Problem Of Interval Estimation Research Articles

Two-stage fixed-width and bounded-width confidence interval estimation methodologies for the common correlation in an equi-correlated multivariate normal distribution

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely, fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014, 2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed sample size and two-stage sampling methodologies for bounded-width confidence interval estimation of are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

Estimation and testing for the integer-valued threshold autoregressive models based on negative binomial thinning

To better describe the characteristics of time series of counts such as overdispersion or structural change, in this paper, we redefines the integer-valued threshold autoregressive models based on negative binomial thinning (NBTINAR(1)) under a weaker condition that the expectation of the innovations is finite. Parameters’ point estimation and interval estimation problems are considered. A method to test the nonlinearity of the data is provided. As an illustration, we conduct a simulation study and empirical analysis of Pittsburgh crime data sets.

An easy design for interval observers

ABSTRACT This paper deals with the problem of interval estimation. The proposed approach is based on the use of successive derivatives in continuous time (or forward measurements in discrete time) in an algebraic (or local) interval observer. The main advantage of such an observer is that it relaxes the Metzler assumption widely used in the literature of cooperative interval observers. Moreover, no prior knowledge on the initial condition is needed, and no stability nor convergence issue do appear as the estimation is local. After introducing such an observer for Linear Time Variant (LTV) systems with bounded disturbances, some extensions are proposed for Linear Parameter Varying (LPV) systems as well as for functional systems and systems affected by Unknown Input (UI). Some examples illustrate the theoretical contributions.

Bayesian inference for the common location parameter of several shifted-exponential populations

Consider k random samples which are independently drawn from k shifted-exponential distributions, with respective scale parameters σ 1 , σ 2 , … , σ k and common location parameter θ . On the basis of the given samples and in a Bayesian framework, we address the problem of point and interval estimation of the location parameter θ under the conjugate priors, which are usually proper priors. Moreover, we also address the problem of testing the equality of the location parameters. We propose Bayesian hypothesis testing procedures for the equality of the location parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Our proposed Bayesian procedures are compared and contrasted, via a comparison study, a simulation study, and a real-world data analysis, to the existing classical exact procedures proposed by Tippett (Tippett’s method (Tippett, 1931)), Fisher (Fisher’s method (Fisher, 1932)), Stouffer (Inverse normal method (Stouffer et al., 1949)), and George (Logit method (George, 1977)), and to the generalized variable procedures proposed by Tsui and Weerahandi (Generalized p -value method (Tsui and Weerahandi, 1989)).

Inference for the reliability function based on progressively type II censored data from the Pareto model: The generalized variable approach

The problem of hypothesis testing and interval estimation, based on the generalized variable method, for the reliability function of Pareto distribution using progressively type II censored data with fixed removals are considered. The confidence limits and testing procedures have to be numerically obtained; however, the required computations are simple and straightforward. There are no exact or approximate testing procedures and confidence intervals for reliability parameter of a Pareto distribution based on the progressively type-II right censored data with random or fixed removals available in the literature. In addition, generalized coverage probabilities, generalized size of the test, and adjusted and unadjusted generalized powers of the test are also discussed. Comparison of the reliability function under the classical paradigm and the generalized paradigm is done citing an illustrative example and an example based on the simulated data to demonstrate the advantages and merits of the proposed generalized variable method over the classical method.

Aggregate modeling and analysis of air conditioning load using coupled\xa0Fokker\u2013Planck equations

Demand response has gained significant attention recently with the increasing penetration of renewable energy sources in power systems. Air conditioning loads are typical thermostatically controlled loads which can play an active role in ancillary services by regulating their aggregated power consumption. The aggregation of air conditioners is essential to the control of air conditioning loads. In this paper, linear state equations are proposed to aggregate air conditioning loads by solving coupled Fokker–Planck equations (CFPEs) using the finite difference method. By analyzing the numerical stability and convergence of the difference scheme, the grid spacings, including temperature step and time step, are properly determined according to the maximal principle. Stationary solutions of the CFPEs are obtained by analytical and numerical methods. Furthermore, a classification method using dimension reduction is proposed to deal with the problem of heterogeneous parameters and interval estimation is applied to describe the stochastic behavior of air conditioning loads. The simulation results verify the effectiveness of the proposed methods.

