Estimation Error Variance Research Articles

Note on the radical inflation in the estimates of error variance in measurement models

This note discusses the radical technical inflation in error variance and the related standard error of test scores from both conceptual and empirical viewpoints. This technical inflation arises as a direct consequence of the technical underestimation of item-score correlation by the product–moment coefficient of correlation (PMC), which is embedded in the traditional estimators of reliability such as coefficients alpha, theta, omega, or rho (maximal reliability). Specifically, in educational settings where compilations usually include both easy and difficult items, the estimate by PMC may substantially deviate from the true association between an item and the score. Consequently, the use of traditional estimators of reliability leads to technically inflated estimates of standard errors, as the error variance related to these traditional measurement models is significantly inflated, resulting in deflated reliability estimates. In educational testing, employing deflation-corrected standard errors, calculated using deflation-corrected reliability estimators, would provide a more accurate measure of the test score’s true precision.

Invariance Principle and Berry–Esseen Bound for Error Variance Estimator for Pth-Order Nonlinear Autoregressive Models

The invariance principle and Berry–Esseen bound for an error variance estimator based on the residuals are established by using a Taylor expansion and the classical invariance principle and Berry–Esseen bound for independent random variables. Some examples are given to illustrate their applications.

COMPARATIVE STUDY OF RESIDUAL VARIANCE ESTIMATORS CONSTRUCTED WITHOUT OR WITH THE PRESENCE OF ATYPICAL VALUES: CASE OF THE PARTIALLY LINEAR MODEL

Error variance is a measure of the goodness of fit for a regression model. Ahoutou et al. [1] proposed estimators of the error variance under the assumption that the dimension of the nonlinear covariate is one. In this paper, assuming that the dimension of the nonlinear covariate is greater than or equal to 2, we propose estimators of error variance in homogeneous and heterogeneous cases. We propose estimators of the error variance that are constructed without taking atypical values into account and others that are constructed by taking all the data into account. Using simulations, we compare the performance of estimators designed without taking into account atypical values with estimators designed taking into account all the data.

A robust decision-making approach for designing coastal groundwater quality monitoring networks.

This paper presents a new approach to the spatiotemporal design of groundwater quality monitoring networks for coastal aquifers. A fusion model combines the outputs of several developed simulation models to make estimates more accurate. A modified GALDIT method is used to incorporate the aquifer vulnerability to saltwater intrusion. The value of information (VOI) theory is applied to determine sufficient monitoring wells. The groundwater quality monitoring network is designed by employing a robust decision-making (RDM) approach under different management strategies and economic considerations. This approach incorporates the deep uncertainties of some critical variables, including water level and total dissolved solids (TDS) concentration at the coastline and pumping flow rates of agricultural wells. The new methodology is implemented in the coastal Qom-Kahak aquifer, Iran. The results illustrate that the combination model has significantly improved evaluation criteria compared to individual prediction models. The fusion model results indicate that thirty monitoring wells would be ideal. The RDM-based analyses in the Qom-Kahak aquifer showed that an optimal network with 30 monitoring wells outperforms the current network regarding various criteria, such as VOI and variance of estimation error. The new well configuration also demonstrates a suitable spatial distribution. Given that the current sampling frequencies are unsuitable for areas with varying vulnerabilities, we recommend sampling every 3months in areas with moderate vulnerabilities and once every three seasons in areas with low vulnerabilities, based on the information transfer index. Finally, a management strategy in which the pumping rate should be less than 60% of the current rate is suggested to prevent saltwater intrusion into the aquifer.

