Nandini Kannan

On the hazard function of Birnbaum-Saunders distribution and associated inference

In this paper, we discuss the shape of the hazard function of Birnbaum-Saunders distribution. Specifically, we establish that the hazard function of Birnbaum-Saunders distribution is an upside down function for all values of the shape parameter. In reliability and survival analysis, as it is often of interest to determine the point at which the hazard function reaches its maximum, we propose different estimators of that point and evaluate their performance using Monte Carlo simulations. Next, we analyze a data set and illustrate all the inferential methods developed here and finally make some concluding remarks.

Point and Interval Estimation for a Simple Step-Stress Model with Type-II Censoring

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress levels, such as temperature, voltage, and load, on the lifetimes of experimental units. Accelerated testing allows the experimenter to increase these stress levels to obtain information on the parameters of the life distributions more quickly than would be possible under normal operating conditions. A special class of accelerated tests are step-stress tests that allow the experimenter to increase the stress levels at fixed times during the experiment. In this article, we consider the simple step-stress model under Type-II censoring. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment-generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions, and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study.

Point and interval estimation for Gaussian distribution, based on progressively Type-II censored samples

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.

Analysis of Progressively Censored Competing Risks Data

Publisher Summary This chapter develops inference for a competing-risk model under a very general censoring scheme. Censoring is inevitable in life-testing and reliability studies because the experimenter is unable to obtain complete information on lifetimes for all individuals. For example, patients in a clinical trial may withdraw from the study, or the study may have to be terminated at a pre-fixed timepoint. In industrial experiments, units may break accidentally. In many situations, however, the removal of units prior to failure is pre-planned to provide savings in terms of time and cost associated with testing. The two most common censoring schemes are termed “type I” and “type II” censoring. The chapter considers competing risk data under progressive type II censoring and focuses on the analysis of the competing risk model when the data are progressively type II censored. The estimation of the different parameters is considered and their distributions are presented. Bayesian analysis of the competing-risk model is discussed in the chapter. Results of a simulation study comparing the coverage probabilities and lengths of the different confidence and credible intervals are also provided and the performance of these different techniques using a real dataset are illustrated.

Goodness-of-fit tests based on spacings for progressively type-II censored data from a general location-scale distribution

There has been extensive research on goodness-of-fit procedures for testing whether or not a sample comes from a specified distribution. These goodness-of-fit tests range from graphical techniques, to tests which exploit characterization results for the specified underlying model. In this article, we propose a goodness-of-fit test for the location-scale family based on progressively Type-II censored data. The test statistic is based on sample spacings, and generalizes a test procedure proposed by Tiku . The null distribution of the test statistic is shown to be approximated closely by a s-normal distribution. However, in certain situations it would be better to use simulated critical values instead of the s-normal approximation. We examine the performance of this test for the s-normal and extreme-value (Gumbel) models against different alternatives through Monte Carlo simulations. We also discuss two methods of power approximation based on s-normality, and compare the results with those obtained by simulation. Results of the simulation study for a wide range of sample sizes, censoring schemes, and different alternatives reveal that the proposed test has good power properties in detecting departures from the s-normal and Gumbel distributions. Finally, we illustrate the method proposed here using real data from a life-testing experiment. It is important to mention here that this test can be extended to multi-sample situations in a manner similar to that of Balakrishnan et al.

Analysis of lognormal survival data.

The failure rate and the mean residual life function (MRLF) of a lognormal distribution are known to be nonmonotonic. It is of interest to study the point at which the monotonicity changes (the change point). In this article we study the change points of the failure rate and the MRLF for the lognormal distribution. It is shown that the change points are the solutions of certain nonlinear equations. We apply these results to estimate the change points for survival data on guinea pigs given by Bjerkedal. The standard deviation of the estimate is obtained using bootstrap and jackknife methods. Finally confidence bands for the failure rate and the MRLF are also provided to illustrate the behavior of the estimates.

Estimating parameters in the damped exponential model

Abstract In this paper we consider the problem of estimation of the frequencies and damping factors of exponential signals in the presence of noise. We propose a non-iterative method based on using forward–backward linear prediction and the notion of extended order modeling. In addition to providing estimators of the unknown parameters in the model, the proposed method can be used to specify initial values in any standard minimization algorithm to obtain the least-squares estimators. For the undamped exponential model, it is well known that any estimator is inconsistent under the usual definition of consistency. We redefine the model so that the sampling interval is finite, and prove the consistency and asymptotic normality of the least-squares estimators under this new assumption. It is observed that the dispersion matrix of the least-squares estimators attains the Cramer–Rao bound.

On modified EVLP and ML methods for estimating superimposed exponential signals

Abstract We consider the estimation procedure of the multiple sinusoidal model for signals, when the damping factor is not present. The solution in the general case depends on the roots of a polynomial, whose coefficients are estimated from the observed data. When the damping factor is absent, the coefficients exhibit a certain symmetry. It reduces in estimating almost half of the total number of unknown parameters. Under these symmetric constraints modified methods have been developed to estimate the coefficients. It is observed that the standard errors for the modified methods are closer to the Cramer-Rao lower bound than before in almost all the situations. It is also observed that the computational cost of the modified maximum likelihood method is lower than the ordinary one. The modified maximum likelihood estimates can be obtained by an iterative process. Theoretical justification has been provided for the convergence of the iterative process.

The generalized exponential cure rate model with covariates

In this article, we consider a parametric survival model that is appropriate when the population of interest contains long-term survivors or immunes. The model referred to as the cure rate model was introduced by Boag 1 in terms of a mixture model that included a component representing the proportion of immunes and a distribution representing the life times of the susceptible population. We propose a cure rate model based on the generalized exponential distribution that incorporates the effects of risk factors or covariates on the probability of an individual being a long-time survivor. Maximum likelihood estimators of the model parameters are obtained using the the expectation-maximisation (EM) algorithm. A graphical method is also provided for assessing the goodness-of-fit of the model. We present an example to illustrate the fit of this model to data that examines the effects of different risk factors on relapse time for drug addicts.

A Test of Exponentiality Based on Spacings for Progressively Type-II Censored Data

There have been numerous tests proposed in the literature to determine whether or not an exponential model is appropriate for a given data set. These procedures range from graphical techniques, to tests that exploit characterization results for the exponential distribution. In this article, we propose a goodness-of-fit test for the exponential distribution based on general progressively Type-II censored data. This test based on spacings generalizes a test proposed by Tiku (1980). We derive the exact and asymptotic null distribution of the test statistic. The results of a simulation study of the power under several different alternatives like the Weibull, Lomax, Lognormal and Gamma distributions are presented. We also discuss an approximation to the power based on normality and compare the results with those obtained by simulation. A wide range of sample sized and progressive censoring schemes have been considered for the empirical study. We also compare the performance of this procedure with two standard tests for exponentiality, viz. the Cramer-von Mises and the Shapiro-Wilk test. The results are illustrated on some real data for the one-and two-parameter exponential models. Finally, some extensions to the multi-sample case are suggested.