Location-scale Family Of Distributions Research Articles

Stochastic Comparisons for Finite Mixtures from Location-scale Family of Distributions

In this study, we consider two finite mixture models (FMMs) with location-scale family distributed components, in which ordering results are established in various stochastic senses. For heterogeneity in one parameter, the comparisons are obtained with respect to usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order. Further, for heterogeneity in two parameters, we derive sufficient conditions for the stochastic comparison of FMMs with respect to usual stochastic order and hazard rate order. Various examples and counterexamples are presented to illustrate the proposed results.

Model checks for two-sample location-scale

Two-sample location-scale refers to a model that permits a pair of standardised random variables to have a common base distribution. Function-based hypothesis testing in these models refers to formal tests based on distributions functions, or direct transformations thereof, that would help decide whether or not two samples come from some location-scale family of distributions. For uncensored data, an approach of testing based on plug-in empirical likelihood (PEL) is carried out with sample means and standard deviations as the plug-ins. The method extends to censored data, where censoring adjusted moment estimators provide the requisite plug-ins. The large sample null distribution of the PEL statistic is derived. Since it is not distribution free, a two-sample location-scale appropriate resampling is employed to obtain thresholds needed for the testing. Numerical studies are carried out to investigate the performance of the proposed method. Real examples are presented for both the uncensored and censored cases.

An uniformly superior exact multi-sample test procedure for homogeneity of variances under location-scale family of distributions

An exact test procedure based on the generalized variable approach for testing homogeneity of variances of several independent populations belonging to the same location-scale family is proposed. An extensive simulation study within the class of location-scale families reveals the uniformly superior performance of the proposed test over existing popularly used parametric and non-parametric tests, both in terms of the type-I error and power function. In normal populations, [Liu X, Xu X. A new generalized p-value approach for testing the homogeneity of variances. Stat Probab Lett. 2010;80:1486–1491] test designed specifically for normal populations is found equally compatible with the proposed test. The proposed test is found to be robust against moderate deviation from model specification and the location-scale assumption. The proposed test is illustrated with two real life examples.

Simultaneous best linear invariant prediction of future order statistics for location-scale and scale families and associated optimality properties

In this article, we first derive an explicit expression for the marginal best linear invariant predictor (BLIP) of an unobserved future order statistic based on a set of early observed ordered statistics. We then derive the joint BLIPs of two future order statistics and prove that the joint predictors are trace-efficient as well as determinant-efficient linear invariant predictors. More generally, the BLIPs are shown to possess complete mean squared predictive error matrix dominance property in the class of all linear invariant predictors of two future unobserved order statistics. Finally, these results are extended to the case of simultaneous BLIPs of any ℓ future order statistics. Both scale and location-scale families of distributions are considered as the parent distribution for the development of results.

Power of goodness-of-fit tests and some competitive proposals based on progressively type-II censored data from a location-scale distribution

In this paper, we review some existing methods for testing goodness-of-fit based on progressively type-II censored samples in the location-scale family of distributions. Also, some similar procedures and new modifications are proposed. Using Monte Carlo simulation, the powers of the reviewed and proposed tests are compared for the normal and Gumbel distributions against several alternatives. Then, we present some results based on the simulation studies. Finally, an application to two datasets is presented for numerical illustration.

Parametric estimation of location and scale parameters based on ranked set sampling with unequal set sizes

Ranked set sampling with unequal set sizes (RSSU) are some of the important variants of ranked set sampling with equal set sizes (RSS). The sets that arise naturally in many applications are typically of different set sizes. One may prefer to use the largest sets available naturally along with the smaller sets rather than an attempt to form artificial equal-sized sets. This article develops maximum likelihood estimators for the location-scale family of distributions based on ranked set sampling with unequal set sizes (RSSU). The closed form expressions for MLE under RSSU do not exist, we have proved the existence of MLE for location and scale parameters for some standard distributions for RSSU data. It is proved that MLE based on MedRSSU are more efficient than their counterparts based on SRS for some standard distributions for location-scale parameters. It is also shown that asymptotic efficiencies of the MLE based on MedRSSU are considerably better than those of the estimators based on RSS with the same number of observations. A simulation study is conducted to compare the performances of the MLE’s from RSS and MedRSSU with the corresponding SRS estimators when the underlying distributions are normal and logistic.