Exact Analysis of MIMO Channel Estimation Based on Superimposed Training

In this paper, channel estimation capabilities of a multiple-input multiple-output (MIMO) system using superimposed training sequences are investigated. A new expression for estimation-error variance is derived. It is shown that the training sequences must be balanced and must have specific correlation properties. The latter are required only in a specific zone. Sequences that satisfy these criteria exist and are referred to as zero-correlation zone (ZCZ) solutions. Consequently, by using balanced ZCZ sequences, harmful direct current (DC) offset can be removed. Owing to their zero-cross correlation, interference from other transmitting antennas may be eliminated. Furthermore, a closed-form expression of the estimation-error variance can be obtained due to their impulse-like autocorrelation. To increase the number of antennas in the MIMO system, a new construction of ZCZ sequence set is proposed, in which all sequences are balanced

Recursive state estimation for two-dimensional systems over decode-and-forward relay channels: A local minimum-variance approach

The paper is concerned with the recursive state estimation problem for a class of two-dimensional systems over decode-and-forward relay channels subject to packet dropouts. Taking into account the sensor's limited transmission ability, a decode-and-forward relay is deployed between the sensor and the remote estimator to enlarge the propagation distance and improve the communication quality. The packet dropout phenomenon is described by two sequences of mutually uncorrelated random variables that obey Bernoulli distributions. The main objective of this paper is to design a recursive state estimator for the underlying system with guaranteed estimation performance. An upper bound on the estimation error variance is firstly derived by resorting to some coupled difference equations, and then such an upper bound is locally optimized at each iteration through proper design of the estimator gain. Furthermore, the impacts of the packet dropouts and the decode-and-forward relay on the estimation performance are evaluated with rigorously theoretical analysis. Finally, an illustrative example is presented to validate the effectiveness of the proposed estimation scheme.

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Constructing group sequential tests with data monitoring

Objective:Group sequential tests are procedures for repeatedly analysing accruing data, and aim to reduce test time by stopping data collection as soon as an effect is deemed statistically present or absent. Data monitoring, on the other hand, aims to control statistical power by repeatedly re-estimating sample variance from the accruing data, and then adjusting the sample size accordingly. The current work presents a numerical framework for combining group sequential tests with data monitoring, thus leveraging the best of both methods. Methods:The approach revolves around numerically convolving truncated probability density functions, which leads to numerically tracktable distributions for the sequential test statistic. This then allows conventional power analysis methodologies to be used during the data monitoring procedure, thus keeping the procedure relatively intuitive and transparent. The test operating characteristics of the procedure were evaluated extensively in simulations and in Auditory Brainstem Response data. Results:A good control over statistical power was observed, which was attributed to data monitoring. In some test conditions, however, data monitoring led to inflated type-I error rates and a reduced control over statistical power. This was attributed to sample variance estimation errors, and was largely overcome by initiating data monitoring after at least ∼40-90 samples had accrued and/or by leaving data blinded. Conclusion/Significance:When there is uncertainty regarding the effect size and/or the population variance, then group sequential tests with data monitoring may offer a solution, providing an efficient means to reach an unambiguous test outcome in terms of effect present/absent.

Beyond a facility: A cross-sectional survey on WASH service levels and informal social accountability in childcare centres in Nairobi's informal settlements

Abstract Access to clean water, sanitation, and hygiene (WASH) services is crucial for a healthy start in life. Social accountability has a potential for enhancing WASH services in childcare centres. However, there are inadequate studies to understand how informal social accountability mechanisms contributes to WASH service provision. To address this gap, we conducted a cross-sectional survey in Korogocho and Viwandani informal settlements in Nairobi, Kenya, to explore the relationship between different levels of WASH services (ranging from basic to limited or nonexistent) and indicators of informal social accountability, including rewards, sanctions, voice, and responsiveness. We employed multinomial regression analysis, utilizing a robust error variance estimator to account for potential biases. Our findings revealed disparities in WASH service provision between the two studied areas, with childcare centres in Korogocho exhibiting higher access to basic WASH services compared to those in Viwandani. Our analysis also highlighted a significant association between informal social accountability mechanisms and the provision of WASH services. Notably, the sanction mechanism exhibited a correlation with all WASH services, suggesting its pivotal role in shaping service delivery outcomes. In light of these findings, it is imperative to prioritize efforts aimed at reinforcing social accountability mechanisms in WASH service delivery frameworks.