Finding the extreme values of operating characteristics for quality control of construction products

The matter of control planning by variables is essentially the inverse problem of interval estimation. Therefore, when determining control plans (“n” is the sample size, “k” is the eligibility criterion), the results of interval estimation are used. The algorithm of interval estimation, proposed by L.N. Bolshev and E.A. Loginov, presupposes finding the upper and lower bounds of the operational characteristic of quality control, which in turn represents the probability of acceptance of the batch with a quality level not exceeding the consumer risk. In this case, the probability of batch rejection is not more than the manufacturer risk. In this article, an attempt is made to find the extreme values of the operating characteristic in an explicit form. As a result, operative characteristics formulas were obtained in an explicit form, which made it possible to significantly simplify the determination of control plans with bilateral tolerance for the controlled parameter that characterizes a specific product. In construction industry, any sign on which the reliability of the construction product is characterized is implied as a controlled parameter.

Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example

Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example

Three-stage confidence intervals for a linear combination of locations of two negative exponential distributions

Mukhopadhyay and Padmanabhan (Metrika 40:121–128, 1993) considered the construction of fixed-width confidence intervals for the difference of location parameters of two negative exponential distributions via triple sampling when the scale parameters are unknown and unequal. Under the same setting, this paper deals with the problem of fixed-width confidence interval estimation for a linear combination of location parameters, using the above mentioned three-stage procedure.

Threshold autoregression analysis for finite-range time series of counts with an application on measles data

ABSTRACTThis article studies the threshold autoregression analysis for the self-exciting threshold binomial autoregressive processes. Parameters' point estimation and interval estimation problems are considered via the empirical likelihood method. A new algorithm to estimate the threshold value of the threshold model is also given. Simulation study is conducted for the evaluation of the developed approach. An application on measles data is provided to show the applicability of the method.

A Statistical Inference Course Based on p-Values

ABSTRACTIntroductory statistical inference texts and courses treat the point estimation, hypothesis testing, and interval estimation problems separately, with primary emphasis on large-sample approximations. Here, I present an alternative approach to teaching this course, built around p-values, emphasizing provably valid inference for all sample sizes. Details about computation and marginalization are also provided, with several illustrative examples, along with a course outline. Supplementary materials for this article are available online.

Estimation of the time delay of coupling between oscillators from time realizations of oscillation phases for different properties of phase dynamics

The problem of interval estimation of the time delay of the coupling between oscillatory systems from observed time series is considered. It is shown that the known asymptotic estimates based on the empirical model in the form of a system of first-order phase oscillators and the maximum likelihood formalism can lead to false inferences of the value of the time delay in two typical situations: 1) nonlinear low-dimensional systems whose phases are well-determined but, as a result of significant amplitude fluctuations, the phase approximation is insufficient for describing the dynamics and 2) systems whose phases are defined not quite well because of too large amplitude fluctuations. A method for empirical diagnostics of problematic situations and its modification (coarse estimation) providing a low probability of false inferences in these situation are proposed. The efficiency of the diagnostic criterion and coarse estimation suggested is demonstrated on reference systems with different dynamic properties (linear stochastic oscillators, van der Pol oscillators, and Ressler and Lorenz chaotic systems).

On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

ABSTRACTFisher’s “Nile” example is a classic that involves a bivariate random variable (X,Y) having a joint probability density function given by f(x,y;𝜃) = exp(−𝜃x−𝜃−1y), 0<x,y<∞, where 𝜃>0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for 𝜃 with a preassigned confidence coefficient. In problem (i), we develop a purely sequential estimation strategy along with its asymptotic properties. In problem (ii), we determine that a fixed-sample-size estimation strategy will suffice and yet the requisite sample size would have to be found. We have done that both exactly as well as approximately and we report that for all practical purposes the approximations nearly provide the exact sample size whether it is small, moderate, or large. The last problem we address is bounded-accuracy fixed-sample-size estimation of P𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.

