Quartic Transmuted Exponential Distribution: Characteristics and Parameter Estimation

The distribution of generalized exponential was invented by Rameshwar D. Gupta and Debasis Kundu in 2007. The distribution was the result of a generalized transformation of the exponential distribution. This paper explained the generalized exponential Marshall-Olkin distribution which is the result of the expansion of the generalized exponential distribution using the Marshall-Olkin method. The generalized exponential Marshall-Olkin distribution has a more flexible form than the previous distribution, especially in its hazard function which has various forms so that it can represent survival data better. The flexibility characteristic is due to the addition of new parameters to the generalized exponential Marshall-Olkin distribution. We explained some characteristics of the Marshall-Olkin generalized exponential distribution such as the probability density function (PDF), cumulative distribution function (CDF), survival function, hazard function, mean, and moments. Parameter estimation was conducted using the maximum likelihood method. In the application, it was shown data with generalized exponential Marshall-Olkin (GEMO) distribution. The GEMO distribution was modelled to the waiting time data until the damage to a lamp. The data was taken from Aarset data (1987). The results of modelling the waiting time data until the damage to a lamp on the distribution of GEMO and was compared to the distribution of alpha power Weibull. A comparison of models using Akaike information criteria (AIC) and Bayesian information criteria (BIC) shows that the distribution of GEMO is more suitable in modelling the lamp damage waiting time data than the distribution of alpha power Weibull.

Two parameters Maxwell – Exponential distribution was proposed using the Maxwell generalized family of distribution. The probability density function, cumulative distribution function, survival function, hazard function, quantile function, and statistical properties of the proposed distribution are discussed. The parameters of the proposed distribution have been estimated using the maximum likelihood estimation method. The potentiality of the estimators was shown using a simulation study. The overall assessment of the performance of Maxwell - Exponential distribution was determined by using two real-life datasets. Our findings reveal that the Maxwell – Exponential distribution is more flexible compared to other competing distributions as it has the least value of information criteria.

The maximum likelihood estimation of shape parameters of Lomax distribution under bilateral Type-I Censored Sample is studied, the explicit expression of shape parameter estimation cannot be obtained, but it is proved that the maximum likelihood estimation of this parameter is unique. Then the EM estimation of shape parameters is obtained by using EM algorithm, and the asymptotic variance and approximate confidence interval of the EM estimation of shape parameters are also given; Finally, the effects of maximum likelihood estimation and EM estimation of shape parameters of Lomax distribution are numerically simulated. By comparing the relative deviation, it shows that the estimation effect of using EM algorithm to calculate shape parameters is relatively good, and the asymptotic variance and asymptotic confidence interval of EM estimation are given.

This paper proposes a weighting of the exponentiated gamma distribution with a polynomial function called the poly-weighted exponentiated gamma distribution (PWEGD). It shows that the modified distribution harnesses the multi-dimensional effects of the distribution. We provided an extensive mathematical treatment of this proposed distribution: obtained its parameters, estimated its statistical properties with applicable, tests and compared the estimates with existing distribution. The study estimated the cumulative distribution function, hazard function, survival function, skewness, kurtosis, mode, median and quartiles of the distribution and evaluated the distribution with Monte Carlo simulated data and the data of the wind direction (degrees) in Lagos, Nigeria. Empirical analysis showed that with increased polynomial function, the estimates and the statistical properties like the expectation, variance, standard error, median, mode, hazard and survival functions, cumulative distribution function (CDF), moments, skewness and kurtosis were significantly better than the existing root distributions. The MSE of the parameters decreased with increased power and the parameter is significant (p <0.05). It is concluded that the proposed distribution does not only provide better fitting but also establishes an efficient structure for lifetime data modelling.

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.

This study introduces the Odd Rayleigh-G (OR-G) family of distribution and explores its mathematical properties, applications, and performance comparisons. The Odd Rayleigh-Weibull distribution (ORWD) is developed by incorporating the "Odd" transformation into the Rayleigh and Weibull distribution, resulting in a flexible model suitable for various real-life and survival data applications. The probability density function (PDF), cumulative distribution function (CDF), hazard function, and survival function of the ORWD are derived and analyzed. Parameter estimation is performed using the Maximum Likelihood Estimation (MLE) method, and the performance of the ORWD is assessed through simulation studies. The simulations for parameter estimates at 100 sample sizes were conducted and the plot of the simulated data on the PDF, CDF, survival and hazard function demonstrate a comprehensive view of the characteristics of the Odd Rayleigh Weibull distribution. This information is useful for understanding the behaviour of the distribution and for applications in reliability analysis and survival studies. The results demonstrate the consistency and efficiency of the MLE method for the ORWD. The ORWD is compared with other distributions, including the Weibull, Power Rayleigh, and Rayleigh distributions, using goodness-of-fit measures such as the Akaike Information Criterion (AIC = 111.0238 and 87.4294), Bayesian Information Criterion (BIC = 117.2564 and 96.0320), and Kolmogorov-Smirnov (KS = 0.9559 and 0.9889) test with p-values (p-val = 7.772e-16 and 2.2e-16). The ORWD shows superior performance in fitting the mortality dataset and the Reddit advertisement dataset, highlighting its potential for modelling complex data structures. Overall, this study provides a comprehensive framework for the...

The inferences of data obtained from periodic inspection and type I censoring for the three step stress accelerated life test are studied in this paper. The failure rate function that a log-quadratic relation of stress and the tampered failure rate model are considered under the exponential distribution. The optimal stress change times which minimize the asymptotic variance of maximum likelihood estimators of parameters is determined and the maximum likelihood estimators of the model parameters are estimated. A numerical example will be given to illustrate the proposed inferential procedures.