Correction to: Every variance function, including Taylor's power law of fluctuation scaling, can be produced by any location-scale family of distributions with positive mean and variance

Correction to: Every variance function, including Taylor's power law of fluctuation scaling, can be produced by any location-scale family of distributions with positive mean and variance

On simultaneous best linear unbiased prediction of future order statistics and associated properties

In this article, the joint best linear unbiased predictors (BLUPs) of two future unobserved order statistics, based on a set of observed order statistics, are developed explicitly. It is shown that these predictors are trace-efficient as well as determinant-efficient BLUPs. More generally, the BLUPs are shown to possess complete mean squared predictive error matrix dominance in the class of all linear unbiased predictors of two future unobserved order statistics. Finally, these results are extended to the case of simultaneous BLUPs of any ℓ future order statistics. Both scale and location-scale family of distributions are considered as the parent distribution for the underlying random variables.

Maximum likelihood estimation in location-scale families using varied L ranked set sampling

Recently, a generalized ranked set sampling (RSS) scheme has been introduced which encompasses several existing RSS schemes, namely varied L RSS (VLRSS), and it provides more precise estimators of the population mean than the estimators with the traditional simple random sampling (SRS) and RSS schemes. In this paper, we extend the work and consider the maximum likelihood estimators (MLEs) of the location and scale parameters when sampling from a location-scale family of distributions. In order to give more insight into the performance of VLRSS with respect to SRS and RSS schemes, the asymptotic relative precisions of the MLEs using VLRSS relative to that using SRS and RSS are compared for some usual location-scale distributions. It turns out that the MLEs with VLRSS are more precise than those with the existing sampling schemes.

Reference priors via [formula omitted]-divergence for a certain non-regular model in the presence of a nuisance parameter

This paper presents reference priors for non-regular model whose support depends on an unknown parameter. A multi-parameter family which includes both regular and non-regular structures is considered. The resulting priors are obtained by asymptotically maximizing the expected α-divergence between the prior and the corresponding posterior distribution. Some examples of reference priors for typical multi-parameter non-regular distributions such as the location-scale family of distributions and the truncated Weibull distribution are also given.

Estimation and prediction in the presence of an outlier under Type-II censoring

ABSTRACTA single-outlier data set containing some independent random variables is considered such that all of observations expect one have the same distribution. To describe the model of interested, a location-scale family of distributions is used and the estimation problem of the parameters is studied when the data are collected under Type-II censoring scheme. Moreover, three different predictors are presented to predict the censored order statistics. They are also compared regarding both of mean squared prediction error and Pitman's measure of closeness criteria. The role of outlier parameter as well as censorship rate is studied on performance of proposed estimator and predictors. The results of the paper are illustrated via a real data set. Finally, some conclusions are stated.

A note on Guo and Xiao's (2016) results on monotonic functions of the Sharpe ratio

In this note, we point out that the recent finding of Guo and Xiao (2016) that several performance measures are monotonic functions of the Sharpe ratio under the family of location-scale distributions is not fully original but has already been covered by the earlier theoretical studies of Schuhmacher and Eling (2011, 2012) and Schuhmacher and Auer (2014).

On stochastic comparisons of maximum order statistics from the location-scale family of distributions

We consider the location-scale family of distributions, which contains many standard lifetime distributions. We give conditions under which the largest order statistic of a set of random variables with different/the same location as well as different/the same scale parameters dominates that of another set of random variables with respect to various stochastic orders. Along with general results, we consider important special cases, namely, the Feller–Pareto, generalized Pareto, Burr, exponentiated Weibull, Power generalized Weibull, generalized gamma, Half-normal and Fréchet distributions.

