Location-scale Research Articles

Asymptotic analysis of minimum volume confidence regions for location-scale families

An asymptotic analysis, when the sample size $n$ tends to infinity, of the optimal confidence region established in Czarnowska and Nagaev (2001) is considered. As a result, two confidence regions, both close to the optimal one when $n$ is sufficiently lar

On optimal schemes in progressive censoring

Optimal censoring schemes in the model of progressive type II censoring are obtained for a location–scale family of distributions which includes exponential, uniform and Pareto distributions. In the one-parameter set-up the variance of the respective BLUE is used as an optimality criterion. In the two-parameter situation two criteria are applied which are based on the covariance matrix of the corresponding BLUEs, namely the trace and the determinant.

Median Moments with Application to Location-Scale Family of Distributions

Median Moments with Application to Location-Scale Family of Distributions

Tests for location-scale families based on the empirical characteristic function

A class of procedures is presented for using random samples to test the fit of location-scale families—distributions F(·;θ1,θ2) such that Z=(X−θ1)/θ2 has distribution Open image in new window Working with empirically standardized data, Open image in new window the test statistic is a measure of Open image in new window distance between the empirical characteristic function, Open image in new windowOpen image in new window and the c.f. of Z under the null hypothesis, ϕ0(t). The closed-form test statistic is derived by integrating over Open image in new window the product of a weight function times Open image in new window Using as weight function for each location-scale family the squared modulus of ϕ0 itself presents a unified test procedure. Included as special cases are well-known tests for normal and Cauchy laws. Small-sample powers are compared with those of Anderson-Darling tests for each of seven univariate location-scale families.

Preferences over Meyer's Location-Scale Family

This paper extends Meyer's (1987) location-scale family with general n random seed sources. Firstly, we clarify and generalize existing results to this multivariate setting. Some useful geometrical and topological properties of the location-scale expected utility functions are obtained. Secondly, we introduce and study some general non-expected utility functions defined over the location-scale (LS) family. Special care is made in characterizing the shape of the indifference curves induced by the LS expected utility functions and non-expected utility functions. Finally, efforts are also made to study several well-defined partial orders and dominance relations defined over the LS family. These include the first-, second- order stochastic dominance, the mean-variance rule, and a newly defined location-scale dominance.

MLE of parameters of location-scale distribution for complete and partially grouped data

This article studies the MLEs of parameters of location-scale distribution functions. It gives the necessary and sufficient conditions under which the MLEs of the location and scale parameters uniquely exist with completely grouped data. The results for the completely grouped data further imply that the Pearson–Fisher test is applicable to location-scale families. It also gives sufficient conditions under which the MLEs of the two parameters uniquely exist with partially grouped data. Here, the partially grouped data include complete data, Type-I censored data and others as special cases.

A One Sided Test Based on Sample Quasi Ranges

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F i (x) = F[(x − μ i )/θ i ], i = 1,…, k (k ≥ 2). Here μ i (−∞ < μ i < ∞) is the location parameter, θ i (θ i > 0) is the scale parameter and F(⋅) is any absolutely continuous cdf, i.e., F i (⋅) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H 0 ≔ θ1 = · = θ k against the simple ordered alternative H A ≔ θ1 ≤ · ≤ θ k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X i1,…, X in be a random sample of size n from the population π i and R ir = X i:n−r − X i:r+1, r = 0, 1,…, [n/2] − 1 be the sample quasi range corresponding to this random sample, where X i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W r = max1≤i<j≤k (R jr /R ir ). The test is reject H 0 for large values of W r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ j /θ i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.

Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances

Weighted L2 functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantile process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in Rn. The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality.

Selection of Good Populations and Related Confidence Intervals Using Sample Quasi-Ranges

SYNOPTIC ABSTRACTWe consider k (k≥ 2) independent populations or treatments such that an observation from population πi follows an absolutely continuous distribution which is a member of a location-scale family with location parameter μi (−∞ <μi < ∞) and scale parameter θi (θi > 0), i = l,…,k. For a given δ1 (δ1 > 1) the population πi is considered ‘good’ if θi ≤δ1θ[1], i = l,…,k, where θ[1] = min (θ1,…,θk). A class of selection procedures, based on sample quasi-ranges, is proposed to select a subset of k populations which includes all the good populations with probability at least P* (a pre-assigned value). Simultaneous confidence intervals for the ratios of scale parameters, which can be derived with the help of proposed procedures, are discussed. We call population πi, ‘bad’ if θi > δ2θ[1] (δ2 > δ1, > 1), i = l,…,k. A class of selection procedures is also proposed to guarantee that the probability of either selecting a bad population or omitting a good population is at most 1- P*. The application of the proposed procedures to uniform and two-parameter exponential probability models is discussed, separately, with necessary tables. Finally, a numerical example, based on real life data, is given.

