Unknown Location Parameters Research Articles

Estimating the scale parameters of several exponential distributions under order restriction

. In the present work, we have investigated the problem of estimating the parameters of several exponential distributions with ordered scale parameters under linex loss function. The study includes the cases of known as well as unknown location parameters. For both cases, we have considered class of equivariant estimators. Then, sufficient conditions have been obtained under which the equivariant class of estimators improves upon the usual estimators. Using the established results, it has been shown that the restricted maximum likelihood estimator is inadmissible. Finally, for both cases, simulation study has been conducted to compare the risk performance of the proposed estimators.

On three-parameter generalized exponential distribution

In this article, we consider the estimation of the three-parameter generalized exponential distribution. In presence of the unknown location parameter the usual maximum likelihood estimators do not exist for The maximum product of spacings method serves as good alternative since it always exists and yields consistent estimators in the entire parametric space. We develop the asymptotic distribution of the proposed estimators along-with a detailed discussion about the computational intricacies involved in implementing the product of spacing method. Extensive simulations have been performed to demonstrate the effectiveness of the proposed method for α in the range (0, 1), in addition to a comparative study with some standard techniques known to provide consistent estimators, even for Furthermore, two real data sets have been analyzed to demonstrate the applicability of the proposed method.

Distribution-free Phase II triple EWMA control chart for joint monitoring the process location and scale parameters

ABSTRACT Distribution-free or nonparametric control charts are used for monitoring the process parameters when there is a lack of knowledge about the underlying distribution. In this paper, we investigate a single distribution-free triple exponentially weighted moving average control chart based on the Lepage statistic (referred as TL chart) for simultaneously monitoring shifts in the unknown location and scale parameters of a univariate continuous distribution. The design and implementation of the proposed chart are discussed using time-varying and steady-state control limits for the zero-state case. The run-length distribution of the TL chart is evaluated by performing Monte Carlo simulations. The performance of the proposed chart is compared to those of the existing EWMA-Lepage (EL) and DEWMA-Lepage (DL) charts. It is observed that the TL chart with a time-varying control limit is superior to its competitors, especially for small to moderate shifts in the process parameters. We also provide a real example from a manufacturing process to illustrate the application of the proposed chart.

Advances in UAV-Assisted Localization: Joint Source and UAV Parameter Estimation

This paper presents efficient estimators to address the problem of source localization aided by unmanned aerial vehicles (UAVs) with unknown location parameters. Leveraging semidefinite relaxation technique and successive weighted least squares estimation, the source and UAV parameters are simultaneously estimated in both mixed line-of-sight/non-line-of-sight (NLOS) and NLOS scenarios. The proposed estimators are shown to approximately achieve the Cramer-Rao bound (CRB) under mild Gaussian noise conditions when the measurement noises are small compared to the associated ranges. Numerical simulations demonstrate that localization without prior UAV location information is still possible with a meter-level positioning and sub-meter per seconds velocity estimation accuracy.

An existence level for the residual sum of squares of the power-law regression with an unknown location parameter

Abstract In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.

Confidence bands for exponential distribution functions under progressive type-II censoring

Based on a progressively type-II censored sample from the exponential distribution with unknown location and scale parameter, confidence bands are proposed for the underlying distribution function by using confidence regions for the parameters and Kolmogorov–Smirnov type statistics. Simple explicit representations for the boundaries and for the coverage probabilities of the confidence bands are analytically derived, and the performance of the bands is compared in terms of band width and area by means of a data example. As a by-product, a novel confidence region for the location-scale parameter is obtained. Extensions of the results to related models for ordered data, such as sequential order statistics, as well as to other underlying location-scale families of distributions are discussed.

Componentwise estimation of ordered scale parameters of two exponential distributions under a general class of loss function

In many real life situations, prior information about the parameters is available, such as the ordering of the parameters. Incorporating this prior information about the order restrictions on parameters leads to more efficient estimators. In the present communication, we investigate estimation of the ordered scale parameters of two shifted exponential distributions with unknown location parameters under a class of bowl-shaped loss functions. We have proved that the best affine equivariant estimator (BAEE) is inadmissible. Various non smooth and smooth estimators has been obtained which improve upon the BAEE. In particular we have derived the improved estimators for some well known loss functions. Finally numerical comparison is carried out to compare the risk performance of the proposed estimators.

