Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.     

On the bayes risk incurred by using asymptotic shapes

Let X₁ X₂…denote Independent and Identically distributed random vectors whose common distributions form a multiparameter exponential family, and consider the problem of sequentially testing separated hypotheses. It is known that the sequential procedure which continues sampling until the likelihood ratio statistic for testing one of the hypotheses exceeds a given level approximates the optimal Bayesian procedure, under general conditions on the loss function and prior distribution. Here we ask whether the approximate procedure is Bayes risk efficient--that is, whether the ratio of the Bayes risk of the approximate procedure to the Bayes risk of the optimal procedure approaches one as the cost of samping approaches zero. We show that the answer depends on the choice of certain parameters in the approximation and the dimensions of the hypotheses.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

Selecting logistic populations using the sample medians

This paper is concerned primarily with subset selection procedures based on the sample mediansof logistic populations. A procedure is given which chooses a nonempty subset from among kindependent logistic populations, having a common known variance, so that the populations with thelargest location parameter is contained in the subset with a pre‐specified probability. Theconstants required to apply the median procedure with small sample sizes (≤= 19) are tabulated and can also be used to construct simultaneous confidence intervals. Asymptotic formulae are provided for application with larger sample sizes. It is shown that, under certain situations, rules based on the median are substantially more efficient than analogous procedures based either on sample means or on the sum of joint ranks.     

A monte carlo study of robust estimators of location

Andrews et al (1972) carried out an extensive Monte Carlo study of robust estimators of location. Their conclusions were that the hampel and the skipped estimates, as classes, seemed to be preferable to some of the other currently fashionable estimators. The present study extends this work to include estimators not previously examined. The estimators are compared over short-tailed as well as long-tailed alternatives and also over some dependent data generated by first-order autoregressive schemes. The conclusions of the present study are threefold. First, from our limited study, none of the so-called robust estimators are very efficient over short-tailed situations. More work seems to be necessary in this situation. Second, none of the estimators perform very well in dependent data situations, particularly when the correlation is large and positive. This seems to be a rather pressing problem. Finally, for long-tailed alternatives, the hampel estimators and Hogg-type adaptive versions of the hampels are the strongest classes. The adaptive hampels neither uniformly outperform nor are they outperformed by the hampels. However, the superiority in terms of maximum relative efficiency goes to the adaptive hampels. That is, the adaptive hampels, under their worst performance.     

Behrens-fisher problem under the assumption of homogeneous coefficients of variation

The powers of the likelihood ratio (LR) test and an “asymptotically (in some sense) optimum” invariant test are examined and compared by simulation techniques with those of several other relevant tests for the problem of testing the equality of two univariate normal population means under the assumption of heterogeneous variances but homogeneous coefficients of variation. It is seen that the LR test is highly satisfactory for all values of the coefficient of variation and the “asymptotically optimum” invariant test, which is computationally much simpler than the LR test, is a reasonably good competitor for cases where the value of the coefficient of variation is greater than or equal to 3. Also, a     

Designing completely randomized experiments to detect ordered treatment effects: an empirical study

In designing experiments the researcher frequently must decide as to how to allocate fixed resources among k factor levels (Cox (1958)). This study investigates the effects on the power of a test caused by changes in: the sample size (n); the number of factor levels (k); the allocation of fixed total observations (N) among k and n: the shift parameter (ø); the type of parent population sampled; and, the type of ordered location alternative involved. Using Monte Carlo methods the powers of eight test procedures specifically devised to detect ordered treatment effects under completely randomized designs were evaluated along with those of the more general one-way F test. The results are of interest to researchers in all fields of application.     

On the sample size of curtailed tests

Formulas are given for the asymptotic distribution, mean, and variance of m^-1N_m,where NN_m is the random sample size of the curtailed version of a fixed-sample most powerful test based on sample size m. The adequacy of the formulas is numerically investigated in some important applications where exact formulas can also be derived     

An extensive power evaluation of some tests for the inverse Gaussian distribution

Many applications of the Inverse Gaussian distribution, including numerous reliability and life testing results are presented in statistical literature. The paper studies the problem of using entropy tests to examine the goodness of fit of an Inverse Gaussian distribution with unknown parameters. Some entropy tests based on different entropy estimates are proposed. Critical values of the test statistics for various sample sizes are obtained by Monte Carlo simulations. Type I error of the tests is investigated and then power values of the tests are compared with the competing tests against various alternatives. Finally, recommendations for the application of the tests in practice are presented.     

Variable screening for survival data in the presence of heterogeneous censoring

Variable screening for censored survival data is most challenging when both survival and censoring times are correlated with an ultrahigh-dimensional vector of covariates. Existing approaches to handling censoring often make use of inverse probability weighting by assuming independent censoring with both survival time and covariates. This is a convenient but rather restrictive assumption which may be unmet in real applications, especially when the censoring mechanism is complex and the number of covariates is large. To accommodate heterogeneous (covariate-dependent) censoring that is often present in high-dimensional survival data, we propose a Gehan-type rank screening method to select features that are relevant to the survival time. The method is invariant to monotone transformations of the response and of the predictors, and works robustly for a general class of survival models. We establish the sure screening property of the proposed methodology. Simulation studies and a lymphoma data analysis demonstrate its favorable performance and practical utility.