Unknown Scale Parameter Research Articles

High breakdown mixture discriminant analysis

Robust S-estimation is proposed for multivariate Gaussian mixture models generalizing the work of Hastie and Tibshirani (J. Roy. Statist. Soc. Ser. B 58 (1996) 155). In the case of Gaussian Mixture models, the unknown location and scale parameters are estimated by the EM algorithm. In the presence of outliers, the maximum likelihood estimators of the unknown parameters are affected, resulting in the misclassification of the observations. The robust S-estimators of the unknown parameters replace the non-robust estimators from M-step of the EM algorithm. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using robust S-estimators as compared to the standard maximum likelihood estimators.

A new method for choosing sample size for confidence interval-based inferences.

Scientists often need to test hypotheses and construct corresponding confidence intervals. In designing a study to test a particular null hypothesis, traditional methods lead to a sample size large enough to provide sufficient statistical power. In contrast, traditional methods based on constructing a confidence interval lead to a sample size likely to control the width of the interval. With either approach, a sample size so large as to waste resources or introduce ethical concerns is undesirable. This work was motivated by the concern that existing sample size methods often make it difficult for scientists to achieve their actual goals. We focus on situations which involve a fixed, unknown scalar parameter representing the true state of nature. The width of the confidence interval is defined as the difference between the (random) upper and lower bounds. An event width is said to occur if the observed confidence interval width is less than a fixed constant chosen a priori. An event validity is said to occur if the parameter of interest is contained between the observed upper and lower confidence interval bounds. An event rejection is said to occur if the confidence interval excludes the null value of the parameter. In our opinion, scientists often implicitly seek to have all three occur: width, validity, and rejection. New results illustrate that neglecting rejection or width (and less so validity) often provides a sample size with a low probability of the simultaneous occurrence of all three events. We recommend considering all three events simultaneously when choosing a criterion for determining a sample size. We provide new theoretical results for any scalar (mean) parameter in a general linear model with Gaussian errors and fixed predictors. Convenient computational forms are included, as well as numerical examples to illustrate our methods.

Non-parametric empirical Bayes procedure

Abstract In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma.

Local asymptotic normality for the scale parameter of stable processes

We consider stable processes given by the density of the Lévy measure αγ| x| −1− γ , where γ∈(0,2) denotes the index which is known and α is the unknown scale parameter. Our aim is to establish local asymptotic normality for the estimation of α, when the process is observed at discrete time points.

Maximum likelihood estimators in regression models with infinite variance innovations

In this paper we consider the problem of maximum likelihood (ML) estimation in the classical AR(1) model with i.i.d. symmetric stable innovations with known characteristic exponent and unknown scale parameter. We present an approach that allows us to investigate the properties of ML estimators without making use of numerical procedures. Finally, we introduce a generalization to the multivariate case.

Convergence rates of MLE in a partly linear model

This paper considers the estimation for a partly linear model with case 1 interval censored data. We assume that the error distribution belongs to a known family of scale distributions with an unknown scale parameter. The sieve maximum likelihood estimator (MLE) for the model's parameter is shown to be strongly consistent, and the convergence rate of the estimator is obtained and discussed.

Quantile estimation of the selected exponential population

Let π 1, π 2,…, π k be k independent exponential populations, where π i has the unknown location parameter ξ i , and the common unknown scale parameter σ. Let X i denote the minimum of a random sample of size n from π i , and X J =max{ X 1,…, X k }. Suppose the population corresponding to X J is selected. The problem of estimating a quantile θ J = ξ J + bσ, b⩾0, of the selected population is considered. The properties of the natural estimators are investigated. We derive a sufficient condition, based on the method of differential inequalities, for an estimator in the class of scale-equivariant estimators to be inadmissible. As a special case, we obtain improved estimators over the natural estimator of θ J , for all values of b⩾0, which is in contrast to the known results for the estimation of θ 1, based on the sample from π 1.

