Estimation of cyclic long-memory parameters

This paper studies cyclic long-memory processes with Gegenbauer-type spectral densities. For a semiparametric statistical model, new simultaneous estimates for singularity location and long-memory parameters are proposed. This generalized filtered method-of-moments approach is based on general filter transforms that include wavelet transformations as a particular case. It is proved that the estimates are almost surely convergent to the true values of parameters. Solutions of the estimation equations are studied, and adjusted statistics are proposed. Monte-Carlo study results are presented to confirm the theoretical findings.     

Nonparametric bayes estimates of estimable parameters with a dirichlet invarant process and inveriant u-statistics

Tne Bayes estimates of estimable parameters of arbitrary degree in the one sample case are obtained against a Dirichlet invariant. process prior and the squared error loss. We also oive the limits of Bayes estimates, which are related to the in- a- variant U-statistics. For a fixed distribution, the limits of Bayes estimates have the asymptotic normal distribution under certain conditjons.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Ordinal ridge regression with categorical predictors

In multi-category response models, categories are often ordered. In the case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) are most commonly used for ordinal responses. In this paper, penalized likelihood with quadratic penalty is used to address these issues with a special focus on POMs. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized-likelihood function penalizes the parameter estimates or differences between the parameter estimates according to the type of predictors. Mean-squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

On the robustness of right and middle censoring schemes in parametric survival models

This article considers the exponential and Weibull distributions for studying the robustness of their parameters estimates in the case of right and middle types of censoring. As might be expected, the simulation studies indicate that the mean square error (MSE) decreases with larger sample sizes, increases with the contamination level and censoring proportion. But somewhat surprisingly, our study shows that, regardless the parametric model and for given values of the other factors, estimates for middle censoring are more robust compared to those for right censoring.     

Residual variance estimation in moving average models

We consider time series models of the MA (moving average) family, and deal with the estimation of the residual variance. Results are known for maximum likelihood estimates under normality, both for known or unknown mean, in which case the asymptotic biases depend on the number of parameters (including the mean), and do not depend on the values of the parameters. For moment estimates the situation is different, because we find that the asymptotic biases depend on the values of the parameters, and become large as they approach the boundary of the region of invertibility. Our approach is to use Taylor series expansions, and the objective is to obtain asymptotic biases with error of o(l/T), where T is the sample size. Simulation results are presented, and corrections for bias suggested.     

Bayesian Estimation of Bivariate Exponential Distributions Based on Linex and Quadratic Loss Functions: A Survival Approach with Censored Samples

In this article, four bivariate exponential (BVE) distributions with subject to right censoring samples are presented. Bayesian estimates of the parameters of BVE are obtained through Linex and quadratic loss functions. Gamma prior distribution has been suggested to reforming the posterior function. The estimations and standard errors of parameters have also been obtained through simulation method. Markov chain Monte Carlo (MCMC) method is employed for the case of Block-Buse bivariate distribution because there was no closed form for estimator criteria. Simulation studies have been conducted to show that the computation parts can be implemented easily and comparing the estimated values due to two methods and with the true values as well.     

Finite mixtures of quantile and M-quantile regression models

In this paper we define a finite mixture of quantile and M-quantile regression models for heterogeneous and /or for dependent/clustered data. Components of the finite mixture represent clusters of individuals with homogeneous values of model parameters. For its flexibility and ease of estimation, the proposed approaches can be extended to random coefficients with a higher dimension than the simple random intercept case. Estimation of model parameters is obtained through maximum likelihood, by implementing an EM-type algorithm. The standard error estimates for model parameters are obtained using the inverse of the observed information matrix, derived through the Oakes (J R Stat Soc Ser B 61:479–482, 1999) formula in the M-quantile setting, and through nonparametric bootstrap in the quantile case. We present a large scale simulation study to analyse the practical behaviour of the proposed model and to evaluate the empirical performance of the proposed standard error estimates for model parameters. We considered a variety of empirical settings in both the random intercept and the random coefficient case. The proposed modelling approaches are also applied to two well-known datasets which give further insights on their empirical behaviour.     

