Unknown Scale Parameter Research Articles

Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods

The non-Gaussian maximum likelihood estimator is frequently used in GARCH models with the intention of capturing heavy-tailed returns. However, unless the parametric likelihood family contains the true likelihood, the estimator is inconsistent due to density misspecification. To correct this bias, we identify an unknown scale parameter ηf that is critical to the identification for consistency and propose a three-step quasi-maximum likelihood procedure with non-Gaussian likelihood functions. This novel approach is consistent and asymptotically normal under weak moment conditions. Moreover, it achieves better efficiency than the Gaussian alternative, particularly when the innovation error has heavy tails. We also summarize and compare the values of the scale parameter and the asymptotic efficiency for estimators based on different choices of likelihood functions with an increasing level of heaviness in the innovation tails. Numerical studies confirm the advantages of the proposed approach.

More powerful tests for the sign testing problem about gamma scale parameters

For i=1, … , p, let denote independent random samples from gamma distributions with unknown scale parameters θi and known shape parameters ηi. Consider testing H0:θi≤θi0 for some i=1, … , p versus H1:θi>θi0 for all i=1, … , p, where θ10, … , θp0 are fixed constants. For any 0<α<0.4, we construct two new tests that have the same size as the likelihood ratio test (LRT) and are uniformly more powerful than it. Power comparisons of our tests with other tests are given. The proposed tests are intersection–union tests. We apply the results to test the variances of normal distributions and scale parameters of two-parameter exponential distributions. Finally, we illustrate our proposed tests with an example.

On the independence Jeffreys prior for skew-symmetric models

Abstract We study the Jeffreys prior of the skewness parameter of a general class of scalar skew-symmetric models. We show that this prior is symmetric, proper, and with tails O ( | λ | − 3 / 2 ) under mild regularity conditions. We also calculate the independence Jeffreys prior for the case with unknown location and scale parameters, and investigate conditions for the propriety of the corresponding posterior distribution.

Selecting the best of two gamma populations having unequal shape parameters

Consider two independent gamma populations π1 and π2, where the population πi has an unknown scale parameter θi>0 and known shape parameter αi>0,i=1,2. Assume that the correct ordering between θ1 and θ2 is not known a priori and let θ[1]≤θ[2] denote the ordered values of θ1 and θ2. Consider the goal of identifying (or selecting) the population associated with θ[2], under the indifference-zone approach of Bechhofer (1954), when the quality of a selection rule is assessed in terms of the infimum of the probability of correct selection over the preference-zone. Under the decision-theoretic framework this goal is equivalent to that of finding the minimax selection rule when (θ1,θ2) lies in the preference-zone and 0–1 loss function is used (which takes the value 0 if correct selection is made and takes the value 1 if correct selection is not made). Based on independent observations from the two populations, the minimax selection rule is derived. This minimax selection rule is shown to be generalized Bayes and admissible. Finally, using a numerical study, it is shown that the minimax selection rule outperforms various natural selection rules.

Computing tolerance limits for the lifetime of a k-out-of-n:F system based on prior information and censored data

Exact guaranteed-coverage and expected-coverage Bayesian tolerance limits for the lifetime distribution of a k-out-of-n:F system are computed by solving nonlinear equations. The bounds are based on exponential component test data and available prior information concerning the expected component lifetime which is described by an inverted gamma distribution. The Bayesian tolerance limits are valid for single (right or left), double and progressive (standard or general) censoring, and even have frequentist validity in the noninformative case. The derived results allow the reliability engineer to judge the quality of a system prior to assembly, which offers obvious practical and economic benefits. Minimum and expected percentages of conforming systems are assessed by constructing suitable tolerance limits. Even though the viewpoints are different, the Bayesian tolerance limits that adopt the natural diffuse prior coincide numerically with recently published conditional tolerance limits in the double censoring case. The proposed Bayesian approach may be deemed as an extension of the existing frequentist methodology under double censoring that also takes into account the presence of prior information and general progressive censoring. The perspective developed simplifies and unifies the computation of tolerance limits with both frequentist and Bayesian interpretations, and also provides a probabilistic way of updating the tolerance limits in the light of new, relevant data, which is especially important in the dynamic analysis of a sequence of data. Moreover, the Bayesian approach is shown to outperform the frequentist viewpoint in terms of accuracy. In most situations, the use of substantial prior information significantly increases the accuracy level and considerably reduces the required number of failures to attain a specified degree of accuracy. Two illustrative numerical examples are studied, including the analysis of a system of water pumps for cooling a reactor. The results developed are extended to the Weibull case with unknown scale parameter and other probability models.

