Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

scholarly journals Statistical interpretation of sterile neutrino oscillation searches at reactors

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Pilar Coloma ◽  
Patrick Huber ◽  
Thomas Schwetz
Keyword(s):  
Neutrino Oscillation ◽  
Sterile Neutrino ◽  
Statistical Significance ◽  
Statistical Interpretation ◽  
Actual Distribution ◽  
Mixing Angle ◽  
Log Likelihood ◽  
Reactor Neutrino ◽  
Neutrino Experiments ◽  
Experimental Effort

AbstractA considerable experimental effort is currently under way to test the persistent hints for oscillations due to an eV-scale sterile neutrino in the data of various reactor neutrino experiments. The assessment of the statistical significance of these hints is usually based on Wilks’ theorem, whereby the assumption is made that the log-likelihood is $$\chi ^2$$ χ 2 -distributed. However, it is well known that the preconditions for the validity of Wilks’ theorem are not fulfilled for neutrino oscillation experiments. In this work we derive a simple asymptotic form of the actual distribution of the log-likelihood based on reinterpreting the problem as fitting white Gaussian noise. From this formalism we show that, even in the absence of a sterile neutrino, the expectation value for the maximum likelihood estimate of the mixing angle remains non-zero with attendant large values of the log-likelihood. Our analytical results are then confirmed by numerical simulations of a toy reactor experiment. Finally, we apply this framework to the data of the Neutrino-4 experiment and show that the null hypothesis of no-oscillation is rejected at the 2.6 $$\sigma $$ σ level, compared to 3.2 $$\sigma $$ σ obtained under the assumption that Wilks’ theorem applies.

scholarly journals Comparing the State-of-the-Art Efficient Stated Choice Designs Based on Empirical Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Li Tang ◽  
Xia Luo ◽  
Yang Cheng ◽  
Fei Yang ◽  
Bin Ran
Keyword(s):  
Discrete Choice ◽  
State Of The Art ◽  
Choice Model ◽  
Orthogonal Design ◽  
Asymptotic Variance ◽  
Parameter Estimates ◽  
Efficient Design ◽  
Stated Choice ◽  
Reliable Parameter ◽  
Efficient Data

The stated choice (SC) experiment has been generally regarded as an effective method for behavior analysis. Among all the SC experimental design methods, the orthogonal design has been most widely used since it is easy to understand and construct. However, in recent years, a stream of research has put emphasis on the so-called efficient experimental designs rather than keeping the orthogonality of the experiment, as the former is capable of producing more efficient data in the sense that more reliable parameter estimates can be achieved with an equal or lower sample size. This paper provides two state-of-the-art methods called optimal orthogonal choice (OOC) andD-efficient design. More statistically efficient data is expected to be obtained by either maximizing attribute level differences, or minimizing theD-error, a statistic corresponding to the asymptotic variance-covariance (AVC) matrix of the discrete choice model, when using these two methods, respectively. Since comparison and validation in the field of these methods are rarely seen, an empirical study is presented.D-error is chosen as the measure of efficiency. The result shows that both OOC andD-efficient design are more efficient. At last, strength and weakness of orthogonal, OOC, andD-efficient design are summarized.

scholarly journals Sampling Errors in the Measurement of Rainfall Parameters Using the Precipitation Occurrence Sensor System (POSS)

2007 ◽  
Vol 24 (2) ◽  
pp. 125-140 ◽  
Author(s):  
B. E. Sheppard
Keyword(s):  
Doppler Radar ◽  
Statistical Significance ◽  
Time Estimation ◽  
Sensor System ◽  
Parameter Estimates ◽  
Problem Analysis ◽  
Sampling Errors ◽  
Precipitation Occurrence ◽  
Occurrence Type ◽  
Raindrop Size

Abstract The Precipitation Occurrence Sensor System (POSS) is a small Doppler radar originally designed by the Meteorological Service of Canada (MSC) to report the occurrence, type, and intensity of precipitation in automated observing stations. It is also used for real-time estimation of raindrop size distributions (DSDs). From the DSD, various rainfall parameters can be calculated and relationships established, such as between the radar reflectivity factor (Z) and the rainfall rate (R). Earlier work presented first-order estimates of the sampling errors for some POSS rainfall parameter estimates. This work combines a Monte Carlo simulation and “inverse problem” analysis to better estimate errors due to the specific sampling problems of this disdrometer type. The uncertainties are necessary to determine the statistical significance of differences between DSD estimates by the POSS and other collocated disdrometers, or between POSS measurements in different climatologies. Additionally, confidence limits can be assigned to regression coefficients for rainfall parameter relationships determined from POSS estimates. An example is given of the uncertainties in the coefficients of measured Z–R relationships.

