Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

On second-order improved estimation of a gamma scale parameter

This paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.     

Bayes designs for multiple linear regression on the unit sphere

We deal with experimental designs minimizing the mean square error of the linear BAYES estimator for the parameter vector of a multiple linear regression model where the experimental region is the k-dimensional unit sphere. After computing the uniquely determined optimum information matrix, we construct, separately for the homogeneous and the inhomogeneous model, both approximate and exact designs having such an information matrix.     

R–estimation of normed bivariate density functions

Normed bivariate density funtions were introduced by HOEFFDING (1940/41). In the present paper estimators for normed bivariate ranks and on a FOURIER series expansion in LEGENDRE polynomials. The estimation of normed bivarate density functions under positive dependence is also described     

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.     

A Likelihood-Based Approximation to the Cumulative Distribution Function of the Noncentral t Distribution

Optimal designs for estimating the parameters and also the optimum factor combinations in multiresponse experiments have been considered by various authors. However, till date, in mixture experiments optimum designs have been studied only in the single response case. In this article, attempt has been made to investigate optimum designs for estimating optimum mixing proportions in a multiresponse mixture experiment.     

A comparison of several analysis of variance programs with unequal cell size

Various programs in statistical packages for analysis of variance with unequal cell size give different results to the same data because of nonorthogonality of the main effects and interactions. This paper explains how these programs treat the problem of analysis of variance of unbalanced data.     

Selecting a Good Stochastic System for the Large Number of Alternatives

In this article, we present the problem of selecting a good stochastic system with high probability and minimum total simulation cost when the number of alternatives is very large. We propose a sequential approach that starts with the Ordinal Optimization procedure to select a subset that overlaps with the set of the actual best m% systems with high probability. Then we use Optimal Computing Budget Allocation to allocate the available computing budget in a way that maximizes the Probability of Correct Selection. This is followed by a Subset Selection procedure to get a smaller subset that contains the best system among the subset that is selected before. Finally, the Indifference-Zone procedure is used to select the best system among the survivors in the previous stage. The numerical test involved with all these procedures shows the results for selecting a good stochastic system with high probability and a minimum number of simulation samples, when the number of alternatives is large. The results also show that the proposed approach is able to identify a good system in a very short simulation time.