Some multiple comparisons using sample quasi ranges on censored data

We consider $k(k \ge 2)$ independent populations (treatments or systems) and an absolutely continuous member of location-scale family of distributions, index by the location parameter $\mu _i ({ - \infty 0}), $ is used to model the observations from the ith population, $ i = 1,...,k .$ The problem of simultaneous selection of two subsets, one containing population associated with the smallest $ \theta $ - value and other containing population with the largest $ \theta $ - value with probability at least a pre-specified value $P^*({\frac{1}{{k(k-1)}} < P^* < 1}) $ is considered when the data are censored. We also construct $100P^*\%$ simultaneous upper and two-sided confidence intervals for $ \frac{{\theta _{[i]}}}{{\theta _[1]}}},i = 2,...,k,\frac{{\theta _{[k]} }}{{\theta _{[i]}}},i = 1,...,k - 1, $ where $ \theta _{[1]} \le .... \le \theta _{[k]} $ denotes the ordered values of $\theta s.$ The proposed procedures, based on sample quasi ranges, are useful when the experimenter has smaller samples or censored samples or there is suspicion of outliers in the samples. The results are applied to exponential populations model and, for this case: (i) the constants have been computed to apply the proposed multiple comparisons; (ii) two members of the proposed class have been compared with the existing procedure. A numerical example is also given.

Goodness-of-Fit Tests Based on Spacings for Progressively Type-II Censored Data From a General Location-Scale Distribution

There has been extensive research on goodness-of-fit procedures for testing whether or not a sample comes from a specified distribution. These goodness-of-fit tests range from graphical techniques, to tests which exploit characterization results for the specified underlying model. In this article, we propose a goodness-of-fit test for the location-scale family based on progressively Type-II censored data. The test statistic is based on sample spacings, and generalizes a test procedure proposed by Tiku . The null distribution of the test statistic is shown to be approximated closely by a s-normal distribution. However, in certain situations it would be better to use simulated critical values instead of the s-normal approximation. We examine the performance of this test for the s-normal and extreme-value (Gumbel) models against different alternatives through Monte Carlo simulations. We also discuss two methods of power approximation based on s-normality, and compare the results with those obtained by simulation. Results of the simulation study for a wide range of sample sizes, censoring schemes, and different alternatives reveal that the proposed test has good power properties in detecting departures from the s-normal and Gumbel distributions. Finally, we illustrate the method proposed here using real data from a life-testing experiment. It is important to mention here that this test can be extended to multi-sample situations in a manner similar to that of Balakrishnan et al.

An asymptotic approach to progressive censoring

Progressive Type-II censoring was introduced by Cohen (Technometrics 5(1963) 327) and has been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2 × 2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.

A note on the sampling properties of the Vincentizing (quantile averaging) procedure

To assess the effect of a manipulation on a response time distribution, psychologists often use Vincentizing or quantile averaging to construct group or “average” distributions. We provide a theorem characterizing the large sample properties of the averaged quantiles when the individual RT distributions all belong to the same location-scale family. We then apply the theorem to estimating parameters for the quantile-averaged distributions. From the theorem, it is shown that parameters of the group distribution can be estimated by generalized least squares. This method provides accurate estimates of standard errors of parameters and can therefore be used in formal inference. The method is benchmarked in a small simulation study against both a maximum likelihood method and an ordinary least-squares method. Generalized least squares essentially is the only method based on the averaged quantiles that is both unbiased and provides accurate estimates of parameter standard errors. It is also proved that for location-scale families, performing generalized least squares on quantile averages is formally equivalent to averaging parameter estimates from generalized least squares performed on individuals. A limitation on the method is that individual RT distributions must be members of the same location-scale family.

A note on the sampling properties of the Vincentizing (quantile averaging) procedure

To assess the effect of a manipulation on a response time distribution, psychologists often use Vincentizing or quantile averaging to construct group or “average” distributions. We provide a theorem characterizing the large sample properties of the averaged quantiles when the individual RT distributions all belong to the same location-scale family. We then apply the theorem to estimating parameters for the quantile-averaged distributions. From the theorem, it is shown that parameters of the group distribution can be estimated by generalized least squares. This method provides accurate estimates of standard errors of parameters and can therefore be used in formal inference. The method is benchmarked in a small simulation study against both a maximum likelihood method and an ordinary least-squares method. Generalized least squares essentially is the only method based on the averaged quantiles that is both unbiased and provides accurate estimates of parameter standard errors. It is also proved that for location-scale families, performing generalized least squares on quantile averages is formally equivalent to averaging parameter estimates from generalized least squares performed on individuals. A limitation on the method is that individual RT distributions must be members of the same location-scale family.