On the variance estimator and its bounds in general linear models under linear restrictions

In the article, the error variance estimator σ ^ 2 and its finite sample size bounds are investigated in the framework of a general linear model under linear restrictions (gLMLR) where no distributions are assumed except for some moment conditions and the covariance matrix of the response may be singular, which extends the results in Lindholm and Wahl (2020). In particular, we overcome the difficulty that the matrix of quadratic form in σ ^ 2 is not idempotent. The estimator is obtained by the proposed restricted and proxy-type generalized least square procedure. Meanwhile, the estimable function of the regression coefficient β is estimated, too. Besides, the finite sample properties of these estimators and their consistency are also established. In the end, a simulation study and a real data analysis are done to examine the finite performance of these estimators.

Reply to Comment by W. Knoben and M. Clark on “The Treatment of Uncertainty in Hydrometric Observations: A Probabilistic Description of Streamflow Records”

AbstractIn this Reply, we address the concerns of Knoben and Clark (2023, https://doi.org/10.1029/2022WR034294) or KC23 that “the assumptions needed to effectively use difference‐based variance estimation methods are not always met by hourly streamflow records.” There should be little doubt that the assumptions of our difference‐based estimator will sometimes be violated in hourly streamflow records but the results from de Oliveira and Vrugt (2022, https://doi.org/10.1029/2022wr032263) and confirmed by the findings in our Reply show that such violations are sporadic enough not to affect much the error variance estimates. Snowmelt as pointed out by KC23 (https://doi.org/10.1029/2022WR034294) may not have received sufficient attention in our paper, yet their 365‐day record is simply not long enough to demonstrate bias of our discharge error variance estimates (and their dependence on flow level). This would require analysis of a much longer, multi‐year, streamflow record of a snowmelt‐driven watershed. The snowmelt catchment analyzed in dOV22 (https://doi.org/10.1029/2022wr032263) did not demonstrate bias in the discharge error variance estimates. We also provide additional clarification on the interpretation of the variance estimates obtained with the nonparametric estimator, and discuss the main issues in the test case presented in Knoben and Clark (2023, https://doi.org/10.1029/2022WR034294)).

Estimating and Calibrating Markov Chain Sample Error Variance

Markov chain Monte Carlo (MCMC) methods are a powerful and versatile tool with applications spanning a wide spectrum of fields, including Bayesian inference, computational biology, and physics. One of the key challenges in applying MCMC algorithms is to deal with estimation error. The main result in this article is a closed form, non-asymptotic solution for the sample error variance of a single MCMC  estimate. Importantly, this result assumes that the state-space is finite and discrete. We demonstrate with examples how this result can help estimate and calibrate MCMC estimation error variance in the more general case, when the state-space is continuous and/or unbounded.

A variance-constrained method to encoding–decoding [formula omitted] state estimation for memristive neural networks with energy harvesting sensor

This paper considers the encoding–decoding-based H∞ state estimation issue under variance constraint for delayed memristive neural networks with energy harvesting sensor. In order to improve the transmission security, the encoding–decoding communication mechanism is introduced in the transmission channel, which can encode the transmitted data with limited bits. In addition, the energy level of the energy harvester is described by a random variable that obeys the specific probability distribution, in which the energy harvesting technology is used to provide the required energy and continuously maintain the operation of the sensor. The main objective is to present new encoding–decoding-based H∞ state estimation method such that, in the presence of time-delay and energy harvesting mechanism, the desirable H∞ performance requirement and the estimation error variance constraints are both ensured by providing some sufficient conditions. In the end, the feasibility of new encoding–decoding-based H∞ state estimation algorithm is demonstrated by a simulation experiment.

