai.

The events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.

The non-line-of-sight (NLOS) propagation caused by intermittently blocking of the direct path between the base station and the mobile station is the main difficulty for mobile user location in cellular wireless communication system. So it is indispensable to investigate the techniques about the NLOS error identification and correction. In this paper, we suggest a robust and accurate approach using probability density function (PDF) estimator to identify the NLOS situation, and then an improved biased Kalman filter which depends on PDF estimator as the nonlinearity weighting coefficient in recursive operation is used to mitigate the NLOS error for time of arrival (TOA) measurements. As the PDF estimator is calculated from actual TOA samples, it is adaptive for the complex practical NLOS environments. The simulation results also indicate that with the less prior information about communication environments, the approach we present has a good location performance even in severe NLOS situations.

Half-Cauchy generalized exponential (HCGE) distribution is a novel distribution that we have proposed on in this paper. The quantiles, the measures of skewness based on quartiles, and the measures of kurtosis based on octiles, survival function, the probability density function, hazard function, cumulative distribution function, cumulative hazard function, are just a few of the crucial statistical properties we have derived for the proposed distribution. To estimate the parameters of the half-Cauchy generalized exponential distribution, the maximum likelihood estimation method has been applied. For the evaluation of the new distribution’s potential, we have considered a real dataset and compared the goodness-of-fit attained by proposed distribution with some competing distribution. The suggested model fits the data much better and is more adaptable than some other models.

In this manuscript, we proposed a new extension of the Rayleigh distribution named as MTI Rayleigh distribution (MTIRD) that provides better fits than the Rayleigh distribution and some of its known generalizations. Various properties of the proposed distribution, including moments, moment generating function, hazard rate, conditional moments, Bonferroni and Lorenz curve, mean residual life, mean waiting time, Renyi entropy and order statistics are derived. Maximum likelihood estimation procedure is employed to estimate the unknown parameters. An extensive simulation study is carried out to illustrate the behaviour of MLEs on the basis of Mean Square Errors. The flexibility of the new distribution is assessed by applying it to two real data sets of COVID-19 mortality. The comparative behaviour of MTIRD with Rayleigh distribution, Power Rayleigh distribution, Transmuted Rayleigh distribution, Exponentiated Rayleigh distribution, Weibull Rayleigh distribution provided the evidence that it outperforms the other competing distributions based on Akaike Information Criterion, Bayesian Information Criterion, Akaike Information criterion Corrected, and other goodness of fit measures.

In this paper, we propose a fast and versatile blind source separation including closed-form estimation of sources' probability density functions (PDFs), where the ICA's activation function is automatically adapted to various noise conditions. In the proposed method, closed-form second-order ICA and closed-form PDF estimation are introduced as a computational-cost-efficient preprocessing to extract sources' PDFs. Compared with various type of conventional ICAs, e.g., fixed activation-function type and ML-based type, our proposed algorithm can give a faster and higher convergence. Experimental assessment reveals that the proposed method is versatile for handling non-speech sound sources.

The research aims to overcome the problem of managing customer data, room data, and transaction data manually at Hotel Fergael. The management of customer data, room data and transaction data that are processed manually, is overcome by building a Front Office Information System for hotel room reservations at Hotel Fergael. The results of the Front Office Information System research for hotel room reservations at the Fergael Hotel can be done quickly, effectively and efficiently at the receptionist's work time, making reports can also assist managers in seeing reservations that have or have occurred at the Fergael Hotel. The system is designed in the form of Visual Basic and uses a MySQL database. With this research, the receptionist can expedite the activities that run at the Fergael Hotel, especially room input, customer input, and room reservations that occur at the Fergael Hotel Pematangsiantar. Keywords: Information System, Visual Basic, MySQL, Reservation, Hotel