A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

A New Control Scheme Always Better Than X-Bar Chart

A new control scheme is proposed by borrowing the idea of the Benjamini–Hochberg procedure for controlling the false discovery rate in multiple testing. It is shown theoretically that the proposed 2-span control scheme outperforms the Shewhart X-bar chart in terms of the average run length under any size of mean shifts. Some simulations are carried out to demonstrate that the proposed scheme having various span sizes always outperforms the X-bar chart in terms of the average run lengths.     

A Simple Approximate Procedure for Constructing Binomial and Poisson Tolerance Intervals

The problems of constructing tolerance intervals for the binomial and Poisson distributions are considered. Closed-form approximate equal-tailed tolerance intervals (that control percentages in both tails) are proposed for both distributions. Exact coverage probabilities and expected widths are evaluated for the proposed equal-tailed tolerance intervals and the existing intervals. Furthermore, an adjustment to the nominal confidence level is suggested so that an equal-tailed tolerance interval can be used as a tolerance interval which includes a specified proportion of the population, but does not necessarily control percentages in both tails. Comparison of such coverage-adjusted tolerance intervals with respect to coverage probabilities and expected widths indicates that the closed-form approximate tolerance intervals are comparable with others, and less conservative, with minimum coverage probabilities close to the nominal level in most cases. The approximate tolerance intervals are simple and easy to compute using a calculator, and they can be recommended for practical applications. The methods are illustrated using two practical examples.     

Adjusted Hazard Rate Estimator Based on a Known Censoring Probability

In most reliability studies involving censoring, one assumes that censoring probabilities are unknown. We derive a nonparametric estimator for the survival function when information regarding censoring frequency is available. The estimator is constructed by adjusting the Nelson–Aalen estimator to incorporate censoring information. Our results indicate significant improvements can be achieved if available information regarding censoring is used. We compare this model to the Koziol–Green model, which is also based on a form of proportional hazards for the lifetime and censoring distributions. Two examples of survival data help to illustrate the differences in the estimation techniques.     

Proposition of a General Formula for Price Indices

In this article, we propose a general formula for aggregative price indices that satisfies most postulates coming from the axiomatic price index theory. We show that the ideal Fisher index, Laspeyres and Paasche formulas, and a lot of other indices are particular cases of the proposed formula. Moreover, using the general formula we can easily define new indices, that would satisfy given postulates. We also present an interesting result for the proposed formula.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

Reliability Properties of Systems with Two Exchangeable Log-Logistic Components

In this article, we study the reliability properties of systems under bivariate log-logistic model which comes out from a particular stress-strength analysis. For this model, we obtain basic reliability characteristics of series and parallel systems and investigate their properties. We also derive distribution and moments of cold standby system under the abovementioned exchangeable model.     

On Sharp Jensen's Inequality and Some Unusual Applications

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.     

Lower and Upper Bounds for Rectangular Probabilities

This article presents some results showing how rectangular probabilities can be studied using copula theory. These results lead us to develop new lower and upper bounds for rectangular probabilities which can be computed efficiently. The new bounds are compared with the ones obtained from the generalized Fréchet–Hoeffding bounds and Bonferroni-type inequalities.