The Scale of Effective Decision-Making in Sport (SEDMS): Validity and Reliability Study

The aim of this study is to develop a measurement tool that measures the level of influence in decision-making of licensed athletes aged 18 years and above who are active in sports. In the stages of creating the candidate form of the scale, writing essays by athletes, scale literature review, and interviews with focus groups were carried out within the scope of "qualitative steps", while hypothesis analyses and validity and reliability analyses were carried out within the scope of "quantitative steps". The expert opinion form prepared with 42 items in the item pool was presented to the experts for their evaluation according to the Lawshe technique. After the expert feedback, 15 items were removed, and 27 items remained. Trial scale form was administered online to 806 active licensed athletes aged 18 and over. Then, the basic assumptions of EFA were tested within the scope of quantitative steps; because of EFA, a 2 dimensional structure consisting of a total of 15 items was obtained. To examine the psychological construct validity of the SECVS, new data were obtained from 493 active licensed athletes by re-collecting data with the 15-item final form formed after EFA and CFA was performed. After CFA, the standardized loadings of the sub-dimensions, the estimated error variances of the variables, the T-values of all items and the goodness of fit values of the model were examined and it was determined that the 2 factor structure of the scale consisting of 15 items was confirmed as a model. In addition, when the convergent and divergent validity and convergent reliability values of the scale were examined after CFA, it was concluded that this scale had values in accordance with the criteria. Cronbach's alpha internal consistency coefficient was calculated as .87 for the ''External Decision Making'' sub-dimension and .85 for the ''Internal Decision Making'' sub-dimension. It was concluded that the SDEQ is a valid and reliable scale for measuring the decision-making effectiveness of athletes aged 18 and over.

Correcting for measurement error under meta-analysis of z-transformed correlations.

This study mainly concerns correction for measurement error using the meta-analysis of Fisher's z-transformed correlations. The disattenuation formula of Spearman (American Journal of Psychology, 15, 1904, 72) is used to correct for individual raw correlations in primary studies. The corrected raw correlations are then used to obtain the corrected z-transformed correlations. What remains little studied, however, is how to best correct for within-study sampling error variances of corrected z-transformed correlations. We focused on three within-study sampling error variance estimators corrected for measurement error that were proposed in earlier studies and is proposed in the current study: (1) the formula given by Hedges (Test validity, Lawrence Erlbaum, 1988) assuming a linear relationship between corrected and uncorrected z-transformed correlations (linear correction), (2) one derived by the first-order delta method based on the average of corrected z-transformed correlations (stabilized first-order correction), and (3) one derived by the second-order delta method based on the average of corrected z-transformed correlations (stabilized second-order correction). Via a simulation study, we compared performance of these estimators and the sampling error variance estimator uncorrected for measurement error in terms of estimation and inference accuracy of the mean correlation as well as the homogeneity test of effect sizes. In obtaining the corrected z-transformed correlations and within-study sampling error variances, coefficient alpha was used as a common reliability coefficient estimate. The results showed that in terms of the estimated mean correlation, sampling error variances with linear correction, the stabilized first-order and second-order corrections, and no correction performed similarly in general. Furthermore, in terms of the homogeneity test, given a relatively large average sample size and normal true scores, the stabilized first-order and second-order corrections had type I error rates that were generally controlled as well as or better than the other estimators. Overall, stabilized first-order and second-order corrections are recommended when true scores are normal, reliabilities are acceptable, the number of items per psychological scale is relatively large, and the average sample size is relatively large.

Estimation in Semi-Varying Coefficient Heteroscedastic Instrumental Variable Models with Missing Responses

This paper studies the estimation problem for semi-varying coefficient heteroscedastic instrumental variable models with missing responses. First, we propose the adjusted estimators for unknown parameters and smooth functional coefficients utilizing the ordinary profile least square method and instrumental variable adjustment technique with complete data. Second, we present an adjusted estimator of the stochastic error variance by employing the Nadaraya–Watson kernel estimation technique. Third, we apply the inverse probability-weighted method and instrumental variable adjustment technique to construct the adaptive-weighted adjusted estimators for unknown parameters and smooth functional coefficients. The asymptotic properties of our proposed estimators are established under some regularity conditions. Finally, numerous simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed estimators.