Generalized inferential procedures for generalized Lorenz curves under the Pareto distribution

ABSTRACTThis paper considers problems of interval estimation and hypotheses testing for the generalized Lorenz curve under the Pareto distribution. Our approach is based on the concepts of generalized test variables and generalized pivotal quantities. The merits of the proposed procedures are numerically carried out and compared with asymptotic and bootstrap methods. Empirical evidence shows that the coverage accuracy of the proposed confidence intervals and the type I error control of the proposed exact tests are satisfactory. For illustration purposes, a real data set on median income of the 20 occupations in the United States Census of Population is analysed.

Interval estimation for a lower-truncated distribution based on the double Type-II censored sample

ABSTRACTThe interval estimation problem is investigated for the parameters of a general lower truncated distribution under double Type-II censoring scheme. The exact, asymptotic and bootstrap interval estimates are derived for the unknown model parameter and the lower truncated threshold bound. One real-life example and a numerical study are presented to illustrate performance of our methods.

Estimation for exponential distribution based on competing risk middle censored data

For a general censoring scheme called “middle censoring” scheme which was proposed by Jammalamadaka and Mangalam (2003) in nonparametric set up. In this article, point and interval estimation problems are considered for the exponential distribution when the failure data is middle censored with two independent competing failure risks. Different methods are introduced to estimate the unknown model parameters such as maximum likelihood estimation, midpoint approximation, equivalent quantities estimation. The Bayesian estimation is also considered with gamma priors. Two numerical examples are analyzed to show the performance of the proposed methods.

Interval observers for PDEs: approximation approach

The problem of interval state estimation is studied for systems described by parabolic Partial Differential Equations (PDEs). The proposed solution is based on a finite-element approximation of PDE, with posterior design of an interval observer for the obtained ordinary differential equation. The interval inclusion of the state function of PDE is obtained using the estimates on the error of discretization. The results are illustrated by numerical experiments with an academic example.

Inference for Surrogate Endpoint Validation in the Binary Case

Surrogate endpoint validation for a binary surrogate endpoint and a binary true endpoint is investigated using the criteria of proportion explained (PE) and the relative effect (RE). The concepts of generalized confidence intervals and fiducial intervals are used for computing confidence intervals for PE and RE. The numerical results indicate that the proposed confidence intervals are satisfactory in terms of coverage probability, whereas the intervals based on Fieller’s theorem and the delta method fall short in this regard. Our methodology can also be applied to interval estimation problems in a causal inference-based approach to surrogate endpoint validation.

Inference on stress-strength reliability of power distribution

This paper deals with inference for the stress-strength reliability when X and Y are two independent power distribution. The problem of hypothesis testing and interval estimation of the reliability parameter in a stress-strength model is considered. Test and interval estimation procedures based on the generalized variable approach are given. Statistical properties of the generalized variable approach and an asymptotic method are evaluated by Monte Carlo simulation.

Accounting for uncertainty in willingness to pay for environmental benefits

Previous literature on the distribution of willingness to pay has focused on its heterogeneity distribution without addressing exact interval estimation. In this paper we derive and analyze Bayesian confidence sets for quantifying uncertainty in the determination of willingness to pay for carbon dioxide abatement. We use two empirical case studies: household decisions of energy-efficient heating versus insulation, and purchase decisions of ultra-low-emission vehicles. We first show that deriving credible sets using the posterior distribution of the willingness to pay is straightforward in the case of deterministic consumer heterogeneity. However, when using individual estimates, which is the case for the random parameters of the mixed logit model, it is complex to define the distribution of interest for the interval estimation problem. This latter problem is actually more involved than determining the moments of the heterogeneity distribution of the willingness to pay using frequentist econometrics. A solution that we propose is to derive and then summarize the distribution of point estimates of the individual willingness to pay under different loss functions.