Bounds on the effect of heteroscedasticity on the chow test for structural change

This paper considers the effect of heteroscedastic regression errors on the size of the Chow test for structural stability. We show that bounds can be placed on the true size of this test in the light of such misspecification, and on the true critical value needed to achieve any desired significance level when using the test under various degrees of heteroscedasticity. These bounds are data-independent, and some cases are tabulated. Examples are given to illustrate the practical application of the critical value bounds.     

Micro-Level Estimation of Optimal Consumption Choice With Intertemporal Nonseparability in Preferences and Measurement Errors

This article investigates the presence of habit formation in household consumption, using data from the Panel Study of Income Dynamics. We develop an econometric model of internal habit formation of the multiplicative specification. The restrictions of the model allow for classical measurement errors in consumption without parametric assumptions on the distribution of measurement errors. We estimate the parameters by nonlinear generalized method of moments and find that habit formation is an important determinant of household food-consumption patterns. Using the parameter estimates, we develop bounds for the expectation of the implied heterogenous intertemporal elasticity of substitution and relative risk aversion that account for measurement errors, and compute confidence intervals for these bounds. Supplementary materials for this article are available online.     

Inequalities for Expected Current Record Statistics

In this article, we obtain sharp distribution-free bounds for the expected value of the gap between the current records and record values as well as upper sharp bounds for the spacings between any two upper current records. We also present two-sided bounds on the errors in approximating the means of current records by inverse hazard functions.     

Testing process capability based on Cpm in the presence of random measurement errors

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process performance. The index C_p provides measures on process precision (or product consistency). The index C_pm, sometimes called the Taguchi index, meditates on process centring ability and process loss. Most research work related to C_p and C_pm assumes no gauge measurement errors. This assumption insufficiently reflects real situations even with highly advanced measuring instruments. Conclusions drawn from process capability analysis are therefore unreliable and misleading. In this paper, we conduct sensitivity investigation on process capability C_p and C_pm in the presence of gauge measurement errors. Due to the randomness of variations in the data, we consider capability testing for C_p and C_pm to obtain lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that the estimator with sample data contaminated by the measurement errors severely underestimates the true capability, resulting in imperceptible smaller test power. To obtain the true process capability, adjusted confidence bounds and critical values are presented to practitioners for their factory applications.     

Probabilistic existence theorems in group testing

For a wide range of combinatorial group testing problems including additive, binary and multiaccess channel models, a probabilistic method is developed to derive upper bounds for the length of optimal nonsequential designs. A general result is proven allowing in many particular cases to compute the asymptotic bounds. The existence theorems are also derived for the situation when several errors in the test results can occur (searching with lies) and for the group testing problem in the binomial sample.     

IMPROVING THE NUMERICAL TECHNIQUE FOR COMPUTING THE ACCUMULATED DISTRIBUTION OF A QUADRATIC FORM IN NORMAL VARIABLES

This paper is concerned with the technique of numerically evaluating the cumulative distribution function of a quadratic form in normal variables. The efficiency of two new truncation bounds and all existing truncation bounds are investigated. We also find that the suggestion in the literature for further splitting truncation errors might reduce computational efficiency, and the optimum splitting rate could be different in different situations. A practical solution is provided. The paper also discusses a modified secant algorithm for finding the critical value of the distribution at any given significance level.     

IMPROVING THE NUMERICAL TECHNIQUE FOR COMPUTING THE ACCUMULATED DISTRIBUTION OF A QUADRATIC FORM IN NORMAL VARIABLES

ABSTRACT

This paper is concerned with the technique of numerically evaluating the cumulative distribution function of a quadratic form in normal variables. The efficiency of two new truncation bounds and all existing truncation bounds are investigated. We also find that the suggestion in the literature for further splitting truncation errors might reduce computational efficiency, and the optimum splitting rate could be different in different situations. A practical solution is provided. The paper also discusses a modified secant algorithm for finding the critical value of the distribution at any given significance level.     

Analysis of least squares regression estimates in case of additional errors in the variables

Estimation of a regression function from independent and identical distributed data is considered. The L₂ error with integration with respect to the design measure is used as error criterion. Upper bounds on the L₂ error of least squares regression estimates are presented, which bound the error of the estimate in case that in the sample given to the estimate the values of the independent and the dependent variables are pertubated by some arbitrary procedure. The bounds are applied to analyze regression-based Monte Carlo methods for pricing American options in case of errors in modelling the price process.     

Truncation bounds for infinite expansions for the stable distributions

A number of authors have presented tabulations for the stable distributions based on infinite expansions for the density functions. In this paper we derive exact bounds for the truncation errors in these expansions and use the results to comment on some of the problems that have arisen in tabulating the stable distribution functions. The derivation of the truncation bounds relies on a little-known result for complex argument Taylor series due to Darboux (1876) which IS of much wider applicability than the present context.     

