Asymptotical problems of sequential interval and point estimation

2020 ◽  
Vol 86 (7) ◽  
pp. 72-80
Author(s):  
A. A. Abdushukurov ◽  
G. G. Rakhimova
Keyword(s):  
Distribution Function ◽  
Confidence Intervals ◽  
Wide Class ◽  
Sequential Analysis ◽  
Interval Estimation ◽  
Sequential Estimation ◽  
Point Estimation ◽  
Minimum Risk ◽  
Unknown Distribution ◽  
Unknown Distribution Function

The accuracy of interval estimation systems is usually measured using interval lengths for given covering probabilities. The confidence intervals are the intervals of a fixed width if the length of the interval is determined, i.e., not random, and tends to zero for a given covering probability. We consider two important directions of statistical analysis -sequential interval estimation with confidence intervals of fixed width and sequential point estimation with asymptotically minimum risk. Two statistical models are used to describe the basis problems of sequential interval estimation by confidence intervals of a fixed width and point estimation. A review of data on nonparametric sequential estimation is carried out and new original results obtained by the authors are presented. Sequential analysis is characterized by the fact that the moment of termination of observations (stopping time) is random and is determined depending on the values of the observed data and on the adopted measure of optimality of the constructed statistical estimate. Therefore, to solve the asymptotic problems of sequential estimation, the methods of summation of random variables are used. To prove the asymptotic consistency of the confidence intervals of a fixed width, we used a method based on application of limit theorems for randomly stopped random processes. General conditions of the consistency and efficiency of sequential interval estimation of a wide class of functionals of an unknown distribution function are obtained and verified by sequential interval estimation of an unknown probability density of asymptotically uncorrelated and linear processes. Conditions of the regularity are specified that provide the property of being an estimate with an asymptotically minimum risk for a wide class of estimates and loss functions. Those conditions are verified by sequential point estimation of an unknown distribution function.

Robust Sequential Confidence Intervals for the Behrens–Fisher Problem*

1971 ◽  
Vol 20 (1-3) ◽  
pp. 77-82 ◽  
Author(s):  
Malay Ghosh
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Confidence Intervals ◽  
Present Note ◽  
Rank Order ◽  
Sample Problem ◽  
Unknown Distribution ◽  
Two Sample Problem ◽  
Unknown Distribution Function ◽  
Rank Order Statistics

Summary The problem of providing a bounded length (sequential) confidence interval for the median of a symmetric (but otherwise unknown) distribution based on a general class of one-sample rank-order statistics was investigated in (Sen & Ghosh, 1971). The purpose of the present note is to indicate how the techniques developed there can be extended to the two-sample problem. It has been shown that in particular for the Behrens- Fisher situation (see e.g., Høyland, 1965 or Ramchandramurty, 1966), when the proposed procedure is based on the “normal-scores” statistic, under very general conditions on the unknown distribution function (d.f.), it is asymptotically at least as efficient as an analogous procedure suggested in (Robbins, Simons & Starr, 1967) and (Srivastava, 1970).

scholarly journals Multistage Estimation of the Rayleigh Distribution Variance

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2084
Author(s):  
Ali Yousef ◽  
Ayman A. Amin ◽  
Emad E. Hassan ◽  
Hosny I. Hamdy
Keyword(s):  
Confidence Interval ◽  
Sample Size ◽  
Interval Estimation ◽  
Sequential Estimation ◽  
Rayleigh Distribution ◽  
Point Estimation ◽  
Asymptotic Results ◽  
Squared Error Loss Function ◽  
Estimation Problems ◽  
Width Confidence Interval

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

Research on the Confidence Interval of Indirect Detection

2011 ◽  
Vol 418-420 ◽  
pp. 532-535
Author(s):  
Hai Bin Chen ◽  
Nan Ge ◽  
Xiao Jun Tong
Keyword(s):  
Confidence Interval ◽  
Confidence Intervals ◽  
Concrete Strength ◽  
Interval Estimation ◽  
Point Estimation ◽  
Indirect Detection ◽  
Detection Methods ◽  
Theoretical Formula ◽  
True Value ◽  
Practical Engineering

Abstract. Using the correlation between the measure value and measured value in the indirect detection, the whole presumption method and theoretical formula of the confidence intervals for measured value are put forward. Based on the different detection methods, the confidence interval of high confidence and high accuracy can be given by the proposed method according to random measurement results. Through the Monte Carlo simulation, using the deducing method and the related theory, it may be concluded that the true value is included within the confidence interval which is obtained by this method. The traditional method can only get the point estimation but not give the confidence intervals in the practical engineering. According to the method, the interval estimation of concrete strength can be give. Moreover, this method is used not only in test concrete strength, especially in the evaluation of earthquake, but also in strength detecting for bridges, the pressure vessel, aircraft wing etc.

scholarly journals Approximation of multivariate distribution functions

2008 ◽  
Vol 58 (5) ◽  
Author(s):  
Margus Pihlak
Keyword(s):  
Distribution Function ◽  
Linear Algebra ◽  
Taylor Expansion ◽  
Simulated Data ◽  
Distribution Functions ◽  
Normal Distribution Function ◽  
Matrix Operation ◽  
Unknown Distribution ◽  
Matrix Integral ◽  
Unknown Distribution Function

AbstractIn the paper the unknown distribution function is approximated with a known distribution function by means of Taylor expansion. For this approximation a new matrix operation — matrix integral — is introduced and studied in [PIHLAK, M.: Matrix integral, Linear Algebra Appl. 388 (2004), 315–325]. The approximation is applied in the bivariate case when the unknown distribution function is approximated with normal distribution function. An example on simulated data is also given.

A Note on Minimum Risk Point Estimation of the Shape Parameter of a Pareto Distribution

1987 ◽  
Vol 36 (1-2) ◽  
pp. 69-78 ◽  
Author(s):  
N. Mukhopadhyay ◽  
M. E. Ekwo
Keyword(s):  
Shape Parameter ◽  
Pareto Distribution ◽  
Numerical Study ◽  
Sequential Estimation ◽  
Estimation Procedure ◽  
Point Estimation ◽  
Minimum Risk ◽  
Squared Error ◽  
Purely Sequential ◽  
Linear Cost

The minimum risk point estimation problem is considered for the shape parameter of a Pareto distribution where the Joss function is taken as squared error plus the linear cost of sampling. A suitable purely sequential procedure is proposed for this problem and the asrmptotic behavior of the “regret” function proposed by Robbins (1959) and many other characteristics are examined. An extensive numerical study is presented in order to look into moderate sample behaviors of the proposed sequential estimation procedure. The procedure is found to be very satisfactory.

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