Confidence Intervals for Comparing Two Life Distributions

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.

In this article, an attempt is made to propose a novel method of lifetime distributions with maximum flexibility using a popular T–X approach together with an exponential distribution, which is known as the New Generalized Logarithmic-X Family (NGLog–X for short) of distributions. Additionally, the generalized form of the Weibull distribution was derived by using the NGLog–X family, known as the New Generalized Logarithmic Weibull (NGLog–Weib) distribution. For the proposed method, some statistical properties, including the moments, moment generating function (MGF), residual and reverse residual life, identifiability, order statistics, and quantile functions, were derived. The estimation of the model parameters was derived by using the well-known method of maximum likelihood estimation (MLE). A comprehensive Monte Carlo simulation study (MCSS) was carried out to evaluate the performance of these estimators by computing the biases and mean square errors. Finally, the NGLog–Weib distribution was implemented on four real biomedical datasets and compared with some other distributions, such as the Alpha Power Transformed Weibull distribution, Marshal Olkin Weibull distribution, New Exponent Power Weibull distribution, Flexible Reduced Logarithmic Weibull distribution, and Kumaraswamy Weibull distribution. The analysis results demonstrate that the new proposed model performs as a better fit than the other competitive distributions.

This paper introduces a novel approach by implementing Type II censoring on both stages of the stage life testing (SLT) experiment. First, we have assumed that the lifetime of the experimental units follows a combination of Weibull distribution on initial stage and an exponential distribution on the accelerated stage. Furthermore, we have also considered a case where the lifetime of the experimental units on both stages follows Weibull distribution with same shape parameter and different scale parameters. The distributions under the two stress levels are connected through the cumulative exposure model. We derive maximum likelihood estimation (MLE) in the SLT model under different lifetime distributions. As an alternative to MLE, we also derive the maximum product of spacings (MPS) estimation method and compare the performance of the MPS and MLE methods through a simulation study. Additionally, we derive asymptotic confidence intervals (CIs) based on the asymptotic normality of MLE, and we also employ bootstrap methods to obtain the boot‐ and boot‐ CIs. Further, we proposed an optimal design for the SLT experiment based on two optimality criteria. Finally, an extensive simulation study and a real data analysis have been carried out to demonstrate the effectiveness of our approach.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

Abstract The existence and some properties of maximum likelihood estimators (MLE's) are studied for a minimum-type distribution function corresponding to a minimum of two independent random variables having exponential and Weibull distributions. It is shown that if all three parameters are unknown, then there is a path in the parameter space along which the likelihood function (LF) tends to infinity. It is also proved that if the Weibull shape parameter is known, then the LF is concave, the MLE's exist, and they can be found by solving the set of likelihood equations. Properties of the MLE's for this case are illustrated by a Monte Carlo experiment. A sufficient condition for the existence of MLE's is given for the case of known Weibull scale parameter. Key Words: Minimum-type distribution functionMaximum likelihood estimatorsWeibull distributionExponential distribution

Simple estimators of the Weibull shape parameter and any quantile in uncensored samples are shown to share the pivotal function properties of maximum likelihood estimators. This allows interval estimation of the Weibull parameters and quantiles once the distributions of the necessary pivotal functions are determined by Monte Carlo sampling for a given sample size and quantile of interest. Free software for this purpose is made available for download. It is shown that the quantile and shape parameter estimates may be readily corrected so as to be median unbiased. The shape parameter estimator is shown to be less precise than the corresponding maximum likelihood estimator. However, the precision of the estimator of the pth quantile is quite comparable to that of the maximum likelihood method over the range from 0.4≤ p≤0.7.

Statistical analysis of adaptive type-II progressively censored competing risks for Weibull models

Univariate Weibull distribution is a well known lifetime distribution and has been widely used in reliability and survival analysis. In this paper, we introduce a new family of bivariate generalized Weibull (BGW) distributions, whose univariate marginals are exponentiated Weibull distribution. Different statistical quantiles like marginals, conditional distribution, conditional expectation, product moments, correlation and a measure component reliability are derived. Various measures of dependence and statistical properties along with aging properties are examined. Further, the copula associated with BGW distribution and its various important properties are also considered. The methods of maximum likelihood and Bayesian estimation are employed to estimate unknown parameters of the model. A Monte Carlo simulation and real data study are carried out to demonstrate the performance of the estimators and results have proven the effectiveness of the distribution in real-life situations.

This paper introduces the new novel four‐parameter Weibull distribution named as the Marshall–Olkin alpha power Weibull (MOAPW) distribution. Some statistical properties of the distribution are examined. Based on Type‐I censored and Type‐II censored samples, maximum likelihood estimation (MLE), maximum product spacing (MPS), and Bayesian estimation for the MOAPW distribution parameters are discussed. Numerical analysis using real data sets and Monte Carlo simulation are accomplished to compare various estimation methods. This novel model’s supremacy upon some famous distributions is explained using two real data sets and it is shown that the MOAPW model can achieve better fits than other competitive distributions.