SEH: Size Estimate Hedging Scheduling of Queues

For a single server system, Shortest Remaining Processing Time (SRPT) is an optimal size-based policy. In this article, we discuss scheduling a single-server system when exact information about the jobs’ processing times is not available. When the SRPT policy uses estimated processing times, the underestimation of large jobs can significantly degrade performance. We propose an index-based policy with a single parameter, Size Estimate Hedging (SEH), that only uses estimated processing times for scheduling decisions. A job’s priority is increased dynamically according to an SRPT rule until it is determined that it is underestimated, at which time the priority is frozen. Numerical results suggest that SEH has desirable performance for estimation error variance that is consistent with what is seen in practice.

Information theoretic waveform design with applications to adaptive‐on‐transmit radar

AbstractThe marginal Fisher information (MFI) metric is used to design waveforms for the sake of informationally optimal adaptive‐on‐transmit radar operation. A framework for MFI waveform design is developed and the Polyphase‐Coded FM (PCFM) waveform model is utilised to produce a constant‐modulus, spectrally contained signal amenable to transmission with high‐power amplifiers. The efficacy of the MFI waveform design and minimum mean square error (MMSE) estimation is experimentally demonstrated and extended into an adaptive and dynamic sensing paradigm. The radar transmit waveform is optimised to maximise the Fisher information with respect to the range profile. Upon observing new information from radar echoes, the iterative MMSE (iMMSE) estimator then minimises the estimation error variance according to prior observations. Sequential information maximisation (via waveform design) and error minimisation (via iMMSE) tends towards the Cramér–Rao lower bound (CRLB) with additional measurements improving radar resolution and accuracy. These concepts maximise the information extracted by a radar operating in a congested spectrum where the available bandwidth is limited.

Homogenization of daily temperatures using covariates and statistical learning—The case of parallel measurements

AbstractA data driven method based on generalized additive modelling (GAM) has been developed for homogenizing daily minimum and maximum temperature (TN, TX) series using parallel measurements and covariates. The method is applied to two coastal and two inland stations in the Netherlands. Between 1950 and 1972, these stations were relocated from cities to airports, accompanied by parallel measurement of at least 5 years at the old and new sites. Separating these parallel measurements in training and test data, the method compares numerous models involving covariates like the wind vector, cloudiness, specific humidity and sea surface temperature, and selects a model for each station. The resulting models offer an improvement compared to models based on temperature and season only: seasonal dependence is largely replaced by dependence on physical quantities. However, quantitatively, the impact is not large in the cases studied. One of the reasons might be that some covariates have only been measured at specific times not coinciding with the occurrences of the temperature minima or maxima. Additional benefits of the method are robustness and estimation of the sampling error variance of the daily homogenized daily temperature values.

Tight Bounds for Uncertain Time-Correlated Errors With Gauss–Markov Structure in Kalman Filtering

Safety-critical navigation applications require that estimation errors be reliably quantified and bounded. This can be challenging for linear dynamic systems if the process noise or measurement errors have uncertain time correlation. In many systems (e.g., in satellite-based or inertial navigation systems), there are sources of time-correlated sensor errors that can be well modeled using Gauss-Markov processes (GMP). However, uncertainty in the GMP parameters, particularly in the correlation time constant, can cause misleading error bounds. In this paper, we develop time-correlated models that ensure tight upper bounds on the estimation error variance, assuming that the actual error is a stationary first-order GMP with a variance and time constant that are only known to reside within an interval. We first use frequency-domain analysis to derive stationary GMP models in both the continuous and discrete time domains, which outperform models previously described in the literature. Then, we derive an even tighter estimation error bound using a non-stationary GMP model, for which we determine the minimum initial variance that guarantees bounding conditions. Both models can easily be implemented in a linear estimator like the Kalman filter.