Independent Normal Random Variables Research Articles

Testing the means of independent normal random variables

Suppose one has available Z 1,…, Z M which are independent and Z i ≈ N( μ i , σ 2 i ) where σ 2 i is known. It is desired to test H 0 : μ i = μ 0, i = 1,…, M. versus H a : μ i ≠ μ 0 for some i, where μ 0 is a known value. Such problems arise in nonparametric component analyses and in time series analysis. A new test, the subset chi-square test method (SCSM), is proposed for this framework. The method examines the sums of squares of all subsets of the normalized random variables. The test is easily calculated and produces a criterion, similar in spirit to Akaike's (1974) AIC, which indicates the means judged to differ from μ 0. This new test is found to compare favorably to existing methods in terms of power.

CONFIDENCE INTERVALS FOR VARIANCE COMPONENTS IN ONE-WAY UNBALANCED DESIGNS

Consider the one way unbalanced components of variance model given by Yij = μ + Ai + Eij, (i = l, ... ,a, j = l, ... ,bi) where μ is an unknown constant parameter, Ai and Eij are independent normal random variables with zero means and variances σ2A and σ2E respectively,

Evaluation of the accuracy of Naslund's approximation for managerial decision making under uncertainty

This paper uses Chance-Constrained Mathematical Programming (CCP) as a tool in project selection when the coefficients involved are independent normal random variables. The main contribution of the paper centers around showing how well chance-constrained programs can be solved using Naslund's approximation instead of other mathematical methods. If a given decision process involves a degree of uncertainty in its parameters and only binary decisions are required, then it is shown that approximation is the only available solution method.

A note on maximum likelihood estimation in the first-order Gaussian moving average model

The likelihoood function of the Gaussian MA(1) zero-mean can be expressed in terms of the variance of the process and the first-order autocorrelation or alternatively in terms of the variance of the unobservable independent normal random variables and the moving average coefficient. The relations between the maximum likelihood estimates of these alternatives pairs are explored. It is noted that in a (finite) sample the maximizing value of the autocorrelation may not correspond to a real value of the moving average coefficient.

Stratified Initial Values and Change Scores

A common question that is encountered in many studies is relating a change in a variable to the variable's initial value. It is well known that regressing the change on the initial value creates a spurious association and misleads the unwary. One attempt to cope with this problem has been to stratify that data based on the initial value. We show that this approach does not solve the problem. Theoretical and simulation results for independent normal and independent exponential random variables are provided.

Comment: On the Relative Stability and Predictability of the Interest Rate and Earnings Growth Rate

Abstract No abstract available.

Moments of variables summing to normal order statistics

We derive means, variances and covariances of components of normal order statistics obtained from a linear combination of several independent normal random variables. This work stems from a problem in hydrology concerning the frequency of extreme lake levels.

Confidence bounds for Pr(X<Y) in 1-way ANOVA random model

Approximate confidence bounds for reliability, R=Pr(X>Y mod X,Y), are obtained, where X and Y are independent normal (Gaussian) random variables, and X and Y are vectors of measurements for X and Y, respectively. Balanced 1-way ANOVA (analysis of variants) random effect models are assumed for the populations of X and Y. Confidence bounds are derived for R under three cases for standard deviations, sigma /sub x/ and sigma /sub y/. An example shows how the results are used. >

A note on the statistical properties of animal locations

A statistical model of the successive locations of an animal in the plane induces a statistical model of the relative positions of successive locations. A common locational model is that the Cartesian coordinates of successive locations in the plane are independent bivariate normal random variables. This note gives the statistical properties of the direction and length of the vector joining successive locations.

Minimax estimation of means of multivariate normal mixtures

Assume X = ( X 1, …, X p )′ is a normal mixture distribution with density w.r.t. Lebesgue measure, f(x)=∫ 1 (2π) p/2δ∣σ∣ 1/2 e −(x−θ)′(σ −1(x−θ))/2σ 2 dF(σ) , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function L Q(θ, a) = ( a − θ)′ Q( a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.

An Ancillarity Paradox Which Appears in Multiple Linear Regression

Consider a multiple linear regression in which $Y_i, i = 1, \cdots, n$, are independent normal variables with variance $\sigma^2$ and $E(Y_i) = \alpha + V'_i\beta$, where $V_i \in \mathbb{R}^r$ and $\beta \in \mathbb{R}^r.$ Let $\hat{\alpha}$ denote the usual least squares estimator of $\alpha$. Suppose that $V_i$ are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix $\theta$. Then $\hat{\alpha}$ is admissible under squared error loss if $r \geq 2$. Several estimators dominating $\hat{\alpha}$ when $r \geq 3$ are presented. Analogous results are presented for the case where $\sigma^2$ or $\theta$ are unknown and some other generalizations are also considered. It is noted that some of these results for $r \geq 3$ appear in earlier papers of Baranchik and of Takada. $\{V_i\}$ are ancillary statistics in the above setting. Hence admissibility of $\hat{\alpha}$ depends on the distribution of the ancillary statistics, since if $\{V_i\}$ is fixed instead of random, then $\hat{\alpha}$ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.

