Steven G. From

Estimation of the Parameters of the Birnbaum–Saunders Distribution

Some alternative estimators to the maximum likelihood estimators of the two parameters of the Birnbaum–Saunders distribution are proposed. Most have high efficiencies as measured by root mean square error and are robust to departure from the model as well as to outliers. In addition, the proposed estimators are easy to compute. Both complete and right-censored data are discussed. Simulation studies are provided to compare the performance of the estimators.

SOME NEW APPROXIMATIONS FOR THE RENEWAL FUNCTION

In this paper, an approximation for computing the (renewal function) solution of renewal-type integral equations is presented. This approximation is a modified rational function near the origin and switches to an asymptotic linear function for larger values of the time variable. This approach is similar to that of Garg and Kalagnanam (IEEE Trans. Reliability 1998, 47 (1), 66–72) but differs near the origin and has a different switch-over-point in general. Examples are presented for the truncated normal and Weibull distributions. Some alternative models are also discussed.

A new goodness of fit test for the equality of multinomial cell probabilities versus trend alternatives

In this paper, a new test statistic is presented for testing the null hypothesis of equal multinomial cell probabilities versus various trend alternatives. Exact asymptotic critical values are obtained, The power of the test is compared with several other statistics considered by Choulakian et al (1995), The test is shown to have better power for certain trend alternatives.

The Effect of Optional Retesting on College Students' Achievement in an Individualized Algebra Course

The sequential nature of mathematics requires that students master each set of concepts before proceeding to the next. Retesting is a strategy for motivating students to relearn to a mastery level concepts and procedures not mastered initially. To test the effectiveness of this strategy, the authors allowed college students in an intermediate algebra course 1 retest of every test on which they had earned a grade less than B. A significant increase in performance between the initial test and retest was found for approximately 90% of the students who had earned a grade less than B on the initial test. Within each group of students eligible to retake a certain number of unit tests, the number of unit tests that they actually retook was correlated with their grade on the final course examination. None of the correlations were statistically significant. Thus, optional retesting appears to affect initial mastery, but not cumulative mastery.

A beta model for estimating the testability and coverage distributions of a VLSI circuit

A relation between fault coverage and testability is employed to predict population coverage. The testability profile is modeled as a mixture of a discrete impulse function and a continuous beta distribution. The parameters of the modeled distribution are estimated from fault coverage data obtained on a sample of faults. The beta distribution is chosen due to its flexible nature. The computed values of the beta parameters are dependent on the distribution of input vectors. Experimental results on three of the large ISCAS-89 circuits reflect the accuracy of the estimated fault coverage. Applications of the presented work include test generation by fault sampling, testability estimation, and test length predictions. >

Approximating the distribution of a renewal process using generalized Poisson distributions

The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.

Confidence Intervals for Gini's Diversity Measure and Shannon's Entropy Using Adjusted Proportions

Abstract Asymptotic confidence intervals are given for two functions of multinomial outcome probabilities: Ginis diversity measure and Shannons entropy. “Adjusted” proportions are used in all asymptotic mean and variance formulas, along with a possible logarithmic transformation. Exact confidence coefficients are computed in some cases. Monte Carlo simulation is used in other cases to compare actual coverages to nominal ones. Some recommendations are made.

A refinement of Hoeffding's inequality

In this paper, we present a refinement of Hoeffdings inequality which is of closed form and which significantly improves on this inequality in many cases. Some numerical comparisons are also presented.

CONFIDENCE INTERVALS FOR EXPECTED COVERAGE FROM A BETA TESTABILITY MODEL

Abstract The testability distribution of a VLSI circuit can be used to predict the fault coverage of a set of test patterns by restricting the standard test pattern generation process to a sample of faults. When testability of a VLSI circuit is modeled as a beta distribution, the random detection counts obtained have a beta-binomial distribution. This paper includes: a) three confidence intervals for the parameters of the beta-binomial distribution; b) a determination of the proper sample size needed such that the theoretical confidence intervals agree with the actual ones; and c) a determination of the effect of the number of generated tests on confidence interval widths. Restricting the test generation process to a sample of faults results in major savings in the overall costs of test generation. Experimental results are given on three of the large combinational benchmark circuits (presented at the 1985 International Symposium on Circuits and Systems).

Estimating parameters from mixed samples using sample fractional moments

Abstract Alternate estimators based on fractional moments for estimating parameters when mixing two independent samples are proposed. These estimators are strongly consistent and asymptotically normal and are obtained by optimally combining several methods of moments estimators, all of closed form. In large samples, these estimators are compared to previously proposed estimators in the exponential scale case by numerically computing asymptotic variances.