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

Assumption of normality in statistical analysis had been a common practice in many literature, but in the event where small sample is obtainable, then normality assumption will lead to erroneous conclusion in the statistical analysis. Taking a large sample had been a serious concern in practice due to various factors. In this paper, we further derived some inferential properties for log student’s t-distribution (simply log-t distribution) which makes it more suitable as substitute to log-normal when carrying out analysis on right-skewed small sample data. Mathematical and Statistical properties such as the moments, cumulative distribution function, survival function, hazard function and log-concavity are derived. We further extend the results to case of multivariate log-t distribution; we obtained the marginal and conditional distributions. The parameters estimation was done via maximum likelihood estimation method, consequently its best critical region and information matrix were derived in order to obtain the asymptotic confidence interval. The applications of log-t distribution and goodness-of-fit test was carried out on two dataset from literature to show when the model is most appropriate.

This paper is devoted to studying the probabilistic and statistical properties of maximum likelihood estimation of parameters in spatially mixed autoregressive models. When the response variable in the mixed autoregressive model obeys a continuous distribution, this paper verifies the monotonicity of the likelihood function of the mixed autoregressive model with respect to the autoregressive parameter P, which proves the existence and uniqueness of the maximum likelihood estimation of the autoregressive parameter. The results show that when the condition n &gt; rank(x+1) is satisfied, the quasi-likelihood function of the mixed autoregressive model has a unique maximum value with probability 1 in the parameter space; when n &gt; rank(x+1) and the regression coefficient When the matrix column is full rank, the maximum likelihood estimates of all parameters in the spatial mixed autoregressive model exist with probability 1 and are unique in the parameter space. In order to detect the strong influence points and abnormal points in the spatial mixed autoregressive model, this paper uses the first derivative method in the local influence analysis to obtain the statistics of the strong influence points and abnormal points in the spatial mixed autoregressive model in the form of variance perturbation. Simulation studies have shown that the first derivative of the maximum likelihood estimate of variance in the spatial mixed autoregressive model should be chosen to obtain the test statistic, which can effectively avoid the mearing and masking effects that often occur and are difficult to handle in local influence analysis. . As application and verification, this paper analyzes a real data to show that the conclusions obtained in this paper are reliable and practical.

This is a new lifetime Exponential using the Topp-Leone generated family of distributions proposed by Rezaei et al. The new distribution is called the Topp-Leone Exponential (TLE) distribution. What is done in this paper is an estimation of the two parameters for Topp-Leone Exponential distribution model by using the maximum likelihood estimator method to get the derivation of the point estimators for all unlabeled parameters according to iterative techniques as Newton $-$ Raphson method, then to derive Ordinary least squares estimator method. Applying all two methods to estimate related probability functions; death density function, cumulative distribution function, survival function and hazard function (rate function). When examining the numerical results for probability survival function by employing mean squares error measure and mean absolute percentage measure, this may lead to work on the best method in modeling a set of real data.

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.

Inference in the additive risk model with time-varying covariates subject to measurement errors

In this paper, a new distribution called Marshall-Olkin Exponentiated Fréchet distribution (MOEFr) is proposed. The goal is to increase the flexibility of the existing Exponentiated Fréchet distribution by including an extra shape parameter, resulting into a more flexible distribution that can provide a better fit to various data sets than the baseline distribution. A generator method introduced by Marshall and Olkin is used to develop the new distribution. Some properties of the new distribution such as hazard rate function, survival function, reversed hazard rate function, cumulative hazard function, odds function, quantile function, moments and order statistics are derived. The maximum likelihood estimation is used to estimate the model parameters. Monte Carlo simulation is used to evaluate the behavior of the estimators through the average bias and root mean squared error. The new distribution is fitted and compared with some existing distributions such as the Exponentiated Fréchet (EFr), Marshall-Olkin Fréchet (MOFr), Beta Exponential Fréchet (BEFr), Beta Fréchet (BFr) and Fréchet (Fr) distributions, on three data sets, namely Bladder cancer, Carbone and Wheaton River data sets. Based on the goodness-of-fit statistics and information criteria values, it is demonstrated that the new distribution provides a better fit for the three data sets than the other distributions considered in the study.

The idea of introducing extra parameters into the existing model in enhancing more flexibility is a giant stride in research. Transmutation map technique is one of the recent methods of introducing additional properties such as skewness, kurtosis and bimodality into the baseline distribution. In this article, a new exponentiated exponential distribution is developed using transmutation map. This model is referred to as exponentiated cubic transmuted exponential distribution (ECTED). The mathematical properties of the model which include survival function, hazard function, central and non- central moments, moment generating function and order statistics are established. The inherent parameters in the model are estimated using method of maximum likelihood estimation (MLE). The system of equations obtained is non-linear in parameters therefore non-linear optimization algorithms are implemented in R package. The distribution is used to model data on infant mortality rate in Nigeria. The performance of the subject model is compared with its baseline exponential distribution (ED), transmuted exponential distribution (TED), exponentiated transmuted exponential distribution (ETED) and cubic transmuted exponential distribution (CTED) using Akaike Information criterion (AIC), Corrected Akaike Information criterion (AICC) and Bayesian Information criterion (BIC). It is hope that this will serve as an alternative distribution in modelling complex real life data arising from various fields of human endeavors.