Minimaxity in Estimation of Restricted Parameters

This paper is concerned with estimation of the restricted parameters in location and/or scale families from a decision-theoretic point of view. A simple method is provided to show the minimaxity of the best equivariant and unrestricted estimators. This is based on a modification of the known method of Girshick and Savage (1951) and can be applied to more complicated cases of restriction in the location-scale family. Classes of minimax estimators are also constructed by using the IERD method of Kubokawa (1994a,b): Especially, the paper succeeds in constructing such a class for estimating a restricted mean in a normal distribution with an unknown variance.

Estimating the parameters of the normal, exponential and gamma distributions using median and extreme ranked set samples

In this paper, we propose maximum likelihood estimators (mle’s) as well as linear un-biased estimators (lue’s) of the parameters of the normal, exponential and gamma distri-butions in the light of the location-scale family of distributions - i.e. distributions with cumulative distribution functions of the form F ((x – µ)/?), using median ranked set sam-pling (MRSS) and extreme ranked set sampling (ERSS). MRSS and ERSS are modifica-tions of ranked set sampling (RSS), which are more practicable and less prone to prob-lems resulting from erroneous ranking. The mle’s of the normal mean and the scale pa-rameters of the exponential and gamma distributions under MRSS are shown to dominate all other estimators, while the mle of the normal standard deviation under ERSS is the most efficient. A similar trend is observed in the lue’s. A modification of ERSS namely partial extreme ranked set sampling (PERSS) is proposed for odd set sizes to generate even-sized samples. The lue of the normal standard deviation under this modification is shown to be the most efficient of all the lue’s of the same parameter. Among the lue’s considered, the PERSS lue’s are the most efficient when the sample size per cycle is two.

Robust joint estimation of location and scale parameters in ranked set samples

A robust estimator is developed for the location and scale parameters of a location-scale family. The estimator is defined as the minimizer of a minimum distance function that measures the distance between the ranked set sample empirical cumulative distribution function and a possibly misspecified target model. We show that the estimator is asymptotically normal, robust, and has high efficiency with respect to its competitors in literature. It is also shown that the location estimator is consistent within the class of all symmetric distributions whereas the scale estimator is Fisher consistent at the true target model. The paper also considers an optimal allocation procedure that does not introduce any bias due to judgment error classification. It is shown that this allocation procedure is equivalent to Neyman allocation. A numerical efficiency comparison is provided.

Weighted correlation tests for location-scale families

A general asymptotic theory of weighted correlation tests was recently developed for location and scale families separately. Here the existing theory is exposed and is extended to location-scale families.

Adaptively truncated maximum likelihood regression with asymmetric errors

We consider robust estimators for the linear regression model with asymmetric (or symmetric) error distribution. We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale is a nuisance parameter. First, we show how to convert an ordinary robust estimate for the usual model with symmetric errors to an estimate for the more general model with asymmetric errors. Then, in order to improve efficiency, we introduce the truncated maximum likelihood or TML-estimator. A TML-estimate is computed in three steps: first, an initial high breakdown point estimate is computed; then, observations that are unlikely under the estimated model are rejected; finally, the maximum likelihood estimate is computed with the retained observations. The rejection rule used in the second step is based on a cut-off parameter that can be tuned to attain the desired efficiency while maintaining the breakdown point of the initial estimator (e.g., 50%). Optionally, one can use a new adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Under the model, the influence function of this adaptive TML-estimator (or ATML-estimator) coincides with the influence function of the maximum likelihood estimator. The ATML-estimator is, therefore, fully efficient at the model; nevertheless, its breakdown point is not smaller than the breakdown point of the initial estimator. We evaluate the TML- and ATML-estimators for finite sample sizes with the help of simulations and discuss an example with real data.

Multivariate targeting with applications to ellipse estimation

The method of target estimation developed by Cabrera and Fernholz to reduce bias and variance is extended to multivariate situations. For a p-dimensional statistic, conditions are given to ensure no bias and smaller variance after targeting. Some applications of multivariate targeting are given for location-scale families. The main application of multivariate targeting is presented for typical applied problems in computer vision such as ellipse estimation. For these cases, a computer-intensive procedure based on stochastic approximation is used to compute the target estimates corresponding to the least squares estimates of the ellipse parameters. In the examples presented, the data are discretized to the image resolution to make the situation more realistic in the context of computer vision. In most of the cases the bias and the mean square error of the least squares estimates are reduced after targeting. Comparisons with the bootstrap show the advantage of using targeting to obtain a more accurate reconstructed ellipse when only an arc of points is available.