Estimation on the Generalized Half Logistic Distribution under Type-II Hybrid Censoring

A hybrid censoring scheme is a mixture of type-I and type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are type-II hybrid censored. The maximum likelihood estimators, and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals. Bayes estimates, and the corresponding highest posterior density credible intervals of the unknown parameters, are obtained using suitable priors on the unknown parameters, and by using Markov chain Monte Carlo techniques. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyse one data set for illustrative purposes.

Background: Reliability analysis for the systems with masked data had been studied by many scholars. However, most researches focused on a system that is either series or parallel only, and the component in the system is mainly exponential or Weibull. In engineering practice, it is often seen that the structure of a system is a combination of series and parallel system, and other types of components are also used in the system. So it is important to study the reliability analysis of hybrid systems with modified Weibull components. Objective: For the hybrid system with masked data, the constant stress partial accelerated life test is performed under type-II progressive hybrid censoring. These data from life test are used to estimate unknown parameters and reliability index of system. The research results will not only provide theoretical basis and reference for system reliability assessment but also favor the patents on partial accelerated life test. Methods: Maximum likelihood estimates of unknown parameters are investigated with the numerical method. The approximate confidence intervals, and bootstrap confidence intervals for parameters are constructed by the asymptotic theory and the bootstrap method, respectively. Results: Maximum likelihood estimations of unknown parameters and reliability index of system are derived. The approximate confidence intervals and bootstrap confidence intervals for unknown parameters are proposed. The performance of estimation of unknown parameters and reliability index are evaluated numerically through Monte Carlo method. Conclusion: The performance on maximum likelihood estimation method is effective and satisfying. For the confidence intervals of parameters, bootstrap method outperforms the approximate method.

This paper develops maximum likelihood estimators (MLEs) of unknown param-eters in an exponentiated half-logistic distribution based on a progressively type-IIcensored sample. We obtain approximate con dence intervals for the MLEs by usingasymptotic variance and covariance matrices. Using importance sampling, we obtainBayes estimators and corresponding credible intervals with the highest posterior densityand Bayes predictive intervals for unknown parameters based on progressively type-IIcensored data from an exponentiated half logistic distribution. For illustration pur-poses, we examine the validity of the proposed estimation method by using real andsimulated data.Keywords: Bayes predictive interval, exponentiated half-logistic distribution, HPD cred-ible interval, importance sampling, progressively type-II censored sample. 1. Introduction Many reliability and survival analysis studies have used half-logistic distributions, partic-ularly for censored data. Several studies have drawn inferences about half logistic distribu-tions. Balakrishnan and Puthenpura (1986) introduce the best linear unbiased estimator oflocation and scale parameters of half-logistic distributions by considering linear functionsof order statistics. Balakrishnan and Wong (1991) obtain approximate maximum likelihoodestimators (AMLEs) of location and scale parameters of half-logistic distributions by usinga type-II right-censored sample. Kang et al. (2008) derive AMLEs and MLEs of scale pa-rameters of half-logistic distributions by using progressively type-II censored samples. Kanget al. (2009) propose AMLEs of scale parameters of half-logistic distributions by consideringdouble-hybrid censored samples. Kang and Seo (2011) recently examine exponentiated half-logistic distributions and propose two types of AMLEs of scale parameters and reliabilityfunctions by using progressively type-II right-censored samples. Kim et al. (2011a) deriveBayesian estimators of shape parameters, reliability functions, and failure rate functions

In orthogonal frequency division multiplexing (OFDM) systems, the carrier frequency offset can he normalized by the subcarrier bandwidth and then he divided into the integer and the fractional part. This paper focuses on the estimation of the integer carrier frequency offset and derives approximate maximum likelihood (ML) estimators in the presence and in the absence of pilot subcarriers. Simplified approximate ML estimators with reduced complexity are also proposed. The performance of the proposed estimators is compared with that of existing estimators, and it is shown by simulation that the proposed estimators outperform existing estimators.

