Statistics with Estimated PARAMETERS1 (original) (raw)
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Statistics with Estimated Parameters
2007
This paper studies a general problem of making inferences for functions of two sets of parameters where, when the first set is given, there exists a statistic with a known distribution. We study the distribution of this statistic when the first set of parameters is unknown and is replaced by an estimator. We show that under mild conditions the variance of the statistic is inflated when the unconstrained maximum likelihood estimator (MLE) is used, but deflated when the constrained MLE is used. The results are shown to be useful in hypothesis testing and confidence-interval construction in providing simpler and improved inference methods than do the standard large sample likelihood inference theories. We provide three applications of our theories, namely Box-Cox regression, dynamic regression, and spatial regression, to illustrate the generality and versatility of our results.
This paper studies the general problem of making inferences for a set of parameters θ in the presence of another set of (nuisance) parameters λ, based on the statistic T (y;λ, θ), where y = {y 1 , y 2 , • • • , y n } represents the data,λ is an estimator of λ and the limiting distribution of T (y; λ, θ) is known. We provide general methods for finding the limiting distributions of T (y;λ, θ) whenλ is either a constrained estimator (given θ) or an unconstrained estimator. The methods will facilitate hypothesis testing as well as confidence-interval construction. We also extend the results to the cases where inferences may concern a general function of all parameters (θ and λ) and/or some weakly exogenous variables. Applications of the theories to testing serial correlation in regression models and confidence-interval construction in Box-Cox regressions are given.
Bias reduction in the estimation of parameters of rare events
Theory of Stochastic Processes
In this paper we consider a class of consistent semi-parametric estimators of a positive tail index -i, parametrized in two f.uning or control parameters a and 6. Such mntrol parameters enable us to have access, for an~' available sample, to an estimator of -; with a null dominant component of asymptotic bias, and with a reasonably flat Mean Squared Error pattC!rn, as a function of k, the number of top order statistics considered. Those control parameters depend on a second order parameter p, which needs to be adequately estimated so that we may achieve a high efficiency relatively to the classical Hill estimator. We then obviously need to have access to a larger number of top order statistics than the number needed for optimal estimation thrnugh the Hill estimator.
Some Improved Classes of Estimators using Auxiliary Information
international journal for research in applied science and engineering technology ijraset, 2020
This paper addresses the problem of estimating the population mean using auxiliary information. A class of linear combination of estimators have been proposed including Srivastava and Walsh type estimators for estimating the population mean. The properties of the suggested family have been discussed. Expressions for the bias and mean square error (MSE) of the suggested family have been derived. It has been shown that the proposed class of estimators has minimum mean square of error as compared to various estimators available in the literature of sampling. An empirical study has been also included at the end to support the fact. Keywords: Multiple auxiliary variable, bias, mean square error, efficiency. I. INTRODUCTION In sampling, the use of auxiliary information has been permeated the important role to improve the efficiency of the estimators. It is well known that the use of auxiliary information results in substantial gain in efficiency over the estimators obtained from those which do not use such information. Out of many, ratio, product and regression methods of estimation are good examples in this context. When the correlation between the study variate y and the auxiliary variate x is positive (high), the ratio method of estimation is quite effective. On the other hand if this correlation is negative (high), the product method of estimation envisaged by Robson (1957) and rediscovered by Murthy (1964), can be employed. Estimators using information of the known population mean of an auxiliary variable have generalized to the cases when such information is available for more than one auxiliary variables by several authors like Olkin (1958), Raj (1965), Rao and Mudholkar (1967), Singh (1967), Srivastava (1965) and Shukla (1966) and Agrawal and Panda (1993) etc. This paper deals with the problem of estimating the population mean of the study variable using single auxiliary information and thereafter, the proposed class of estimators has been extended to the use of multiple auxiliary information. Many authors have made use of linear combination of various estimators available in literature, Singh and Solanki (2011) is one example from the list. In this paper, we have suggested an alternative class of estimators using a linear combination of Srivastava and Walsh estimators in section 2. Section 3 deals with the extension of the proposed class of estimators using two auxiliary variables, related bias and mean square error are obtained up to the first order of approximation. Furthermore, section 4 gives the ultimate extension of the proposed class of estimators using multiple auxiliary information along with the bias and minimum mean square error of the proposed one. Theoretical comparisons with some known estimators of the literature like, mean per unit, ratio, product and some special cases of the proposed class of estimators are given under section 5 and 6 respectively. An illustration, to support the theoretical comparisons, is given as an empirical study in section 7.