The identification of outliers in exponential samples

In this paper, the task of identifying outliers in exponential samples is treated conceptionally in the sense of DAVIES and GATHER (1989, 1993) by means of a so‐called outlier region. In case of an exponential distribution, an empirical version of such a region – also called an outlier identifier – is mainly dependent on some estimator of the unknown scale parameter. The worst‐case behaviour of several reasonable outlier identifiers is investigated thoroughly and it is shown that only robust estimators of scale should be used to construct reliable identifiers. These findings lead to the recommendation of an outlier identifier that is based on a standardized version of the sample median.

Rationalization of Sequential Monitoring of Reliability

A discussion is given of the problem of increasing the efficiency of monitoring the quality of industrial products for which the moments of occurrence of failures are distributed in accordance with a binomial law or a Weibull law with unknown scale parameters. A mathematical modeling method is used to perform a comparison of the properties of different criteria and a sequential criterion with parabolic boundaries of the acceptance and rejection zones is proposed which is most economic in a number of cases.

Simultaneous Multiple Comparisons with the Worst and Best

SYNOPTIC ABSTRACTSuppose that there are k > 2 independent populations (treatments or systems), where the worth of the i-th population is measured in terms of an unknown location parameter μi, i = 1,… k. For a pre-assigned level P*, we construct 100P*% simultaneous two-sided confidence intervals for parameters minj≠i μj – μi, μi – maxj≠i μj i = 1,…, k, for the location-scale probability models by assuming that the k populations have the same scale parameter θ. We call these comparisons as multiple comparisons with the worst and best. We also construct 100P*% simultaneous upper and two-sided confidence intervals for parameters μ[i] – μ[1], i ≠ 1, μ[k] – μ[i], i ≠ k, where μ[1] < … < μ[k] denote the ranked values of μis. The cases of known and unknown scale parameter θ are dealt with separately. Results are applied to exponential populations model and, for this case, tables for implementation of proposed confidence intervals are provided. Finally, implementation of proposed confidence intervals is illustrated through an example.

On Empirical Bayes Estimators for the Means of the Selected Gamma Populations

SYNOPTIC ABSTRACTLet π1, π2, …, πk be k independent populations, where πi ~ G(θi, pi), the gamma distribution with θi as the unknown scale parameter, and pi, as the known shape parameter. Let {X1,…, Xk} be a random sample, where Xι is from πi. Suppose a subset of the above populations is selected using Gupta's subset selection procedure, according to which πi is selected iff Xi > cX[1], where X[1] = max{Xi,…, Xk}, and c is a suitable constant. In this paper, we obtain the empirical Bayes estimators for the means of the selected populations. For the squared error loss, it is shown that the empirical Bayes estimators are inadmissible and the dominating estimators are also obtained, using the technique of solving differential inequalities.

NATURAL ESTIMATORS FOR THE LARGER OF TWO EXPONENTIAL LOCATION PARAMETERS WITH A COMMON UNKNOWN SCALE PARAMETER

Article NATURAL ESTIMATORS FOR THE LARGER OF TWO EXPONENTIAL LOCATION PARAMETERS WITH A COMMON UNKNOWN SCALE PARAMETER was published on April 1, 2002 in the journal Statistics & Risk Modeling (volume 20, issue 1-4).

Numerical study of the drainage system at a fly ash landfill

Abstract The paper presents an analysis of the drainage system for an existing fly ash landfill formed on the ground surface by a confining embankment. A finite element model of seepage through the embankment was developed. The unknown field scale parameters, found by model calibration, were the values of the hydraulic conductivity of fly ash, the degree of anisotropy of fly ash, and the reduction of hydraulic conductivity of a layer below the landfill, due to the load. Sensitivity analysis showed that the output water levels and the discharges from the drainage system are very sensitive to the fly ash characteristics, while the water levels in the aquifer are sensitive to the characteristics of the layer below the landfill.