The problem of estimating propagation delays of orthogonally modulated signals in asynchronous DS‐CDMA systems over time‐varing Rayleigh‐fading channels is treated in this paper. The maximum likelihood (ML) estimator and its unaffordable complexity for implementation are discussed. Some suboptimal solutions, e.g., whitened sliding correlator, MMSE estimator, subspace‐based estimator, approximate ML estimator, are proposed to combat the multiple access interference in the fading channels. The performance of these estimators are evaluated with the computer simulations and shown to have better acquisition performance than the standard sliding correlator. They also achieve reduced computational complexity compared with the ML estimator, while maintaining an acceptable performance degradation. Copyright © 2003 AEI

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

Estimating Spatial Autocorrelation with Sampled Network Data

Spatial autocorrelation is a parameter of importance for network data analysis. To estimate spatial autocorrelation, maximum likelihood has been popularly used. However, its rigorous implementation requires the whole network to be observed. This is practically infeasible if network size is huge (e.g., Facebook, Twitter, Weibo, WeChat, etc.). In that case, one has to rely on sampled network data to infer about spatial autocorrelation. By doing so, network relationships (i.e., edges) involving unsampled nodes are overlooked. This leads to distorted network structure and underestimated spatial autocorrelation. To solve the problem, we propose here a novel solution. By temporarily assuming that the spatial autocorrelation is small, we are able to approximate the likelihood function by its first-order Taylor’s expansion. This leads to the method of approximate maximum likelihood estimator (AMLE), which further inspires the development of paired maximum likelihood estimator (PMLE). Compared with AMLE, PMLE is computationally superior and thus is particularly useful for large-scale network data analysis. Under appropriate regularity conditions (without assuming a small spatial autocorrelation), we show theoretically that PMLE is consistent and asymptotically normal. Numerical studies based on both simulated and real datasets are presented for illustration purpose.

We consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. Since the exact maximum likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low signal to-noise ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML/sub 1/) estimator is derived. The performance of the AML/sub 1/ estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML/sub 2/ solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis.

We propose some estimators of the location parameter and derive the approximate maximum likelihood estimators (AMLEs) of the scale parameter in the exponential distribution based on multiply Type-II censored samples. We calculate the moments for the proposed estimators of the location parameter, and the AMLEs which are the linear functions of the order statistics. We compare the proposed estimators in the sense of the mean squared error (MSE) for various censored samples.

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.

One application for time-difference-of-arrival (TDOA) estimation is in emitter localization. A signal from an emitter reaching a group of sensors, each in a separate location, will have different arrival times. Finding the TDOAs between the output of pairs of sensors will provide the necessary measurements for the hyperbolic localization of the emitter. When the sensors acquire the signal by compressed sensing (CS), their outputs are reduced dimension linear transformation of the time samples of the signal. This shuffling of the time samples breaks up their time relation. Thus, a cross correlation of the CS output of two sensors cannot determine the TDOA. To apply cross correlation, it is necessary to reconstruct the time samples. This letter proposes an alternative that uses only the coefficients of the discrete Fourier transform (DFT) of the CS samples. It begins with the derivation of the maximum likelihood (ML) equation and the ML estimator. This estimator requires known values of signal and noise powers. Substituting these values by their estimates lead to the approximate ML estimator. The phase of the product of two DFT coefficients from each sensor is proportional to the unknown TDOA. Hence, these coefficients can provide an estimation of the TDOA. Simulation results show that although ML is the best, as expected, all these estimators have very close performance.

In this paper, given a progressively type II censored sample from a generalized half logistic distribution, the Bayesian and E-Bayesian (expectation of the Bayesian estimate) estimators are obtained under LINEX and squared-error loss functions, for the parameter and reliability function. Monte Carlo simulation method is used to generate a progressive Type-II censored data from generalized half logistic distribution, then these data is used to compute the estimations of the parameter and compare both the methods used when different random schemes.

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.