Case Of Unknown Variance Research Articles

Regression Analysis of Data from a Cluster Sample

Abstract For the case of different regression relationships in different subgroups of a finite population, only part of which are sampled, the extended least squares estimator of any weighted average of the distinct coefficients is derived, under assumptions relating only to the first two moments of the distribution of the coefficients. Under these assumptions, the estimator is shown to be the best linear ξ-unbiased estimator, while under further distributional assumptions, it is also the Bayesian estimator for a quadratic loss function. For the case of unknown variances a method for estimating them from the sample is proposed. The empirical estimator thus obtained is shown to perform well by a simulation comparison with the optimal estimator and with other proposed empirical estimators.

Estimation following sequential tests

SUMMARY A method is given for obtaining confidence intervals following sequential tests. It involves defining an order relation among points on the stopping boundary and computing the probability of a deviation more extreme in this order relation than the observed one. Particular attention is given to the case of a normal mean with known or unknown variance. A comparison with the customary fixed sample size interval based on the same data is given. The purpose of this paper is to describe a method for obtaining confidence intervals following sequential tests. An order relation is defined among the points on the stopping boundary of the test. The confidence limits are determined by finding those values of the unknown parameter for which the probabilities of more extreme deviations in the order relation than the one observed have prescribed values. To facilitate understanding the proposed procedures, most of the paper is restricted to estimating the mean of a normal population with known variance following the class of sequential tests recommended by Armitage (1975) for clinical trials. The case of unknown variance is discussed briefly in ? 4. It is easy to see that the proposed method is valid more generally, although the probability calculations required to implement it depend on the specific parent distribution and stopping rule. A closely related method was proposed by Armitage (1958), who studied the case of binomial data numerically by enumeration of sample paths. Let xl, x2, ... be independent and normally distributed with unknown mean ,u and known variance cr2. Let sn =x1 + .. . + xn, and for given b > 0 consider the stopping rule

Repeated significance tests for a normal mean

SUMMARY Repeated significance tests for a normal mean, with variance known, are studied asymptotically. Approximations are given for the significance level, power and expected sample size. These approximations are shown to be accurate enough for practical numerical purposes over a wide range of parameter values. An analogous test for the case of unknown variance is suggested and an approximation to its significance level obtained.

Repeated Significance Tests for a Normal Mean

SUMMARY Repeated significance tests for a normal mean, with variance known, are studied asymptotically. Approximations are given for the significance level, power and expected sample size. These approximations are shown to be accurate enough for practical numerical purposes over a wide range of parameter values. An analogous test for the case of unknown variance is suggested and an approximation to its significance level obtained.

A Family of Minimax Estimators of the Mean of a Multivariate Normal Distribution

A family of estimators, each of which dominates the one, is given for the problem of simultaneously estimating means of three or more independent normal random variables which have a common unknown variance. Charles Stein [4] established the existence of such estimators (for the case of a known variance) and later, with James [3], exhibited some, both for the case of unknown common variances considered here and for other cases as well. Alam and Thompson [1] have also obtained estimators which dominate the usual one. The class of estimators given in this paper contains those of James and Stein and also those of Alam and Thompson.

A two-stage procedure on comparing several experimental treatments and a control—the common and unknown variance case

This paper introduces a two-stage selection rule to compare several experimental treatments with a control when the variances are common and unknown. The selection rule integrates the indifference zone approach and the subset selection approach in multiple-decision theory. Two mutually exclusive subsets of the parameter space are defined, one is called the preference zone (PZ) and the other, the indifference zone (IZ). The best experimental treatment is defined to be the experimental treatment with the largest population mean. The selection procedure opts to select only the experimental treatment which corresponds to the largest sample mean when the parameters are in the PZ, and selects a subset of the experimental treatments and the control when the parameters fall in the IZ. The concept of a correct decision is defined differently in these two zones. A correct decision in the preference zone (CD1) is defined to be the event that the best experimental treatment is selected. In the indifference zone, a selection is called correct (CD2) if the selected subset contains the best experimental treatment. Theoretical results on the lower bounds for P(CD1) in PZ and P(CD2) in IZ are developed. A table is computed for the implementation of the selection procedure.

Analysis of Outliers with Adjusted Residuals

Many statistical procedures designed to guard against the occurrence of outliers or spurious observations in normal theory are based upon examining the magnitude of the residuals. A major difficulty involved is caused by the fact that the residuals are correlated. It is shown that one way to avoid such difficulty is to adjust the residuals using information from an auxiliary experiment so that the adjusted residuals become uncorrelated. For the problem of making inferences about the unknown mean of a normal population N(μ, σ2) with known σ2, this leads to a set of estimation procedures by which the observation(s) associated with the largest adjusted residual(s) in magnitude will be excluded. Certain properties of the procedures are discussed and exact numerical results are given for the cases of one and two spurious observations. Generalization to the case of unknown variance and to the general linear model is also given.

Analysis of Outliers with Adjusted Residuals

Many statistical procedures designed to guard against the occurrence of outliers or spurious observations in normal theory are based upon examining the magnitude of the residuals. A major difficulty involved is caused by the fact that the residuals are correlated. It is shown that one way to avoid such difficulty is to adjust the residuals using information from an auxiliary experiment so that the adjusted residuals become uncorrelated. For the problem of making inferences about the unknown mean of a normal population N(μ, σ2) with known σ2, this leads to a set of estimation procedures by which the observation(s) associated with the largest adjusted residual(s) in magnitude will be excluded. Certain properties of the procedures are discussed and exact numerical results are given for the cases of one and two spurious observations. Generalization to the case of unknown variance and to the general linear model is also given.