APPROXIMATION TO OPTIMAL STOPPING RULES FOR GAMMA RANDOM VARIABLES WITH UNKNOWN LOCATION AND SCALE PARAMETERS

Let X 1, X 2, …, X n ,… be independent, identically distributed gamma random variables with unknown location and scale parameters. If we define the reward sequence Y n = max{X 1,X 2, …, X n } − cn, for c > 0, the optimal stopping rule for Y n depends on the unknown parameters. In this paper, we propose an adaptive stopping rule that does not depend on the parameters and show that the difference between the optimal expected reward and the expected reward using the proposed adaptive stopping rule vanishes as c goes to zero.

EXPERIMENTAL DESIGNS FOR SELECTING THE BETTER OF TWO QUANTAL RESPONSE FUNCTIONS

This paper presents an approach for the selection of the better of two quantal response functions. The procedure assumes that two random variables X and Y are described by logistic models, conditioned on the same continuous random variable Z, have the same but unknown scale parameter and different location parameters. The probability of correctly selecting the random variable with the largest location parameter is estimated and tabulated for several commonly used experimental designs, using computer simulation. The selection rule is based on the maximum likelihood estimators of the location parameters.

Goodness-of-fit tests for the Cauchy distribution

This article presents several modified goodness-of-fit tests for the Cauchy distribution with unknown location and scale parameters. Monte Carlo studies are performed to calculate critical values for several tests based on the empirical distribution function. Power studies suggest that the modified Kuiper (V) test is the most powerful standard test against most alternate distributions over a full range of sample sizes. A reflection technique is also employed which yields substantial improvement in the power of this test against symmetric distributions.

ON THE COMMON SCALE PARAMETER OF SEVERAL PARETO POPULATIONS IN CENSORED SAMPLES

The problem of estimation of an unknown common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters in censored samples is considered. A new class of estimators which includes both the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) is proposed and examined under a squared error loss.

Estimating the True Probability of Correct Selection: Two Exponential Distributions

SYNOPTIC ABSTRACTIndependent random samples of same size are drawn from two exponential populations, having unknown scale parameters. For the goal of selecting the population associated with the larger scale parameter, we consider the natural decision rule, which selects the population corresponding to the larger sample mean, and investigate the problem of estimating the true probability of the correct selection (PCS) using the methods of unbiased estimation and maximum likelihood estimation. The non-existence of the unbiased estimators for the true PCS is established and two approaches to find the maximum likelihood estimators (MLEs) of many-to-one parametric functions are discussed. Using these two approaches, the MLEs for the true PCS are derived and performances of the MLEs are compared, using the Monte Carlo simulations, in terms of the absolute error, the mean squared error and the bias.

Goodness-of-fit test based on empirical characterisitc function

This paper discusses a goodness-of-fit test that uses the integral of the squared modulus of the difference between the empirical characteristic function of the sample data and thecharacteristic function of the hypothesized distribution. Monte Carlo procedures are employed to obtain the empirical percentage points for testing the fit of normal, logistic and exponential distributions with unknown location and scale parameters. Results of Monte Carlo power comparisons with other well-developed goodness-of-fit tests are summarized. Tne proposed test is shown to have superior power for testing the fit of the logistic distibotion (for moderate sample sizes) against a wide range of alternative distributions.

The predictive-sequential goodness-of-fit approach: a study of the exponential distribution case

The predictive-sequential prequential approach is often used to measure the predictive quality of reliability-growth models eg: software-reliability models. Many practitioners use this approach as a goodness-of-fit test whereas no theoretical results justify such a use. This paper presents a theoretical study of the prequential approach in the case of the exponential distribution which is used in many reliability-growth models. We show that the prequential approach leads to a new goodness-of-fit test for the exponential distribution with unknown scale parameter. Monte Carlo simulations are used to compare the power of the prequential test to the standard exponentiality test. The obtained mathematical derivations represent a first step in the theoretical study of the prequential approach as used for validating and comparing reliability-growth models