Estimation of Weibull distribution parameters for irregular interval group failure data with unknown failure times

SUMMARY This paper presents three methods for estimating Weibull distribution parameters for the case of irregular interval group failure data with unknown failure times. The methods are based on the concepts of the piecewise linear distribution function (PLDF), an average interval failure rate (AIFR) and sequential updating of the distribution function (SUDF), and use an analytical approach similar to that of Ackoff and Sasieni for regular interval group data. Results from a large number of simulated case problems generated with specified values of Weibull distribution parameters have been presented, which clearly indicate that the SUDF method produces near-perfect parameter estimates for all types of failure pattern. The performances of the PLDF and AIFR methods have been evaluated by goodness-of-fit testing and statistical confidence limits on the shape parameter. It has been found that, while the PLDF method produces acceptable parameter estimates, the AIFR method may fail for low and high shape parameter values that represent the cases of random and wear-out types of failure. A real-life application of the proposed methods is also presented, by analyzing failures of hydrogen make-up compressor valves in a petroleum refinery.     

Bayesian inference of the inverse Weibull mixture distribution using type-I censoring

A large number of models have been derived from the two-parameter Weibull distribution including the inverse Weibull (IW) model which is found suitable for modeling the complex failure data set. In this paper, we present the Bayesian inference for the mixture of two IW models. For this purpose, the Bayes estimates of the parameters of the mixture model along with their posterior risks using informative as well as the non-informative prior are obtained. These estimates have been attained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the former case, Bayes estimates are obtained under three loss functions while for the latter case only the squared error loss function is used. Simulation study is carried out in order to explore numerical aspects of the proposed Bayes estimators. A real-life data set is also presented for both cases, and parameters obtained under case when shape parameter is known are tested through testing of hypothesis procedure.     

Second-order biases of the maximum likelihood estimates in von Mises regression models

This paper discusses issues related to the improvement of maximum likelihood estimates in von Mises regression models. It obtains general matrix expressions for the second-order biases of maximum likelihood estimates of the mean parameters and concentration parameters. The formulae are simple to compute, and give the biases by means of weighted linear regressions. Simulation results are presented assessing the performance of corrected maximum likelihood estimates in these models.     

An iterative procedure for the estimation of parameters in a dose-response model

The least squares estimates of the parameters in the multistage dose-response model are unduly affected by outliers in a data set whereas the minimum sum of absolute errors, MSAE estimates are more resistant to outliers. Algorithms to compute the MSAE estimates can be tedious and computationally burdensome. We propose a linear approximation for the dose-response model that can be used to find the MSAE estimates by a simple and computationally less intensive algorithm. A few illustrative ex-amples and a Monte Carlo study show that we get comparable values of the MSAE estimates of the parameters in a dose-response model using the exact model and the linear approximation.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

Inference on unknown parameters of a Burr distribution under hybrid censoring

Based on hybrid censored data, the problem of making statistical inference on parameters of a two parameter Burr Type XII distribution is taken up. The maximum likelihood estimates are developed for the unknown parameters using the EM algorithm. Fisher information matrix is obtained by applying missing value principle and is further utilized for constructing the approximate confidence intervals. Some Bayes estimates and the corresponding highest posterior density intervals of the unknown parameters are also obtained. Lindley’s approximation method and a Markov Chain Monte Carlo (MCMC) technique have been applied to evaluate these Bayes estimates. Further, MCMC samples are utilized to construct the highest posterior density intervals as well. A numerical comparison is made between proposed estimates in terms of their mean square error values and comments are given. Finally, two data sets are analyzed using proposed methods.     

Asymptotic distribution of regression type estimators of parameters of stable laws

Asymptotic distributions of regression-type estimators for the parameters of stable distributions am obtained. The asymptotic normalized standard deviations of the estimators are computed for various values of the parameters and various choices of the number of points used in getting the regression estimates.     

Using the singly truncated normal distribution to analyze satellite data

This paper proposes the singly truncated normal distribution as a model for estimating radiance measurements from satellite-borne infrared sensors. These measurements are made in order to estimate sea surface temperatures which can be related to radiances. Maximum likelihood estimation is used to provide estimates for the unknown parameters. In particular, a procedure is described for estimating clear radiances in the presence of clouds and the Kolmogorov-Smirnov statistic is used to test goodness-of-fit of the measurements to the singly truncated normal distribution. Tables of quantile values of the Kolmogorov-Smirnov statistic for several values of the truncation point are generated from Monie Carlo experiment Mnally a numerical emample using satetic data is presented to illustrate the application of the procedures.     

Maximum likelihood estimation of Burr XII distribution parameters under random censoring

In this paper, we consider the maximum likelihood estimation of the parameters of Burr XII distribution using randomly right censored data. We provide necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates. Under such conditions, it is shown that the maximum likelihood estimates are strongly consistent for the true values of the parameters and are asymptotically bivariate normal. An application to leukemia free-survival times for allogeneic and autologous transplant patients is given.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.