Jie Mi

Existence and uniqueness of the MLEs for normal distribution based on general progressively Type-II censored samples

In this paper, we prove that the MLEs of the parameters [mu] and [sigma] of a normal population, based on general progressively Type-II censored samples, exist and are unique.

Shape and crossing properties of mean residual life functions

In addition to the CDF or density function, the failure rate and mean residual life functions are the other important functions that can be used to characterize a lifetime. In the literature, much work on the properties of these quantities has been done. This paper points out and corrects some errors in the literature by giving a counterexample and new proofs. Moreover, it gives a more complete discussion on the shape of the mean residual life function when the associated failure rate function has a roller-coaster shape. It also discusses the crossing properties of two mean residual life functions in detail, when the associated failure rate functions have several crossing points.

Age-smooth properties of mixture models

The tail behavior or age-smoothness of a lifetime distribution is an important property from both the theoretical and practical points of view. There are several ways of describing tail behavior. One is to use the failure rate, another uses the mean residual life, and a third uses the conditional survival probability. This article discusses whether a mixture of distribution functions reflects the age-smoothness property of the components making up the mixture for each of the above three type descriptors of tail behavior. Using the criteria of tail behavior, the article further defines strong and weak subpopulations, and shows that the strong subpopulation tail behavior will asymptotically dominate the mixture population and hence the reliability quality of mixture population can be improved by applying burn-in.

A comparison of the Bonferroni and Scheffé bounds

Abstract For the general linear model there are many ways to obtain simultaneous confidence intervals. Among them are the Bonferroni, Scheffe and several Sidak-type procedures. The comparisons of these procedures have been extensively studied. However, previously the comparison between the Bonferroni and Scheffe procedures has only been based on numerical inspection or on a comparison with infinite error degrees of freedom. In this paper, we show surprisingly that for one or two degrees of freedom for error. Scheffes procedure is better, that is, produces shorter intervals, than the Bonferronis procedure. For more than two degrees of freedom for error. Bonferroni is better than Scheffe for standard levels of confidence, but the reverse can be true for nonstandard confidence levels.

Order-preserving property of maximum likelihood estimator

Various properties of the maximum likelihood estimator have been studied. However, the order preserving property of MLE has not been studied in the literature. This is precisely the subject matter of this paper. General sufficient conditions are derived under which the MLE possesses the property of preserving the usual stochastic order. In particular, sufficient conditions are obtained for exponential family, location family, scale family, and failure rate models. Examples are finally presented to illustrate the results obtained in this paper.

Inferences about the scale parameter of the gamma distribution based on data mixed from censoring and grouping

This paper studies the MLE of the scale parameter of the gamma distribution based on data mixed from censoring and grouping when the shape parameter is known. The study shows that under Type I mixed data, the MLE of the scale parameter exists, is unique, and converges almost surely to the true value provided the number of items that fail in the last interval is less than the total number of items placed on test. Under Type II mixed data, these properties hold unconditionally. The relationship between the MLEs based on mixed data and censored data is also examined. An upper bound on the MLE under both Type I and II mixed data is derived to simplify the search for the MLE.

Crossing properties of F distributions

The crossing properties of F distributions are discussed. The results show that the F distribution stochastically decreases in its denominator degrees of freedom if the numerator degrees of freedom m is either one or two, and the difference of two F distribution functions with the same numerator degrees of freedom m [greater-or-equal, slanted] 3 has number of sign changes of one. It is also shown that the upper percentile of Student t distribution decreases in its degrees of freedom.