On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

Asymptotic cumulants of the estimator of the canonical parameter in the exponential family

Asymptotic cumulants of the maximum likelihood estimator of the canonical parameter in the exponential family are obtained up to the fourth order with the added higher-order asymptotic variance. In the case of a scalar parameter, the corresponding results with and without studentization are given. These results are also obtained for the estimators by the weighted score, especially for those using the Jeffreys prior. The asymptotic cumulants are used for reducing bias and mean square error to improve a point estimator and for interval estimation to have higher-order accuracy. It is shown that the kurtosis to squared skewness ratio of the sufficient statistic plays a fundamental role.     

Approximations to the powers of the likelihood ratio tests: the loop ordering and slippage alternatives

Approximations to the power functions of the likelihood ratio tests of homogeneity of normal means against the simple loop ordering at slippage alternatives are considered. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a simple loop ordering is appropriate. The accuracy of the several moment approximations are studied for the case of known variances and it is found that for powers in the range typically of interest, the two-moment approximation seems quite adequate. Approximations based on mixtures of noncentral F variables are developed for the case of unknown variances. The critical values of the test statistics are also tabulated for selected levels of significance.     

On the experience of projecting the family composition of the population of the USSR

The joint efforts in the USSR by the Central Planning Commission and the State Statistical Commission to develop projections of family characteristics of the Soviet population are described. The projections, based on official data for the rural and urban populations in 1987, are for the years 1991, 1996, 2001, and 2006.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

On the Bayesian estimation of the weighted Lindley distribution

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.     

On the estimation of the spectral density for continuous spatial processes

In this work, we study the asymptotic properties of smoothed nonparametric kernel spectral density estimators for the spatial spectral density. We consider the case of continuous stationary spatial processes under a shrinking asymptotic framework. Expressions for the bias and the covariance structure are obtained and the implications for the edge effect bias of the choice of the kernel, bandwidth and spacing parameter in the design are also discussed, both for tapered and untapered estimates. Results are illustrated with a simulation study.     

On the Skew-Normal-Cauchy Distribution

A fixed effects one-way layout model of analysis of variance is considered where the variances are taken to be possibly unequal. Conservative single-stage procedures based on Banerjee’s method for the solution of the Behrens-Fisher problem are proposed for the following multiple comparisons problems: 1) all pairwise comparisons with a control population mean, and 2) all pairwise comparisons and all linear contrasts among the means. Since these procedures are likely to be very conservative in practice, approximate procedures based on Welch’s method for the solution of the Behrens-Fisher problem are suggested as alternatives. Monte Carlo studies indicate that the latter are much less conservative and hence may be better in practice. Both these sets of procedures need only the tables of the Student’s t-distribution for their application and are very simple to use. Exact two-stage procedures are proposed for the following multiple comparisons problems: 1) all pairwise comparisons and all linear contrasts among the means, and 2) all linear combinations of the means.     

On the estimates of the parameters of the first order autoregressive process

Two methods of estimation for the parameters of an AR(1) process which are based on a non-linear least-squares approach are presented. On the basis of some simulation results they are compared with two maximum likelihood estimates and their relative merits are discussed.     

On the behavior of mle of the scale parameter of the student family

This paper contains a study of the qualitative properties of the maximum likelihood estimate of the scale parameter of the Student family as a function of its (continuous) degrees of freedom. Sharp bounds are provided for the estimate and its (continuous) degrees of freedom. Sharp bounds are provided for the estimate and its monotonicity as a function of the degrees of freedom is established. Inferential implications are also discussed.     

Bias-Corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution

The two-parameter weighted Lindley distribution is useful for modeling survival data, whereas its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters. We adopt a “corrective” approach to derive modified MLEs that are bias-free to second order. We also consider an alternative bias-correction mechanism based on Efron’s bootstrap resampling. Monte Carlo simulations are conducted to compare the performance between the proposed and two previous methods in the literature. The numerical evidence shows that the bias-corrected estimators are extremely accurate even for very small sample sizes and are superior than the previous estimators in terms of biases and root mean squared errors. Finally, applications to two real datasets are presented for illustrative purposes.     

