Optimal and Robust Designs for Estimating the Concentration Curve and the AUC

The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with an autocorrelated error process. We introduce a simple linear unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from a sampling design generated by an appropriate density is asymptotically optimal in the sense that it has exactly the same asymptotic performance as the best linear unbiased estimator. Moreover, we prove that the optimal design is robust with respect to a minimax criterion. When repeated observations are available, this estimator is consistent and has an asymptotic normal distribution. Finally, a simulated annealing algorithm is applied to a pharmacokinetic model with correlated errors.     

The precision of an estimator due to ryan

Ryan (1974) presents a method of estimating the mean of a symmetric distribution when observations are recorded in classes of common width h, and when some of the observations may be repeats of earlier observations (the extent of which is unknown). (In other words, if a class contains two observations, the investigator is unsure as to whether they arose from two distinct subjects or the same subject twice.) Ryan’s results, as presented, do not permit one to construct asymptotic confidence intervals for the mean. In this note, we solve this problem.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Operational ridge regression estimators under the prediction goal

In this paper we study the Mean Square Error and Conditional Mean Forecasting of Operational Ordinary Ridge Estimator. We use the G( ) functions to provide both the exact and the approximate bias and Mean Square Error of ordinary ridge estimator (ORE), We show, among other things, that ORE dominates OLS up to a certain order of approximation under the conditional mean forecasting sense.     

TESTING NEW BETTER THAN RENEWAL USED LIFE DISTRIBUTIONS BASED ON U-TEST

In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.     

On the exact distribution of the likelihood ratio criterion for testing

The exact null distribution of the likelihood ratio criter- 2 ion for testing the hypothesis H: y = y~; z = a I, a unknown and UQ a given known vector against the alternative A =f H in a p-vari- ate normal population N (y,z) has been derived in the form of Meijer's G-function using mellin integral transform and also in a chisquare series form. Asymptotic behavior of the distribution of -2 log L has also been discussed. Percentage points for p=2(l)10for various level of significance and various degrees of freedom have been computed, but only selected tables have been presented in this paper.     

ON ASYMPTOTIC PROPERTIES OF A TWO DIMENSIONAL FREQUENCY ESTIMATOR

Estimating parameters of a two dimensional frequency model is an important problem in statistical signal processing. In this paper, we consider the two-dimensional frequency model in presence of an additive stationary noise. We consider two different estimators and obtain their asymptotic properties. The asymptotic properties can be used to construct confidence intervals of the unknown parameters and for testing purposes also. The small sample performances of these estimators are observed using numerical simulations.     

Partially-adaptive robust estimators of location via exponential embedding

A general method is presented for constructing a location estimator which is asymptotically efficient at any two different location-scale families of symmetric distributions as well as at an appropriately defined class of distributions lying in between. The method works by embedding the two families in a comprehensive parametric model and identifying the estimator with the MLE. The case when the families are Normal and Double exponential is examined in detail.     

On two asymptotic normal distributions for the generalized wilks lambda statistic

In this paper we.present a Normal asymptotic distribution for the logarithm of the generalized Wilks Lambda statistic based on an asymptotic distribution for the determinant of a Wishart matrix. This distribution is obtained through the combined use of Taylor expansions of random variables whose exponentials have chi-square distributions and the Lindeberg-Feller version of the Central Limit Theorem, Another asymptotic Normal distribution for the logarithm of the generalized Wilks Lambda statistic for the case when at most one of the sets has an odd number of variables is derived directly from the exact distribution. Both distributions are non-degenerate and non-singular. The first Normal distribution compares favorably with other known approximations and asymptotic distributions namely for large numbers of variables and small sample sizes, while the second Normal distribution, which has a more restricted application, compares in most cases highly favorably with other known asymptotic distributions and approximations. Finally, a method to compute approximate quantiles which lay very close and converge steadily to the exact ones is presented.     

Repeated chi-square testing

A sequentialized version of the x²; goodness of fit test, called repeated x,²; test, is introduced. The form of the asymptotic distribution of the repeated x² test statistic is given under the null hypothesis as well as under local alternatives. For various numbers of cells Monte Carlo results are given for critical values, power and distribution of stopping time. Finally, the perfor-mance of the repeated and the fixed sample x² test are compared.