The Asymptotic Variance of the ML Estimator of MA(1) Coefficient

1993 ◽  
Vol 9 (3) ◽  
pp. 527-530
Author(s):  
Young-Ho Chang ◽  
Eric Iksoon Im
Keyword(s):  
Asymptotic Variance ◽  
Ml Estimator

Reliability of Pharmacodynamic Analysis by Logistic Regression

2000 ◽  
Vol 92 (4) ◽  
pp. 985-992 ◽  
Author(s):  
Wei Lu ◽  
James M. Bailey
Keyword(s):  
Logistic Regression ◽  
Confidence Intervals ◽  
Coefficient Of Variation ◽  
Parameter Estimates ◽  
True Parameter ◽  
True Value ◽  
Coefficients Of Variation ◽  
Effect Relation ◽  
Log Normal ◽  
Multiple Patients

Background Many pharmacologic studies record data as binary yes-or-no variables, and analysis is performed using logistic regression. This study investigates the accuracy of estimation of the drug concentration associated with a 50% probability of drug effect (C50) and the term describing the steepness of the concentration-effect relation (gamma). Methods The authors developed a technique for simulating pharmacodynamic studies with binary yes-or-no responses. Simulations were conducted assuming either that each data point was derived from the same patient or that data were pooled from multiple patients in a population with log-normal distributions of C50 and gamma. Coefficients of variation were calculated. The authors also determined the percentage of simulations in which the 95% confidence intervals contained the true parameter value. Results The coefficient of variation of parameter estimates decreased with increasing n and gamma. The 95% confidence intervals for C50 estimation contained the true parameter value in more than 90% of the simulations. However, the 95% confidence intervals of gamma did not contain the true value in a substantial number of simulations of data from multiple patients. Conclusion The coefficient of variation of parameter estimates may be as large as 40-50% for small studies (n < or = 20). The 95% confidence intervals of C50 almost always contain the true value, underscoring the need for always reporting confidence intervals. However, when data from multiple patients is naively pooled, the estimates of gamma may be biased, and the 95% confidence intervals may not contain the true value.

Relating Latent Class Assignments to External Variables: Standard Errors for Correct Inference

2014 ◽  
Vol 22 (4) ◽  
pp. 520-540 ◽  
Author(s):  
Zsuzsa Bakk ◽  
Daniel L. Oberski ◽  
Jeroen K. Vermunt
Keyword(s):  
Latent Class ◽  
Asymptotic Variance ◽  
Latent Class Model ◽  
Monte Carlo Study ◽  
Correction Method ◽  
Standard Errors ◽  
Classification Error ◽  
Parameter Estimates ◽  
Science Literature ◽  
External Variables

Latent class analysis is used in the political science literature in both substantive applications and as a tool to estimate measurement error. Many studies in the social and political sciences relate estimated class assignments from a latent class model to external variables. Although common, such a “three-step” procedure effectively ignores classification error in the class assignments; Vermunt (2010, “Latent class modeling with covariates: Two improved three-step approaches,” Political Analysis 18:450–69) showed that this leads to inconsistent parameter estimates and proposed a correction. Although this correction for bias is now implemented in standard software, inconsistency is not the only consequence of classification error. We demonstrate that the correction method introduces an additional source of variance in the estimates, so that standard errors and confidence intervals are overly optimistic when not taking this into account. We derive the asymptotic variance of the third-step estimates of interest, as well as several candidate-corrected sample estimators of the standard errors. These corrected standard error estimators are evaluated using a Monte Carlo study, and we provide practical advice to researchers as to which should be used so that valid inferences can be obtained when relating estimated class membership to external variables.