Introduction and objectives The objective measurement of cough frequency has changed the standards by which we evaluate anti-tussive treatments and study the mechanisms driving cough. The distribution of individual cough events over time can vary substantially between patients with identical cough frequencies but this has received little attention. The aim of this study was to explore the differences in the temporal distribution of cough in 24 hour acoustic recordings across a range of respiratory conditions using the Weibull Distribution. Methods We studied 24 hour acoustic recordings made using the VitaloJAK™ cough monitoring system from 62 participants [healthy volunteers (n=11), chronic cough (n=15), asthma (n=10) chronic obstructive pulmonary disease (COPD) (n=16), Idiopathic Pulmonary Fibrosis (IPF) n=10)]. Using the exact location of the explosive phase of each cough sound, the lengths of inter-cough intervals were calculated in each recording and the fit to Weibull and exponential distributions assessed; the shape parameter which describes the ‘clustering’ of events was examined. Results The number of coughs per 24 recording ranged from 0 to 3133, median 158.0 coughs and was increased compared with healthy controls in all disease groups [healthy volunteers median 13.0(IQR 1.0–22.0) coughs/24 hour versus chronic cough 509.0 (296.0–916.0), asthma 35.0 (13.5–144.0) COPD 158.0 (41.0–791.5), Idiopathic Pulmonary Fibrosis 193.5 (101.3–542.8), all p Conclusions The intervals between cough events are not random and can be described by a Weibull distribution. Significant differences in the Weibull shape parameter suggest differences in the temporal clustering of coughs between different respiratory conditions and therefore may be an important novel parameter from objective cough monitoring.

The response variable in an experiment follows a 2-parameter Weibull distribution having a scale parameter that varies inversely with a power of a deterministic, externally controlled, variable generically termed a stress. The shape parameter is invariant with stress. A numerical scheme is given for solving a pair of nonlinear simultaneous equations for the maximum likelihood (ML) estimates of the common shape parameter and the stress-life exponent. Interval and median unbiased point estimates for the shape parameter, stress-life exponent and a specified percentile at any stress, are expressed in terms of percentage points of the sampling distributions of pivotal functions of the ML estimates. A numerical example is given.

The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

Introduction and objectives Cough frequency is known to correlate well with perceived cough severity and is commonly used as an endpoint to evaluate the efficacy of new anti-tussive agents. However, little is known about the distribution of cough and its impact on cough severity. The Weibull distribution is a probability distribution model often used to predict time between events, as applied in COPD exacerbations. The shape parameter describes if coughs are more or less likely to occur over time. We investigated if cough temporal distribution is modulated by an effective anti-tussive agent (morphine) and how this distribution influences perceived cough severity. Methods Cough recordings, made using the VITALOJAK™ cough monitoring system were analysed from 22 chronic cough patients (18 female, mean age 61.7 years, mean cough duration 14 years) who had participated in a previously reported randomised controlled trial of low dose morphine sulphate. Coughs were identified and the cough intervals recorded and fit to an exponential (random) and Weibull distribution. The goodness of fit was checked with the Akaike information criterion (AIC). The shape parameter was calculated and using generalised estimating equations compared for morphine versus placebo. Relationships between cough severity visual analogue scale (VAS), cough frequency and the shape parameter were also explored. Results The AIC indicated that cough distribution better fitted the Weibull equation than the exponential in 38/41 (93%) recordings. The Weibull shape parameter was significantly reduced by morphine compared with placebo (geometric mean 0.418 versus 0.548, p=0.04), suggesting more coughs occurred early, with a fall in the rate of subsequent coughs over time with morphine. Furthermore, cough severity VAS was significantly predicted by a combination of cough frequency (p Conclusions In chronic cough patients, low dose morphine, in addition to reducing cough frequency may also reduce the temporal clustering of cough events. This may be important as temporal clustering appears to be a component of perceived cough severity, independent of cough frequency.

Poisson point processes provide a versatile framework for modeling the distributions of random points in space. When the space is partitioned into cells, each associated with a single generating point from the Poisson process, there appears a geometric structure known as Poisson Voronoi tessellation. These tessellations find applications in various fields such as biology, material science, and communications, where the statistical properties of the Voronoi cells reveal patterns and structures that hold key insights into the underlying processes generating the observed phenomena. In this paper, we investigate a distance measure of Poisson Voronoi tessellations that is emerging in the literature, yet for which its statistical and geometrical properties remain explored only in the asymptotic case when the density of seed points approaches infinity. Our work, specifically focused on homogeneous Poisson point processes, characterizes the cumulative distribution functions governing the maximum and minimum distances between the points generating the Voronoi regions and their respective vertices for an arbitrary density of points in R 2 . For that, we conduct a Monte-Carlo type simulation with 1 0 8 Voronoi cells and fit the resulting empirical cumulative distribution functions to the Generalized Gamma, Gamma, Log-normal, Rayleigh, and Weibull distributions. Our analysis compares these fits in terms of root mean-squared error and maximum absolute variation, revealing the Generalized Gamma distribution as the best-fit model for characterizing these distances in homogeneous Poisson Voronoi tessellations. Furthermore, we provide the maximum likelihood estimates and the 95% confidence intervals of the parameters of the Generalized Gamma distribution along with the algorithm implemented to calculate the maximum and minimum distances. • Generator-vertex distances studied in finite Poisson Voronoi tessellations. • Empirical CDFs of max/min distances obtained via Monte Carlo simulation. • Generalized Gamma provides best fit across multiple error metrics • Maximum likelihood estimates and confidence intervals provided for all parameters.