A Test for an Abrupt Change in Weibull Hazard Functions with Staggered Entry and Type I Censoring

We consider a test for an abrupt change in a Weibull hazard function with unknown scale parameter when the data are subject to staggered entry and Type I censoring. We formulate the profile log-likelihood ratio test statistic as a function of the change point and show that it converges weakly to an Ornstein-Uhlenbeck process. We determine critical values from an approximation to the tail probability of the supremum of the process. We study the power of the test through simulation, and demonstrate its applicability using real data from a clinical study for the treatment of chronic granulomatous disease.

Estimation of the stress–strength reliability for the generalized logistic distribution

Ragab [A. Ragab, Estimation and predictive density for the generalized logistic distribution, Microelectronics and Reliability 31 (1991) 91–95] described the Bayesian and empirical Bayesian methods for estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent random variables from two generalized logistic (GL) distributions having the same known scale but different shape parameters. In this current paper, we consider the estimation of R, when X and Y are both two-parameter GL distribution with the same unknown scale but different shape parameters or with the same unknown shape but different scale parameters. We also consider the general case when the shape and scale parameters are different. The maximum likelihood estimator of R and its asymptotic distribution are obtained and it is used to construct the asymptotic confidence interval of R. We also implement Gibbs and Metropolis samplings to provide a sample-based estimate of R and its associated credible interval. Finally, analyzes of real data set and Monte Carlo simulation are presented for illustrative purposes.

A Bayesian Approach for Estimating Onset Time of Nephropathy for Type 2 Diabetic Patients Under Various Health Conditions

Nephropathy is a life-threatening complication of diabetes mellitus and is the leading cause of end-stage renal disease. The rate of rise in Serum Creatinine (SrCr) is a well-accepted marker for the progression of Diabetic Nephropathy (DN). The objective of this paper is to estimate the DN onset times and for this a retrospective data from 132 type 2 diabetic patients were collected as per American Diabetes Association (ADA) standards. This data is divided into four groups: Less Advance DN (LADN) group (1.4mg/dl $\leq$ SrCr $\leq$ 1.9 mg/dl), Advance DN (ADN) group (SrCr $>$ 1.9 mg/dl), Non-Informative group (SrCr $<$ 1.4 mg/dl \& duration of diabetes $\leq$ 15 years) and Controlled group (SrCr $<$ 1.4 mg/dl \& duration of diabetes $>$15 years). For estimating the time of onset of DN for each group we have applied Bayesian approach and Bayes estimate is obtained under squared error loss function for the unknown scale parameter. We found that patients with DN as complication will reach less advance and advance stages of DN in approximately 15.012 years and 16.890 years respectively. And it was also found that patients with duration of diabetes less than 15 years (non-informative group) will be free from DN for at least 7.781 years. And, patients with duration of diabetes greater than 15 years (controlled group) will not develop any complication up to 20.109 years. This estimation can be used to predict the future onset times of new type 2 diabetic patients.

Comparisons among some predictors of exponential distributions using Pitman closeness

The bias, mean squared error and likelihood optimality criteria are used frequently to compare estimators and predictors. Recently, the probability of nearness around a statistic (estimator or predictor) has received a considerable attention in the literature. In this article, we adopt the Pitman's measure of closeness (PMC) as an optimality criterion to compare the maximum likelihood, best linear unbiased, best linear invariant, median unbiased and conditional median predictors of a future ordered statistic based on a type II censored sample from an exponential distribution with unknown scale parameter. Numerical computations of the PMC for all comparisons among these predictors are performed and presented.