Error analysis of stochastic flight trajectory prediction models

This paper focuses on the analysis of errors between a flight trajectory prediction model and flight data. A novel stochastic prediction flight model is compared with the popular fly-by and fly-over turn models. The propagated error is measured using either spatial coordinates or angles. Depending on the case, the distribution of error is estimated and confidence bounds for the linear and directional mean are provided for all three stochastic flight models.     

Note on a linear model with errors in the variables and with trend

The least squares estimate of the slope parameter of a simple linear model with errors in the variables is typically biased. However the bias vanishes asymptotically for increasing sample size if the regressor variable follows a linear trend. For this case asymptotic expansion formulas for bias and variance of the least squares estimator are derived from exact expressions presented by Richardson and Wu (1970) and certain bounds to these expressions given by Friedmann (1990).     

An efficient algorithm for estimating the parameters of superimposed exponential signals

An efficient computational algorithm is proposed for estimating the parameters of undamped exponential signals, when the parameters are complex valued. Such data arise in several areas of applications including telecommunications, radio location of objects, seismic signal processing and computer assisted medical diagnostics. It is observed that the proposed estimators are consistent and the dispersion matrix of these estimators is asymptotically the same as that of the least squares estimators. Moreover, the asymptotic variances of the proposed estimators attain the Cramer–Rao lower bounds, when the errors are Gaussian.     

Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

Measuring Measurement Error in Economic Time Series

A spectral decomposition method is described for obtaining an upper bound on the amount of measurement error in a time series. The method is applied to generated data and to M1b, real GNP, and the CPI. The bounds provide insight into both the amount of measurement error in these series and the stochastic specification of the errors.     

Asymptotic Bias in Parameter Estimation of AR-Processes Using Recursive Least Squares with Exponential Forgetting

The recursive least squares technique is often extended with exponential forgetting as a tool for parameter estimation in time-varying systems. The distribution of the resulting parameter estimates is, however, unknown when the forgetting factor is less than one. In this paper an approximative expression for bias of the recursively obtained parameter estimates in a time-invariant AR( n_a ) process with arbitrary noise is given, showing that the bias is non-zero and giving bounds on the approximation errors. Simulations confirm the approximation expressions.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Exceedance probabilities for parametric control charts

Common control charts assume normality and known parameters. Quite often, these assumptions are not valid and large relative errors result in the usual performance characteristics such as the false alarm rate or the average run length. A fully nonparametric approach can form an attractive alternative but requires more Phase I observations than usually available. Sufficiently general parametric families then provide realistic intermediate models. In this article, the performance of charts based on such families is considered. Exceedance probabilities of the resulting stochastic performance characteristics during in-control are studied. Corrections are derived to ensure that such probabilities stay within prescribed bounds. Attention is also devoted to the impact of the corrections for an out-of-control process. Simulations are presented both to illustrate and to demonstrate that the approximations obtained are sufficiently accurate for practical usage.     

Bounds on Sample Variation Measures Based on Majorization

Using majorization theory, upper and lower bounds are derived for different measures of variation as progressively more items of information are available about the sample data. As a convenient starting point, bounds are first established for a one-parameter family of variation measures, which is a generalized mean difference measure of which Gini's mean difference, the standard deviation, and the range are particular cases. While, as pointed out, some of the derived bounds are well known, others do not appear to have been published and are tighter than established bounds. Some 40 different bounds are derived, besides any number of bounds given for the generalized family of variation measures. A number of interesting inequalities are also derived on the basis of some of the bounds. While the bounds have been developed in terms of real-valued sample data generally, the paper concludes with a brief discussion of the bounds for categorical data when the sample data consists of frequencies (counts).     

Counting by weighing: construction of two-sided confidence intervals

Counting by weighing is widely used in industry and often more efficient than counting manually which is time consuming and prone to human errors especially when the number of items is large. Lower confidence bounds on the numbers of items in infinitely many future bags based on the weights of the bags have been proposed recently in Liu et al. [Counting by weighing: Know your numbers with confidence, J. Roy. Statist. Soc. Ser. C 65(4) (2016), pp. 641–648]. These confidence bounds are constructed using the data from one calibration experiment and for different parameters (or numbers), but have the frequency interpretation similar to a usual confidence set for one parameter only. In this paper, the more challenging problem of constructing two-sided confidence intervals is studied. A simulation-based method for computing the critical constant is proposed. This method is proven to give the required critical constant when the number of simulations goes to infinity, and shown to be easily implemented on an ordinary computer to compute the critical constant accurately and quickly. The methodology is illustrated with a real data example.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.