Minimum Profile Hellinger Distance Estimation for A Two-Sample Location-Shifted Model

Minimum Hellinger distance estimation (MHDE) for parametric model is obtained by minimizing the Hellinger distance between an assumed parametric model and a nonparametric estimation of the model. MHDE receives increasing attention for its efficiency and robustness. Recently, it has been extended from parametric models to semiparametric models. This manuscript considers a two-sample semiparametric location-shifted model where two independent samples are generated from two identical symmetric distributions with different location parameters. We propose to use profiling technique in order to utilize the information from both samples to estimate unknown symmetric function. With the profiled estimation of the function, we propose a minimum profile Hellinger distance estimation (MPHDE) for the two unknown location parameters. This MPHDE is similar to but dif- ferent from the one introduced in Wu and Karunamuni (2015), and thus the results presented in this work is not a trivial application of their method. The difference is due to the two-sample nature of the model and thus we use different approaches to study its asymptotic properties such as consistency and asymptotic normality. The efficiency and robustness properties of the proposed MPHDE are evaluated empirically though simulation studies. A real data from a breast cancer study is analyzed to illustrate the use of the proposed method.

Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

AbstractIn the present article we study the average of Lipschitz‐Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on . The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters of X, we introduce novel fundamental quantities called effective level and effective spectral moment. We propose unbiased and asymptotically normal estimators of these parameters. From these asymptotic results, we build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms. Meanwhile, we establish the consistency of the empirical variance estimators of the third LK curvature under a weak condition on the correlation function of X.

Quantifying time-varying sources in magnetoencephalography—A discrete approach

We study the distribution of brain source from the most advanced brain imaging technique, Magnetoencephalography (MEG) which measures the magnetic fields outside of the human head produced by the electrical activity inside the brain. Common time-varying source localization methods assume the source current with a time-varying structure and solve the MEG inverse problem by mainly estimating the source moment parameters. These methods use the fact that the magnetic fields linearly depend on the moment parameters of the source and work well under the linear dynamic system. However, magnetic fields are known to be nonlinearly related to the location parameters of the source. The existing work on estimating the time-varying unknown location parameters is limited. We are motivated to investigate the source distribution for the location parameters based on a dynamic framework, where the posterior distribution of the source is computed in a closed form discretely. The new framework allows us not only to directly approximate the posterior distribution of the source current, where sequential sampling methods may suffer from slow convergence due to the large volume of measurement, but also to quantify the source distribution at any time point from the entire set of measurements reflecting the distribution of the source, rather than using only the measurements up to the time point of interest. Both a dynamic procedure and a switch procedure are pro- posed for the new discrete approach, balancing estimation accuracy and computational efficiency when multiple sources are present. In both simulation and real data, we illustrate that the new method is able to provide comprehensive insight into the time evolution of the sources at different stages of the MEG and EEG experiment.

On estimating the location parameter of the selected exponential population under the LINEX loss function

Suppose that $\pi_{1},\pi_{2},\ldots ,\pi_{k}$ be $k(\geq2)$ independent exponential populations having unknown location parameters $\mu_{1},\mu_{2},\ldots,\mu_{k}$ and known scale parameters $\sigma_{1},\ldots,\sigma_{k}$. Let $\mu_{[k]}=\max \{\mu_{1},\ldots,\mu_{k}\}$. For selecting the population associated with $\mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $\mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $\delta_{N,1},\delta_{N,2}$ and $\delta_{N,3}$ of $\mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $\mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $\mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $\mu_{S}$ is derived. Using this result, estimator better than the natural estimator $\delta_{N,1}$ is obtained. We also shown that the estimator $\delta_{N,1}$ is dominated by the natural estimator $\delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $\mu_{S}$.