Interval Estimation for the Sortino Ratio and the Omega Ratio

In this article, asymptotic confidence intervals (CIs) for the Sortino and Omega ratios are proposed and analyzed. First, the CIs are derived under the assumption of temporal independence and identical distribution of returns. Later they are obtained assuming that the returns process is strictly stationary and α-mixing of a certain size. In order to evaluate the minimum sample size for a good coverage accuracy of the asymptotic CIs, a simulation study is performed. It is obtained that the minimum sample sizes are very high, especially under the more realistic assumption of not-iid returns.     

Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan

This article presents the goodness-of-fit tests for the Laplace distribution based on its maximum entropy characterization result. The critical values of the test statistics estimated by Monte Carlo simulations are tabulated for various window and sample sizes. The test statistics use an entropy estimator depending on the window size; so, the choice of the optimal window size is an important problem. The window sizes for yielding the maximum power of the tests are given for selected sample sizes. Power studies are performed to compare the proposed tests with goodness-of-fit tests based on the empirical distribution function. Simulation results report that entropy-based tests have consistently higher power than EDF tests against almost all alternatives considered.     

On the Convergence of the Horvitz-Thompson Estimator

We investigate the degree to which the Horvitz-Thompson estimator approximates the population mean in survey sampling. Specifically, conditions for (mean square) consistency are given, along with rates of convergence. The conditions involve only first and second-order inclusion probabilities, and for various sampling designs they are easy to check.     

Three modifications of the principle of the minque

Nonnegative estimators for the variance components of a linear model are obtained by ignoring the condition for unbiasedness in the principle of the MINQUE. An estimator is derived when the priori weights are proportional to the variance components. The ordinary sample variance is shown to be the nonnegative MINQUE. Efficiencies of the three estimators are examined for some special cases of the model.     

Estimation for the parameters of the generalized half-normal distribution

As an applicable and flexible lifetime model, the two-parameter generalized half-normal (GHN) distribution has been received wide attention in the field of reliability analysis and lifetime study. In this paper maximum likelihood estimates of the model parameters are discussed and we also proposed corresponding bias-corrected estimates. Unweighted and weighted least squares estimates for the parameters of the GHN distribution are also presented for comparison purpose. Moreover, the likelihood ratio test is provided as complementary. Simulation study and illustrative examples are provided to compare the performance of the proposed methods.     

Adjusting the degrees of freedom for the jackknife

The rule of thumb given by Mosteller and Tukey for obtaining the degrees of freedom which are appropriate for the approximate t-statistic associated with the jackknife is discussed.Examples of possible misapplication of this rule are given.     

Testing the similarity of two normal populations with application to the bioequivalence problem

The problem of testing the similarity of two normal populations is reconsidered, in this article, from a nonclassical point of view. We introduce a test statistic based on the maximum likelihood estimate of Weitzman's overlapping coefficient. Simulated critical points are provided for the proposed test for various sample sizes and significance levels. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing tests. Furthermore, Type-I error robustness of the proposed and the existing tests are studied via simulation studies when the underlying distributions are non-normal. Two data sets are analyzed for illustration purposes. Finally, the proposed test has been implemented to assess the bioequivalence of two drug formulations.     

Inferences on the difference and ratio of the means of two inverse Gaussian distributions

Methods for interval estimation and hypothesis testing about the ratio of two independent inverse Gaussian (IG) means based on the concept of generalized variable approach are proposed. As assessed by simulation, the coverage probabilities of the proposed approach are found to be very close to the nominal level even for small samples. The proposed new approaches are conceptually simple and are easy to use. Similar procedures are developed for constructing confidence intervals and hypothesis testing about the difference between two independent IG means. Monte Carlo comparison studies show that the results based on the generalized variable approach are as good as those based on the modified likelihood ratio test. The methods are illustrated using two examples.     

On the analysis of long-term experiments

Summary.  Long-term experiments are commonly used tools in agronomy, soil science and other disciplines for comparing the effects of different treatment regimes over an extended length of time. Periodic measurements, typically annual, are taken on experimental units and are often analysed by using customary tools and models for repeated measures. These models contain nothing that accounts for the random environmental variations that typically affect all experimental units simultaneously and can alter treatment effects. This added variability can dominate that from all other sources and can adversely influence the results of a statistical analysis and interfere with its interpretation. The effect that this has on the standard repeated measures analysis is quantified by using an alternative model that allows for random variations over time. This model, however, is not useful for analysis because the random effects are confounded with fixed effects that are already in the repeated measures model. Possible solutions are reviewed and recommendations are made for improving statistical analysis and interpretation in the presence of these extra random variations.