Malwane M. A. Ananda

A Generalization of the Half-Normal Distribution with Applications to Lifetime Data

A two-parameter family of lifetime distribution which is derived from a model for static fatigue is presented. This derivation follows from considerations of the relationship between static fatigue crack extension and the failure time of a certain specimen. The cumulative distribution function (cdf) of this new family is quite similar to the cdf of the half-normal distribution, and therefore this density is referred to as the generalized half-normal distribution (GHN). Furthermore, this GHN family is a special case of the three-parameter generalized gamma distribution. Even though the GHN distribution is a two-parameter distribution, the hazard rate function can form variety of shapes such as monotonically increasing, monotonically decreasing, and bathtub shapes. Some properties of this family are given, and examples are cited to compare with other commonly used failure time distributions such as Weibull, gamma, lognormal, and Birnbaum-Saunders.

Modeling actuarial data with a composite lognormal-Pareto model

The actuarial and insurance industries frequently use the lognormal and the Pareto distributions to model their payments data. These types of payment data are typically very highly positively skewed. Pareto model with a longer and thicker upper tail is used to model the larger loss data, while the larger data with lower frequencies as well as smaller data with higher frequencies are usually modeled by the lognormal distribution. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to its monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, we were motivated to look for a new avenue to remedy the situation. Here we introduce a two-parameter smooth continuous composite lognormal-Pareto model that is a two-parameter lognormal density up to an unknown threshold value and a two-parameter Pareto density for the remainder. The resulting two-parameter smooth density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation techniques and properties of this new composite lognormal-Pareto model are discussed and we compare its performance with the other commonly used models. A simulated example and a well-known fire insurance data set are analyzed to show the importance and applicability of this newly proposed composite lognormal-Pareto model.

Risk behavior of variance estimators in multivariate normal distribution

In this paper we consider the estimation problem of unknown variance of a multivariate normal vector under quadratic loss and entropy loss. The behavior of risk functions of the Brewster--Zidek estimator and the original Stein estimator is examined. Numerical studies show that an asymptotically inadmissible Stein estimator provides a larger degree of improvement than an admissible Brewster--Zidek estimator.

Confidence intervals for steady state availability of a system with exponential operating time and lognormal repair time

Long-run availability of a system assuming lognormal repair times and exponential failure times is considered. With many repairable systems, the time to repair is well characterized by a lognormal distribution. However, due to the difficulties associated with the lognormal distribution, repair times are often modeled using the lognormal distribution assuming that the variance of the lognormal distribution is known or by using some other distribution such as the exponential distribution instead of the lognormal distribution. In this paper using the generalized p-value approach, we propose confidence intervals and exact tests for steady state availability of systems using the two-parameter lognormal distribution for repair time and exponential distribution for operating time. Here, both parameters of the lognormal distribution are assumed to be unknown. A couple of examples are given to illustrate the proposed procedures. A simulation study is given to demonstrate the performance of the proposed procedure. Results are extended for the long-run availability of a system consisting of several independent parallel sub-systems.

On steady state availability of a system with lognormal repair time

Assuming two-parameter lognormal distribution for repair times, statistical inference for the steady state availability of a system is considered. For the failure time distribution, weibull, gamma, and lognormal distributions were considered. Using the generalized p-value approach, we propose confidence intervals and exact tests for the steady state availability of a system. A couple of examples are given to illustrate the proposed procedures.

PERFORMANCE OF TWO-WAY ANOVA PROCEDURES WHEN CELL FREQUENCIES AND VARIANCES ARE UNEQUAL

A fixed effect two-way ANOVA model with unequal cell frequencies and unequal error variances is considered. Under the Neyman-Pearson theory, exact tests for testing the interaction effect and main effect do not exist for this problem. When variances are unequal, classical F-tests which are calculated under the equal error variance assumption will provide only approximate solutions. For testing the interaction effect, we compare the performance of the generalized F-test and the classical F-test. Generalized F-test (generalized p-value) is a recently developed exact test which is based on an extended definition of the p-values. Using simulation, size, power and robustness comparisons are made. According to the simulation study, when heteroscedasticity is present under the normality, the size of the generalized F-test does not exceed the intended level allthough the size of the classical F-test exceeds the intended level. When the size is adjusted at the same level, for all the studied cases, the power of the generalized test is nearly equal or better than the power of the classical F-test. Both procedures are quite robust in terms of the size when heteroscedasticity is present under non-normality. Gamma distribution was used for non-normal comparisons. When the size is adjusted, for all the considered cases, the power of the generalized F-test is as good or better than the power of the classical F-test.

Adaptive Bayes estimators for parameters of the Gompertz survival model

The two-parameter Gompertz model is a commonly used survival time distribution in actuarial science and reliability and life testing. The estimation of the parameters of this model is numerically involved. We consider the estimation problem in a Bayesian framework and give the Bayesian estimators of parameters in terms of single numerical integrations. We propose an adaptive Bayesian estimation procedure by putting a prior only on one parameter and finding the other parameter by minimizing the distance between empirical and parametric cumulative distribution functions. This easily computable (even for large samples) adaptive Bayesian procedure is compatible with the exact Bayesian procedure. In particular, numerical integration for computing the exact Bayesian procedure is difficult for large samples. Furthermore, for the no prior information situation, a noninformative adaptive Bayes procedure is given. Some examples of the proposed adaptive method along with a comparison with other existing methods are given. Monte Carlo simulation has been used to compare the existing procedures with the proposed procedures.

Estimation and testing of availability of a parallel system with exponential failure and repair times

In this paper we consider the long-run availability of a parallel system having several independent renewable components with exponentially distributed failure and repair times. We are interested in testing availability of the system or constructing a lower confidence bound for the availability by using component test data. For this problem, there is no exact test or confidence bound available and only approximate methods are available in the literature. Using the generalized p-value approach, an exact test and a generalized confidence interval are given. An example is given to illustrate the proposed procedures. A simulation study is given to demonstrate their advantages over the other available approximate procedures. Based on type I and type II error rates, the simulation study shows that the generalized procedures outperform the other available methods.

GeneralizedF-tests for unbalanced nested designs under heteroscedasticity

Two-factor fixed-effect unbalanced nested design model without the assumption of equal error variance is considered. Using the generalized definition ofp-values, exact tests under heteroscedasticity are derived for testing “main effects” of both factors. These generalizedF-tests can be utilized in significance testing or in fixed level testing under the Neyman-Pearson theory. Two examples are given to illustrate the proposed test and to demonstrate its advantages over the classicalF-test. Extensions of the procedure for three-factor nested designs are briefly discussed.

Testing the difference of two exponential means using generalized p-values

There are no exact fixed-level tests for testing the null hypothesis that the difference of two exponential means is less than or equal to a prespecified value θ0. For this testing problem, there are several approximate testing procedures available in the literature. Using an extended definition of p-values, Tsui and Weerahandi (1989) gave an exact significance test for this testing problem. In this paper, the performance of that procedure is investigated and is compared with approximate procedures. A size and power comparison is carried out using a simulation study. Its findings show that the test based on the generalized p-value guarantees the intended size and that it is either as good as or outperforms approximate procedures available in the literature, both in power and in size.