Tests based on EDF statistics for randomly censored normal distributions when parameters are unknown

Goodness-of-fit techniques are an important topic in statistical analysis. Censored data occur frequently in survival experiments; therefore, many studies are conducted when data are censored. In this paper we mainly consider test statistics based on the empirical distribution function (EDF) to test normal distributions with unknown location and scale parameters when data are randomly censored. The most famous EDF test statistic is the Kolmogorov-Smirnov; in addition, the quadratic statistics such as the Cramer-von Mises and the Anderson- Darling statistic are well known. The Cramer-von Mises statistic is generalized to randomly censored cases by Koziol and Green (Biometrika, 63, 465-474, 1976). In this paper, we generalize the Anderson-Darling statistic to randomly censored data using the Kaplan-Meier estimator as it was done by Koziol and Green. A simulation study is conducted under a particular censorship model proposed by Koziol and Green. Through a simulation study, the generalized Anderson-Darling statistic shows the best power against almost all alternatives considered among the three EDF statistics we take into account.

Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation

Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) schemes are widely used in monitoring small and persistent shifts in specific process characteristics. Nonparametric (Distribution-free) CUSUM and EWMA schemes are useful in detecting such changes when the underlying process distribution is unknown or complicated. The CUSUM-Wilcoxon rank-sum (CUSUM-WRS), CUSUM-Precedence, EWMA-WRS, and EWMA-Precedence are well-known distribution-free CUSUM and EWMA schemes for Phase-II monitoring of a shift in the unknown location parameter of a process. In this article, we compare their performances with two robust CUSUM and EWMA schemes based on the Hogg-Fisher-Randle (HFR) type statistic. We investigate the accomplishments of these CUSUM and EWMA schemes in detecting shifts of different sizes for various process distributions and show that the proposed CUSUM and EWMA HFR schemes perform favourably for highly skewed distributions. In this article, we also discuss the efficacy of these nonparametric CUSUM and EWMA schemes when the test sample size is small. We also outline the implementation strategies various plans with an illustration using a dataset from the iron ore mining plant. We offer some concluding remarks and future research problems.

Goodness-of-Fit Tests Based on a Characterization of Logistic Distribution

The logistic family of distributions belongs to the class of important families in the theory of probability and mathematical statistics. However, the goodness-of-fit tests for the composite hypothesis of belonging to the logistic family with unknown location parameter against the general alternatives have not been sufficiently explored. We propose two new goodness-of-fit tests: the integral and the Kolmogorov-type, based on the recent characterization of the logistic family by Hua and Lin. Here we discuss asymptotic properties of new tests and calculate their Bahadur efficiency for common alternatives.

Estimating the common hazard rate of several exponential distributions

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.

Testing a Multivariate Distribution for Generalized Skew Ellipticity

We consider the problem of testing whether a sample comes from a family of the multivariate generalized skew-elliptical distributions with an unknown location parameter, an unknown scaling matrix, ...

A double generally weighted moving average exceedance control chart

AbstractSince the inception of control charts by W. A. Shewhart in the 1920s, they have been increasingly applied in various fields. The recent literature witnessed the development of a number of nonparametric (distribution‐free) charts as they provide a robust and efficient alternative when there is a lack of knowledge about the underlying process distribution. In order to monitor the process location, information regarding the in‐control (IC) process median is typically required. However, in practice, this information might not be available due to various reasons. To this end, a generalized type of nonparametric time‐weighted control chart labeled as the double generally weighted moving average (DGWMA) based on the exceedance statistic (EX) is proposed. The DGWMA‐EX chart includes many of the well‐known existing time‐weighted control charts as special or limiting cases for detecting a shift in the unknown location parameter of a continuous distribution. The DGWMA‐EX chart combines the better shift detection properties of a DGWMA chart with the robust IC performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is IC or out‐of‐control. An extensive simulation study reveals that the proposed DGWMA‐EX chart, in many cases, outperforms its counterparts.

Efficient reliability estimation in two-parameter exponential distributions

This article concerns reliability estimation in two-parameter exponential distributions setup with known scale parameters, and unknown location parameters. Based on the uniformly minimum variance unbiased estimator, we propose a new estimator and study its theoretical properties. Simulation results reveal that the suggested estimator could be highly efficient.

Natural (Non‐)Informative Priors for Skew‐symmetric Distributions

AbstractIn this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew‐symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the total variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our non‐informative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale‐invariant and location‐invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.

A Baumgartner control chart for joint monitoring of unknown location and scale parameters

A Baumgartner control chart for joint monitoring of unknown location and scale parameters