Optimal Nonlinear Estimation of Linear Stochastic Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 529-536 ◽  
Author(s):  
M. A. Hopkins ◽  
H. F. VanLandingham
Keyword(s):  
Stochastic Systems ◽  
Linear Time ◽  
Minimum Mean Square Error ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Error Estimator ◽  
Nonlinear Method ◽  
True Parameter ◽  
Single Input Single Output ◽  
Persistent Excitation

This paper presents a new nonlinear method of simultaneous parameter and state estimation called pseudo-linear identification (PLID), for stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer function must be strictly proper. Under standard gaussian assumptions, for completely controllable and observable systems, it is proved that PLID is the minimum mean-square-error estimator of the states and model parameters, conditioned on the input and output measurements. It is also proved, given persistent excitation, that the parameter estimates converge a.e. to the true parameter values. All results have been extended to the multiple-input, multiple-output case, but the single-input, single-output case is presented here to simplify notation.

Statistical spatial series modelling II: Some further results on unilateral lattice processes

1983 ◽  
Vol 15 (3) ◽  
pp. 562-584 ◽  
Author(s):  
Dag Tjøstheim
Keyword(s):  
Time Series ◽  
Half Space ◽  
Special Interest ◽  
Asymptotic Theory ◽  
Innovation Process ◽  
Asymptotic Results ◽  
Practical Applications ◽  
Spatial Series ◽  
Lattice Processes ◽  
Series Criteria

An asymptotic theory of estimation is developed for classes of spatial series F(x1, · ··, xn), where (x1, · ··, xn) varies over a regular cartesian lattice. Two classes of unilateral models are studied, namely half-space models and causal (quadrant-type) models. It is shown that a number of asymptotic results are common for these models. Of special interest for practical applications is the problem of determining how many parameters should be included to describe the degree of dependence in each direction. Here we are able to obtain weakly consistent generalizations of familiar time-series criteria under the assumption that the generating variables of the model are independently and identically distributed. For causal models we introduce the concepts of spatial innovation process and lattice martingale and use these to extend some of the asymptotic theory to the case where a certain type of dependence is permitted in the generating variables.

scholarly journals A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

2021 ◽  
Author(s):  
Oliver Lüdtke ◽  
Alexander Robitzsch ◽  
Esther Ulitzsch
Keyword(s):  
Bayesian Approach ◽  
Prior Distribution ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Reciprocal Effects ◽  
Ml Estimation ◽  
Model Parameterization ◽  
Lagged Effects ◽  
The Bayesian Approach ◽  
Over Time

The bivariate Stable Trait, AutoRegressive Trait, and State (STARTS) model provides a general approach for estimating reciprocal effects between constructs over time. However, previous research has shown that this model is difficult to estimate using the maximum likelihood (ML) method (e.g., nonconvergence). In this article, we introduce a Bayesian approach for estimating the bivariate STARTS model and implement it in the software Stan. We discuss issues of model parameterization and show how appropriate prior distributions for model parameters can be selected. Specifically, we propose the four-parameter beta distribution as a flexible prior distribution for the autoregressive and cross-lagged effects. Using a simulation study, we show that the proposed Bayesian approach provides more accurate estimates than ML estimation in challenging data constellations. An example is presented to illustrate how the Bayesian approach can be used to stabilize the parameter estimates of the bivariate STARTS model.

scholarly journals Bound lengths for type-I progressive hybrid burr type-XII censored data under SS-PALT

2021 ◽  
Vol 10 (1) ◽  
pp. 4-22
Author(s):  
Gyan Prakash
Keyword(s):  
Confidence Intervals ◽  
Asymptotic Variance ◽  
Real Data ◽  
Accelerated Life Test ◽  
Type I ◽  
Unknown Parameters ◽  
Data Set ◽  
Ml Estimation ◽  
Hybrid Censoring ◽  
Burr Type Xii

Our main focus on combining two different approaches, Step-Stress Partially Accelerated Life Test and Type-I Progressive Hybrid censoring criteria in the present article. The fruitfulness of this combination has been investigated by bound lengths for unknown parameters of the Burr Type-XII distribution. Approximate confidence intervals, Bootstrap confidence intervals and One-Sample Bayes prediction bound lengths have been obtained under the above scenario. Particular cases of Type-I Progressive Hybrid censoring (Type-I and Progressive Type-II censoring) has also evaluated under SS-PALT. Optimal stress change time also measured by minimizing the asymptotic variance of ML Estimation. A simulation study based on Metropolis-Hastings algorithm have carried out along with a real data set example.

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