Recursive confidence band construction for an unknown distribution function

2014 ◽  
Vol 57 (1) ◽  
pp. 39-51 ◽  
Author(s):  
Seksan Kiatsupaibul ◽  
Anthony J. Hayter
Keyword(s):  
Distribution Function ◽  
Confidence Band ◽  
Unknown Distribution ◽  
Unknown Distribution Function

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.

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.

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).

Semiparametric estimation and inference

2019 ◽  
pp. 139-164
Author(s):  
M. D. Edge
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Permutation Test ◽  
Cumulative Distribution ◽  
Permutation Testing ◽  
Unknown Distribution ◽  
Regression Parameters ◽  
Independent Observations

Nonparametric and semiparametric statistical methods assume models whose properties cannot be described by a finite number of parameters. For example, a linear regression model that assumes that the disturbances are independent draws from an unknown distribution is semiparametric—it includes the intercept and slope as regression parameters but has a nonparametric part, the unknown distribution of the disturbances. Nonparametric and semiparametric methods focus on the empirical distribution function, which, assuming that the data are really independent observations from the same distribution, is a consistent estimator of the true cumulative distribution function. In this chapter, with plug-in estimation and the method of moments, functionals or parameters are estimated by treating the empirical distribution function as if it were the true cumulative distribution function. Such estimators are consistent. To understand the variation of point estimates, bootstrapping is used to resample from the empirical distribution function. For hypothesis testing, one can either use a bootstrap-based confidence interval or conduct a permutation test, which can be designed to test null hypotheses of independence or exchangeability. Resampling methods—including bootstrapping and permutation testing—are flexible and easy to implement with a little programming expertise.

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