Estimation of critical points in the mixture inverse Gaussian model

The maximum likelihood estimation for the critical points of the failure rate and the mean residual life function are presented in the case of mixture inverse Gaussian model. Several important data sets are analyzed from this point of view. For each of the data sets, Bootstrapping is used to construct confidence intervals of the critical points.     

On prediction from the location-scale model with a compound error distribution

The prediction distribution of future response(s) given a set of data from a location-scale model with a compound error distribution has been derived by utilizing the structural relations of the model. The compound error distribution has been specialized to cover the case of multivariate t-distribution.     

A sequential sampling plan for the inverse gaussian mean

The inverse gaussian distribution is a flexible model which has been extensively applied in the theory of generalized linear models and accelerated life testing where early failure times predominate. More recently it has received attention in areas such as quality control, and as an underlying model that provides an alternative to the analysis of variance. In reliability testing and acceptance sampling data acquisition is often in the face of scarce resources and may be both costly and time-consuming. In such settings it is desirable to reach a statistically sound decision as quickly as possible. Based on sequential probability ratio tests (SPRT), sequential sampling plans provide one method of arriving at a timely, statistically based decision. A sequential sampling plan for the inverse gaussian process mean when the value of the shape parameter of the density is known is presented in this paper.     

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).     

Inference for a simultaneous linear partially explosive model with polynomial regression components of different degrees

This paper is concerned with the estimation of the coefficients of simultaneous partially explosive model with polynomial regression components of different degrees in its equations. Since the least squares method breaks down in this case, a three stage estimation procedure is suggested for obtaining CAN estimates of the coefficients.     

Least squares estimation of the coefficients of a partially explosive model,with polynomial regressions of same degree,generating a pair of related time series

This paper is concerned with a partially explosive linear model with polynomial regression components generating a pair of related time series. The least squares estimates of the coefficients are shown to be √N-consistent and asymptotically singular normal, when the degrees of polynomial regression components are same, thus generalising a result due to Venkataraman (1974).     

On the estimation of coefficients of a simultaneous linear explosive model of higher orders with moving average errors generating a pair of time series

In this paper, estimation of coefficients of simultaneous linear partially explosive model of higher orders with moving average errors is considered. It has been shown that the above model can be decomposed into a purely explosive model and an autoregressive model. A two stage estimation, procedure is carried out towards proposing estimators for the partially explosive model. The asymptotic properties of these estimators are also studied.     

On confidence intervals for nonmonotone parametric functions and an application to the squared mean of the normal distribution

Consider the problem of obtaining a confidence interval for some function g(θ) of an unknown parameter θ, for which a (1-α)-confidence interval is given. If g(θ) is one-to-one the solution is immediate. However, if g is not one-to-one the problem is more complex and depends on the structure of g. In this note the situation where g is a nonmonotone convex function is considered. Based on some inequality, a confidence interval for g(θ) with confidence level at least 1-α is obtained from the given (1-α) confidence interval on θ. Such a result is then applied to the n(μ, σ ²) distribution with σ known. It is shown that the coverage probability of the resulting confidence interval, while being greater than 1-α, has in addition an upper bound which does not exceed Θ(3z_1−α/2)-α/2.     

On the accuracy of approximate studentization

With a parametric model, a measure of departure for an interest parameter is often easily constructed but frequently depends in distribution on nuisance parameters; the elimination of such nuisance parameter effects is a central problem of statistical inference. Fraser & Wong (1993) proposed a nuisance-averaging or approximate Studentization method for eliminating the nuisance parameter effects. They showed that, for many standard problems where an exact answer is available, the averaging method reproduces the exact answer. Also they showed that, if the exact answer is unavailable, as say in the gamma-mean problem, the averaging method provides a simple approximation which is very close to that obtained from third order asymptotic theory. The general asymptotic accuracy, however, of the method has not been examined. In this paper, we show in a general asymptotic context that the averaging method is asymptotically a second order procedure for eliminating the effects of nuisance parameters.     

Estimation of system reliability

In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common stress which is independent of the strengths of these k components. If (X ₁,X ₂,…,X _k ) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability is given byR=P[Y<X _(k−s+1)] whereX _(k−s+1) is (k−s+1)-th order statistic of (X ₁,…,X _k ). We estimate R when (X ₁,…,X _k ) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution of the proposed estimator.     

Estimation of the signal-to-noise in the linear regression model

In the present paper estimators of the signal-to-noise are given. A simulation study is conducted in order to see how the proposed estimators perform relative to the naive estimator by way of scalar risk comparison. The results favour our suggested estimators.     

Admissibility of estimators in the non-regular family under entropy loss function

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities. The admissibility of generalized Bayes and Pitman estimates of functions of parameters have been established under entropy loss function.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

Testing whether the survival function is multivariate new better than used

A test is proposed to test that a life distribution is multivariate exponential (MVE) against the alternative that it is multivariate new better than used (MNBU) class of alternatives. We also show that the proposed test is consistent for the alternatives of multivariate new better than used in expectations (MNBUE).     

Diagnostics in multivariate generalized Birnbaum-Saunders regression models

Birnbaum–Saunders (BS) models are receiving considerable attention in the literature. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this paper, we formulate multivariate generalized BS regression models and carry out a diagnostic analysis for these models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. We also consider the local influence approach and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate data to show their potential applications.     

A note on a result for two SUR models

Revankar (1974, p. 190, equation (4.4)) obtains a result for the covariance matrices of the “Aitken” estimators of the regression coefficients parameter matrices of two SUR models. The present note supplies a simpler derivation of this result. It is obtained by using a known result in multivariate statistical analysis, see e.g., Sarkar (1981, p. 560, Theorem 3.1).     

A generalization and Bayesian interpretation of ridge-type estimators with good prior means

Pliskin (1987) and Trenkler (1988) compared ridge-type estimators with good prior means. From a Bayesian viewpoint, these estimators are special cases of Bayesestimators and the mean square error matrix comparisons can be made in the more general case.     

A relation for the covariances of order statistics from n independent and non-identically distributed random variables

We derive a simple relation satisfied by the covariances of order statistics in the i.i.d. case and then generalize it to the case when the variables are independent and non-identically distributed. This relation could be employed successfully either to check the calculations or to reduce the amount of direct computations involved in evaluating the covariances of order statistics from an outlier model.     

Higher order moments of order statistics from INID exponential random variables

In this paper we develop recurrence relations for the third and fourth order moments of order statistics from I.NI.D exponential random variables. Recurrence relations for the p-outlier model (with a slippage of p observations) are derived as a special case. Applications of these results will also be described.