Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

An Autoregressive Model for Time Series of Circular Data

This article focuses on estimating an autoregressive regression model for circular time series data. Simulation studies have shown the difficulties involved in obtaining good estimates from low concentration data or from small samples. It presents an application using real data.     

Inference for a Simple Step-Stress Model with Type-I Censoring and Lognormally Distributed Lifetimes

Accelerated life-testing (ALT) is a very useful technique for examining the reliability of highly reliable products. It allows the experimenter to obtain failure data more quickly at increased stress levels than under normal operating conditions. A step-stress model is one special class of ALT, and in this article we consider a simple step-stress model under the cumulative exposure model with lognormally distributed lifetimes in the presence of Type-I censoring. We then discuss inferential methods for the unknown parameters of the model by the maximum likelihood estimation method. Some numerical methods, such as the Newton–Raphson and quasi-Newton methods, are discussed for solving the corresponding non-linear likelihood equations. Next, we discuss the construction of confidence intervals for the unknown parameters based on (i) the asymptotic normality of the maximum likelihood estimators (MLEs), and (ii) parametric bootstrap resampling technique. A Monte Carlo simulation study is carried out to examine the performance of these methods of inference. Finally, a numerical example is presented in order to illustrate all the methods of inference developed here.     

Dependent Insurance Risk Model: Deterministic Threshold

This article considers a dependent insurance risk model. We assume that the inter-arrival time depends on the previous claim size through a deterministic threshold structure. Adjustment coefficient and Lundberg-type upper bound for the ruin probability are obtained. In case of exponential claim size, an explicit solution for the ruin probability is obtained by solving a system of ordinary delay differential equations. Some numerical results are included for illustration purposes.     

Linear Subspace Spanned by Principal Points of a Mixture of Spherically Symmetric Distributions

For each positive integer k, a set of k-principal points of a distribution is the set of k points that optimally represent the distribution in terms of mean squared distance. However, explicit form of k-principal points is often difficult to obtain. Hence a theorem established by Tarpey et al. (1995 Tarpey , T. , Li , L. , Flury , B. D. ( 1995 ). Principal points and self-consistent points of elliptical distributions . Ann. Statist. 23 : 102 – 112 .[Crossref], [Web of Science ®] , [Google Scholar]) has been influential in the literature, which states that when the distribution is elliptically symmetric, any set of k-principal points is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem is called a “principal subspace theorem”. Recently, Yamamoto and Shinozaki (2000b Yamamoto , W. , Shinozaki , N. ( 2000b ). Two principal points for multivariate location mixtures of spherically symmetric distributions . J. Japan Statist. Soc. 30 : 53 – 63 .[Crossref] , [Google Scholar]) derived a principal subspace theorem for 2-principal points of a location mixture of spherically symmetric distributions. In their article, the ratio of mixture was set to be equal. This article derives a further result by considering a location mixture with unequal mixture ratio.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

Interval Estimation of the Stress-Strength Reliability with Independent Normal Random Variables

This article develops a procedure to obtain highly accurate confidence interval estimates for the stress-strength reliability R = P(X > Y) where X and Y are data from independent normal distributions of unknown means and variances. Our method is based on third-order likelihood analysis and is compared to the conventional first-order likelihood ratio procedure as well as the approximate methods of Reiser and Guttman (1986 Reiser, B., Guttman, I. (1986). Statistical inference for Pr(Y < X): the normal case. Technometrics 28: 253257.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Guo and Krishnamoorthy (2004 Guo, H., Krishnamoorthy, K. (2004). New approximate inferential methods for the reliability parameter in a stress-strength model: the normal case. Commun. Statist. Theor. Meth. 33: 17151731.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The use of our proposed method is illustrated by an empirical example and its superior accuracy in terms of coverage probability and error rate are examined through Monte Carlo simulation studies.     

Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions

In this article, we introduce new asymptotic expansions for probability functions of sums of independent and identically distributed random variables. Results are obtained by efficiently employing information provided by lower-order convolutions. In comparison with Edgeworth-type theorems, advantages include improved asymptotic results in the case of symmetric random variables and ease of computation of main error terms and asymptotic crossing points. The first-order estimate can perform quite well against the corresponding renormalized saddlepoint approximation and, pointwise, requires evaluation of only a single convolution integral. While the new expansions are fairly straightforward, the implications are fortuitous and may spur further related work.