A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Statistical estimation of measured quantities from strongly discretized observations with unknown scale parameter of the random component

The problem of selecting the best statistical estimate of strongly discretized observations is considered for the practical case of an unknown scale parameter of the random component. A method of estimation of measured quantities in conjunction with an unknown scale parameter of the random component that makes it possible to decrease the estimation error by comparison with traditional estimates is proposed.

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $$cX\,{+}\,d$$ , when $$X\sim \varGamma (\alpha ,\eta )$$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.

Robust estimation by expectation maximization algorithm

A mixture of normal distributions is assumed for the observations of a linear model. The first component of the mixture represents the measurements without gross errors, while each of the remaining components gives the distribution for an outlier. Missing data are introduced to deliver the information as to which observation belongs to which component. The unknown location parameters and the unknown scale parameter of the linear model are estimated by the EM algorithm, which is iteratively applied. The E (expectation) step of the algorithm determines the expected value of the likelihood function given the observations and the current estimate of the unknown parameters, while the M (maximization) step computes new estimates by maximizing the expectation of the likelihood function. In comparison to Huber’s M-estimation, the EM algorithm does not only identify outliers by introducing small weights for large residuals but also estimates the outliers. They can be corrected by the parameters of the linear model freed from the distortions by gross errors. Monte Carlo methods with random variates from the normal distribution then give expectations, variances, covariances and confidence regions for functions of the parameters estimated by taking care of the outliers. The method is demonstrated by the analysis of measurements with gross errors of a laser scanner.

척도모수가 미지인 임의중도절단자료의 EDF 통계량을 이용한 지수 검정

수명시간 분석에서 가장 간단하고 또한 자주 이용되는 분포는 지수분포이다. Koziol과 Green (1976)은 Cram<TEX>$\acute{e}$</TEX>r-von Mises 통계량을 Kaplan-Meier의 product limit 경험분포함수를 이용하여 임의중도절단자료에 대해서 일반화하였다. 그러나 이 통계량은 모수의 값이 주어진 단순귀무가설을 가정하고 있으므로 실제 자료에 적용하기에는 어려운 점이 있다. 본 논문에서는 척도모수가 미지인 지수분포의 적합도 검정에 모수를 추정하여 Koziol-Green 통계량을 적용하였다. 그리고 같은 방법으로, 전통적인 Kolmogorov-Smirnov 검정통계량을 일반화하고 두 가지 통계량의 검정력을 모의실험을 통하여 비교하였다. 그 결과 전반적으로 일반화된 Koziol-Green 통계량이 Kolmogorov-Smirnov 통계량보다 지수분포의 검정에 있어서는 좀 더 좋은 검정력을 보여주었다. The simplest and the most important distribution in survival analysis is exponential distribution. Koziol and Green (1976) derived Cram<TEX>$\acute{e}$</TEX>r-von Mises statistic's randomly censored version based on the Kaplan-Meier product limit estimate of the distribution function; however, it could not be practical for a real data set since the statistic is for testing a simple goodness of fit hypothesis. We generalized it to the composite hypothesis for exponentiality with an unknown scale parameter. We also considered the classical Kolmogorov-Smirnov statistic and generalized it by the exact same way. The two statistics are compared through a simulation study. As a result, we can see that the generalized Koziol-Green statistic has better power in most of the alternative distributions considered.

Stein Estimation for Spherically Symmetric Distributions: Recent Developments

This paper reviews advances in Stein-type shrinkage estimation for spherically symmetric distributions. Some emphasis is placed on developing intuition as to why shrinkage should work in location problems whether the underlying population is normal or not. Considerable attention is devoted to generalizing the "Stein lemma" which underlies much of the theoretical development of improved minimax estimation for spherically symmetric distributions. A main focus is on distributional robustness results in cases where a residual vector is available to estimate an unknown scale parameter, and, in particular, in finding estimators which are simultaneously generalized Bayes and minimax over large classes of spherically symmetric distributions. Some attention is also given to the problem of estimating a location vector restricted to lie in a polyhedral cone.

Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

This paper is concerned with time-optimal navigation for flight vehicles in a planar, time-varying flow-field, where the objective is to find the fastest trajectory between initial and final points. The primary contribution of the paper is the observation that in a point-symmetric flow, such as inside vortices or regions of eddie-driven upwelling/downwelling, the rate of the steering angle has to be equal to one-half of the instantaneous vertical vorticity. Consequently, if the vorticity is zero, then the steering angle is constant. The result can be applied to find the time-optimal trajectories in practical control problems, by reducing the infinite-dimensional continuous problem to numerical optimization involving at most two unknown scalar parameters.

STATISTICAL INFERENCE ABOUT AVAILABILITY OF SYSTEM WITH GAMMA LIFETIME AND REPAIR TIME

Availability is an important measure of performance of repairable system. The steady state system availability has special importance since it demonstrates the performance of a system after it has been operated for long. The statistical inference about the steady state availability are particularly useful for practitioners. Much work has been done in this regard. Most of these researches proposed certain pivotal quantities for constructing confidence intervals of the steady state availability. Assuming both the lifetime and repair time follow gamma distribution with known shape parameters and unknown scale parameters, we propose a pivotal quantity for making inferences, and further derive the likelihood ratio tests. Tables of critical values are given for the convenience of applying the two-sided likelihood ratio test. Confidence intervals are also obtained by converting the acceptance regions.

The simultaneous confidence intervals for all distances from the extreme populations for two-parameter exponential populations based on the multiply type II censored samples

Among k independent two-parameter exponential distributions which have the common scale parameter, the lower extreme population (LEP) is the one with the smallest location parameter and the upper extreme population (UEP) is the one with the largest location parameter. Given a multiply type II censored sample from each of these k independent two-parameter exponential distributions, 14 estimators for the unknown location parameters and the common unknown scale parameter are considered. Fourteen simultaneous confidence intervals (SCIs) for all distances from the extreme populations (UEP and LEP) and from the UEP from these k independent exponential distributions under the multiply type II censoring are proposed. The critical values are obtained by the Monte Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration.

Design of progressively censored group sampling plans for Weibull distributions: An optimization problem

Optimization algorithms provides efficient solutions to many statistical problems. Essentially, the design of sampling plans for lot acceptance purposes is an optimization problem with several constraints, usually related to the quality levels required by the producer and the consumer. An optimal acceptance sampling plan is developed in this paper for the Weibull distribution with unknown scale parameter. The proposed plan combines grouping of items, sudden death testing in each group and progressive group removals, and its decision criterion is based on the uniformly most powerful life test. A mixed integer programming problem is first solved for determining the minimum number of failures required and the corresponding acceptance constant. The optimal number of groups is then obtained by minimizing a balanced estimation of the expected test cost. Excellent approximately optimal solutions are also provided in closed-forms. The sampling plan is considerably flexible and allows to save experimental time and cost. In general, our methodology achieves solutions that are quite robust to small variations in the Weibull shape parameter. A numerical example about a manufacturing process of gyroscopes is included for illustration.

Fusion of IMU and Vision for Absolute Scale Estimation in Monocular SLAM

The fusion of inertial and visual data is widely used to improve an object's pose estimation. However, this type of fusion is rarely used to estimate further unknowns in the visual framework. In this paper we present and compare two different approaches to estimate the unknown scale parameter in a monocular SLAM framework. Directly linked to the scale is the estimation of the object's absolute velocity and position in 3D. The first approach is a spline fitting task adapted from Jung and Taylor and the second is an extended Kalman filter. Both methods have been simulated offline on arbitrary camera paths to analyze their behavior and the quality of the resulting scale estimation. We then embedded an online multi rate extended Kalman filter in the Parallel Tracking and Mapping (PTAM) algorithm of Klein and Murray together with an inertial sensor. In this inertial/monocular SLAM framework, we show a real time, robust and fast converging scale estimation. Our approach does not depend on known patterns in the vision part nor a complex temporal synchronization between the visual and inertial sensor.