A general algorithm for computing simultaneous prediction intervals for the (log)-location-scale family of distributions

ABSTRACTMaking predictions of future realized values of random variables based on currently available data is a frequent task in statistical applications. In some applications, the interest is to obtain a two-sided simultaneous prediction interval (SPI) to contain at least k out of m future observations with a certain confidence level based on n previous observations from the same distribution. A closely related problem is to obtain a one-sided upper (or lower) simultaneous prediction bound (SPB) to exceed (or be exceeded) by at least k out of m future observations. In this paper, we provide a general approach for computing SPIs and SPBs based on data from a particular member of the (log)-location-scale family of distributions with complete or right censored data. The proposed simulation-based procedure can provide exact coverage probability for complete and Type II censored data. For Type I censored data, our simulation results show that our procedure provides satisfactory results in small samples. We use three applications to illustrate the proposed simultaneous prediction intervals and bounds.

Two-sided tolerance intervals for members of the (log)-location-scale family of distributions

In this paper, we propose methods to calculate exact factors for two-sided control-the-centre and control-both-tails tolerance intervals for the (log)-location-scale family of distributions, based on complete or Type II censored data. With Type I censored data, exact factors do not exist. For this case, we developed an algorithm to compute approximate factors. Our approaches are based on Monte Carlo simulations. We also provide algorithms for computing TIs that control the probability in both tails of a distribution. A simulation study for Type I censored data shows that the estimated coverage probability is close to the nominal confidence level when the expected number of uncensored observations is moderate to large. We illustrate the methods with applications using different combinations of distributions and types of censoring.

Nonparametric two-sample estimation of location and scale parameters from empirical characteristic functions

ABSTRACTTwo random variables X and Y are said to belong to the same location-scale family when with unknown constants and . Given iid observations , and , satisfying this location-scale assumption, we wish to estimate μ and σ with high efficiency in the absence of knowledge of the functional form of the underlying common family of distributions of X and Y. Here, ‘high efficiency’ means that the estimator is asymptotically unbiased and that its asymptotic variance is close to the asymptotic variance of the maximum likelihood estimator that would be used had the form of the underlying location-scale family of distributions been known. We propose in the present paper two methods for estimating these parameters based on the empirical characteristic function (ECF). The first approach considered minimizes a weighted distance between the ECFs of the X and Y data. The second approach constructs a quadratic form comparing the real and imaginary parts of the X- and Y-sample ECFs at a preselected number of points. In both approaches, the constructed distance metric is minimized to estimate μ and σ. The asymptotic distributions of the estimators are found, and small sample performance is investigated via a Monte Carlo simulation study.

Statistical Inference of Component Lifetimes With Location-Scale Distributions From Censored System Failure Data With Known Signature

Statistical inference of the component lifetime distribution is developed when Type-II censored system lifetime data are observed with a known system structure. The component lifetime distributions are assumed to be from either the log-location-scale family of distributions or the location-scale family of distributions. Two estimation methods, the maximum likelihood method, and the regression-based method, are proposed for the model parameters, and the corresponding computational formulae are provided. Construction of confidence intervals for the model parameters is also considered. The methodologies are illustrated with two commonly used lifetime distributions: the Weibull, and the lognormal. Monte Carlo simulations are used to study the performances of the point and interval estimation methods proposed here. Finally, some recommendations are made based on the obtained simulation results.

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.

Univariate and multivariate process yield indices based on location-scale family of distributions

Several measures of process yield, defined on univariate and multivariate normal process characteristics, have been introduced and studied by several authors. These measures supplement several well-known Process Capacity Indices (PCI) used widely in assessing the quality of products before being released into the marketplace. In this paper, we generalise these yield indices to the location-scale family of distributions which includes the normal distribution as one of its member. One of the key contributions of this paper is to demonstrate that under appropriate conditions, these indices converge in distribution to a normal distribution. Several numerical examples will be used to illustrate our procedures and show how they can be applied to perform statistical inferences on process capability.

On the nearness of record values to order statistics from Pitman’s measure of closeness

In the two-sample prediction problem, record values from the present sample may be used as predictors of order statistics from a future sample. In this paper, we investigate the nearness of record statistics (upper and lower) to order statistics from a location-scale family of distributions in the sense of Pitman closeness and discuss the corresponding monotonicity properties. We then determine the closest record value to a specific order statistic from a future sample. Even though in general it depends on the parent distribution, exact and explicit expressions are derived for the required probabilities in the case of exponential and uniform distributions, and some computational results are presented as well. Finally, we consider the mean squared error criterion and examine the corresponding results in the exponential case.