On the Lower Bound of Confidence Coefficients for a Confidence Interval on Variance Components

Consider the one-factor nested components-of-variance model Yij = A + ai + eij, where a2 and eij are jointly independent normal random variables with zero means and variances 2 and a2, respectively, for i = 1, 2, . .. , I; and j = 1, 2, . . ., J. The confidence interval on the among variance component a2 is of interest to the investigators. Except for an artificial method (see Healy, 1961) that uses a set of random numbers which is of no use in practical situations, an exact-size confidence interval on a 2 has not yet been derived. Several approximate procedures are available for obtaining a confidence interval on 2. Satterthwaite (1946) approximates the distribution of the estimate of a linear combination of variances by a chi-squared distribution times a constant. Satterthwaite urges caution in using the method when any of the coefficients of the linear combination are negative, as is the case when estimating a2. Gaylor and Hopper (1969) provide a criterion based on the F-statistics from the ANOVA table for determining when Satterthwaite's procedure may be safely used for setting confidence limits on the among variance components. Welch (1956) offers a series approximation, using the Cornish-Fisher (1937) expansion, for setting approximate confidence intervals on the linear combinations of several variances. Another approximation is given by Moriguti (1954) and repeated by Bulmer (1957). They derive a confidence interval on a2 by assuming the confidence limit is of certain form and solving for it by forcing the confidence coefficient to be exact under certain limiting conditions. Williams (1962) constructed an approximate confidence interval with a guaranteed lower confidence coefficient of 1 2a by a geometrical projection. This confidence interval is identical to the one proposed by Tukey (1951). Howe (1974) also derives confidence limits on a2 through a modified Cornish-Fisher expansion. Boardman (1974) presents a simulation study that investigates the actual probability coverage of the Satterthwaite, Welch, Moriguti-Bulmer, and Tukey-Williams intervals. The results of the study show that the Moriguti-Bulmer and Tukey-Williams procedures give adequate coverage at the nominal 95% confidence coefficient. Both Satterthwaite and Welch procedures behaved poorly in the simulation, giving inadequate coverage at the nominal 95% level. Although Howe's interval was not included in the study, the method

Asymptotic properties for the first-order bilinear time series model

For the first-order bilinear time series model where {et} is a sequence of independent normal random variables with mean 0 and variance σ2, the asymptotic distribution of the sample autocorrelation function is obtained and shown to follow a normal distribution. The variance of the asymptotic distribution is of a complicated form and hence a bootstrap estimate of the variance is proposed for large sample inference. This result can be used to distinguish between different bilinear models. Finally, we obtain moment estimators of the parameters and derive their asymptotic distribution.

Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator

We study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1986) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. (Note that the approximation is carried out on the original probability space, which should be distinguished from the Hungarian embedding approach.) Pointwise strong consistency and a law of iterated logarithm are derived, as well as the mean squared error expression and asymptotic normality, which is obtain using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983).

A unified method to calculate the moments of the least squares estimators in nonlinear regression

A unified method and a set of regularity conditions are presented in this paper to calculate the moments of the least squares estimators, residuals, and fitting errors in nonlinear regression. The results given by Box, Clarke and other authors have been greatly improved and developed. The key point of the method is that we find a series of stochastic expansions related to the estimators. These expansions consist of curvature measures and independent standard normal random variables; therefore they are easy to understand and to deal with from statistical and geometrical point of view.

Stochastic Reduction of Loss in Estimating Normal Means by Isotonic Regression

Consider the problem of estimating the ordered means $\mu_1 \leq \mu_2 \leq \cdots \leq \mu_k$ of independent normal random variables, $Y_1, Y_2, \cdots, Y_k$. It is shown that the absolute error of each component $\hat{\mu}_i$ of the isotonic regression estimator is stochastically smaller than that of the usual estimator $Y_i$. Thus $\hat{\mu}_i$ is superior to $Y_i$ under any nonconstant loss which is a nondecreasing function of absolute error.

On the pitman estimator op ordered normal means

Let be independent normal random variables with means and the common variance unity. It is assumed that We show that the Pitman estimator the generalized Bayes estimator of with respect to the uniform prior on is minimax. When k = 2, the components of for estimating θ1and θ2 are minimax (Cohen and Sackrowitz (1970)). However, for k = 3, we prove that the similar result does not hold for θ1 and θ3 Admississibility of in certain classes of estimators is also discussed.

Bounds on the mean of the maximum of a number of normal random variables

A simple method producing lower and upper bounds on E max(X1,...,Xn) is presented under assumption that the Xi's are independent normal random variables. Furthermore the upper bounds are determined when the Xi's are normal and positively correlated

Homogeneity Tests Against Central-Mixture Alternatives

Abstract A test of homogeneity is derived for a general sampling model. The alternative hypothesis is a mixture over a parameter of the sampling model, centered at the null hypothesis. The test statistic is derived through a score test. Its functional form is independent of the particular mixing distribution. Suppose Yi (i = 1, …, n) are independent random variables with respective probability density (or mass) functions fi (yi ‖ λi). Under the null hypothesis, suppose all λ i homogeneous and equal to the common value λ0. Under the alternative hypothesis, suppose the λ i behave as random samples taken from a distribution with mean λ0 and finite third moment. The score statistic for testing these hypotheses rejects the null for large values of , where is the maximum likelihood estimate of λ0 under the null hypothesis and . When the sample size n is large, S is normally distributed under both the null hypothesis and a sequence of alternative hypotheses in which the variance of the mixing distribution tends to 0 as n −1/2. For example, reject the null hypothesis of Poisson observations yi in favor of a mixture of Poissons for large values of , where . In addition, suppose the Yi are independent normal random variables with respective means α + βxi and variance σ2. Reject the null hypothesis of a constant slope β in favor of a mixture of slopes for large values of , where , and are the usual maximum likelihood estimates of α, β, and σ, respectively.

The law of large numbers for moving averages of independent random variables

The law of large numbers for moving averages of independent random variables