A note on nonparametric estimations
Asymptotic Methods in Probability and Statistics, 1998
We give an informal explanation with the help of a Taylor expansion about the most important properties of the maximum likelihood estimate in the parametric case. Then an analogous estimate in two nonparametric models, in the estimate of the empirical distribution function from censored data and in the Cox model is investigated. It is shown that an argument very similar to the proof in the parametric case yields analogous properties of the estimates in these cases too. There is an important non-trivial step in the proofs which is discussed in more detail. A double stochastic integral with respect to a standardized empirical process has to be estimated. This corresponds to the estimate of the second term of the Taylor expansion in the parametric case. We think that the method explained in this paper is applicable in several other models.
An Alternative Perspective on Estimators
Lobachevskii Journal of Mathematics, 2021
The present article aims to shows that obtaining the best estimator does not depend on estimators’ terms and functions. To prove this situation, it is shown that there is no difference between the existing estimators and the proposed estimator by proposing new sophisticated estima- tors. Besides, theoretical, graphical, empirical and simulation studies are conducted to confirm the efficiency of these proposed estimators with the help of the method used by Wolter (2007). Also, the MSE values are evaluated by using the power coefficient of the interested estimators. In data sets with the known correlation coefficient, it is possible to determine which power coefficient estimator is better without computing the MSE value.
Evaluating Statistical Hypotheses Using Weakly-Identifiable Estimating Functions
Scandinavian Journal of Statistics, 2013
Many statistical models arising in applications contain non-and weakly-identified parameters. Due to identifiability concerns, tests concerning the parameters of interest may not be able to use conventional theories and it may not be clear how to assess statistical significance. This paper extends the literature by developing a testing procedure that can be used to evaluate hypotheses under non-and weakly-identifiable semiparametric models. The test statistic is constructed from a general estimating function of a finite dimensional parameter model representing the population characteristics of interest, but other characteristics which may be described by infinite dimensional parameters, and viewed as nuisance, are left completely unspecified. We derive the limiting distribution of this statistic and propose theoretically justified resampling approaches to approximate its asymptotic distribution. The methodology's practical utility is illustrated in simulations and an analysis of quality-of-life outcomes from a longitudinal study on breast cancer.
The Indirect Method: Inference Based on Intermediate Statistics?A Synthesis and Examples
Statistical Science, 2004
This paper presents an exposition and synthesis of the theory and some applications of the so-called "indirect" method of inference. These ideas have been exploited in the field of econometrics, but less so in other fields such as biostatistics and epidemiology. In the indirect method, statistical inference is based on an intermediate statistic, which typically follows an asymptotic normal distribution, but is not necessarily a consistent estimator of the parameter of interest. This intermediate statistic can be a naive estimator based on a convenient but misspecified model, a sample moment, or a solution to an estimating equation. We review a procedure of indirect inference based on generalized method of moments, which involves adjusting the naive estimator to be consistent and asymptotically normal. The objective function of this procedure is shown to be interpretable as an 'indirect likelihood' based on the intermediate statistic. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. This method is often more convenient computationally than maximum likelihood estimation when handling such model complexities as random effects and measurement error, for example; and it can also serve as a basis for robust inference and model selection, with less stringent assumptions on the data generating mechanism. Many familiar estimation techniques can be viewed as examples of this approach. We describe applications to measurement error, omitted covariates, and recurrent events. A data set concerning prevention of mammary tumors in rats is analyzed using a Poisson regression model with overdispersion. A second data set from an epidemiological study is analyzed using a logistic regression model with mismeasured covariates. A third data set of exam scores is used to illustrate robust covariance selection in graphical models.
A new approach to the estimator ’ s selection
2017
In the framework of an abstract statistical model we discuss how to use the solution of one estimation problem (Problem A) in order to construct an estimator in another, completely different, Problem B. As a solution of Problem A we understand a data-driven selection from a given family of estimators A(H) = { Âh, h ∈ H } and establishing for the selected estimator so-called oracle inequality. If ĥ ∈ H is the selected parameter and B(H) = { B̂h, h ∈ H } is an estimator’s collection built in Problem B we suggest to use the estimator B̂ĥ. We present very general selection rule led to selector ĥ and find conditions under which the estimator B̂ĥ is reasonable. Our approach is illustrated by several examples related to adaptive estimation. AMS 2000 subject classifications: Primary 60E15; secondary 62G07, 62G08.