Linear Invariant Estimators Research Articles

Different Methods for Estimating Default Parameters of Alpha Power-Transformed Power Distributions Using Record-Breaking Data

The current study addresses the estimation of the default parameters of alpha power-transformed power (APTPO) distributions. For the location and scale parameters of the APTPO distributions, we provide coefficients for both the best linear unbiased estimators (BLUE) and the best linear invariant estimators (BLIE) methods. Furthermore, we establish a forecast for future records. The parameters of the APTPO distribution are estimated using the maximum likelihood estimation method (MLE). The goodness-of-fit test (using Akaike information criterion (AIC)) is computed using both the inter-record time sequence and the entire sample. Also, we utilize a simulation approach to demonstrate the practicality and benefits of our perspective. Finally, we demonstrate the accuracy of these parameters and the performance of estimators through a real-life example.

On the order statistics of exponentiated moment exponential distribution and associated inference

Explicit algebraic expressions for both single and product moments of order statistics from the exponentiated moment exponential (EME) distribution are derived. By using these expressions, we have computed the means, variances and covariances of order statistics for and for arbitrarily chosen parameter values. Further, these moments are used to determine the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) of the location and scale parameters based on complete as well as Type-II right censored samples. Prediction of unobserved order statistics in Type-II right censored samples is also discussed. A simulation study and a real data example are presented for the sake of comparison and illustration. The concluding remarks are given at the end.

A comparative assessment of reliability indicators and parameters of the bearings using different estimation methods

The paper depicts the method of statistical processing of experimental results, obtained by testing bearings on the test stands, using the maximum likelihood estimators and/or Best Linear Invariant Estimator. To verify statistical assumptions regarding parameters and reliability indicators of bearings, as well as, to determine confidence intervals, the article contains two original algorithms that allow obtaining values of some specific random variables by Monte Carlo numerical simulation. The statistical inferences are made through these values. The paper also, contains two case studies using data from failures and time truncated tests of bearings.

Linear Regression Estimation Methods for Inferring Standard Values of Snow Load in Small Sample Situations

The aim of this paper is to establish a new method for inferring standard values of snow load in small sample situations. Due to the incomplete meteorological data in some areas, it is often necessary to infer the standard values of snow load in the conditions of small samples in engineering, but the point estimation methods of classical statistics adopted till now do not take into account the influences of statistical uncertainty, and the inference results are always aggressive. In order to overcome the above shortcomings, according to the basic principle of optimal linear unbiased estimation and invariant estimation of the minimum type I distribution parameters and the tantile, using the least square method, the linear regression estimation methods for inferring standard values of snow load in small sample situations are proposed, which can take into account two cases such as parameter-free and known coefficient of variation, and the predicted formulas of snow load standard values are given, respectively. Through numerical integration and Monte Carlo numerical simulation, the numerical table of correlation coefficients is established, which is more convenient for the direct application of inferential formulas. According to the results of theoretical analysis and examples, when using the indirect point estimation methods to infer the standard values of snow load in the conditions of small samples, the inference results are always small. The linear regression estimation method is suitable for inferring standard values of snow load in the conditions of small samples, which can give more reasonable results. When using the linear regression estimation to infer standard values of snow load in practical application, even if the coefficient of variation is unknown, it can set the upper limit value of the coefficient of variation according to the experience; meanwhile, according to the parameter-free and known coefficient of variation, the estimation is carried out, respectively, and the smaller value of the two is taken as the final estimate. The method can be extended to the statistical inference of variable load standard values such as wind load and floor load.

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACTIn this paper, we use the idea of order statistics from independent and non-identically distributed random variables to propose ordered partially ordered judgment subset sampling (OPOJSS) and then develop optimal linear parametric inferences. The best linear unbiased and invariant estimators of the location and scale parameters of a location-scale family are developed based on OPOJSS. It is shown that, despite the presence or absence of ranking errors, the proposed estimators with OPOJSS are uniformly better than the existing estimators with simple random sampling (SRS), ranked set sampling (RSS), ordered RSS (ORSS) and partially ordered judgment subset sampling (POJSS). Moreover, we also derive the best linear unbiased estimators (BLUEs) of the unknown parameters of the simple linear regression model with replicated observations using POJSS and OPOJSS. It is found that the BLUEs with OPOJSS are more precise than the BLUEs based on SRS, RSS, ORSS and POJSS.

On the Lindley Record Values and Associated Inference

In this paper, we discuss the record values arising from the Lindley distribution. We compute the means, variances and covariances of the record values. These values are used to compute the best linear unbiased estimators (BLUEs) and the best linear invariant estimators (BLIEs) of the location and scale parameters. By using the BLUEs and BLIEs, we construct confidence intervals for the location and scale parameters through Monte Carlo simulations. Prediction for the future records is also discussed.

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.

Pooled parametric inference for minimal repair systems

Consider two independent and identically structured systems, each with a certain number of observed repair times. The repair process is assumed to be performed according to a minimal-repair strategy. In this strategy, the state of the system after the repair is the same as it was immediately before the failure of the system. The resulting pooled sample is then used to obtain best linear unbiased estimators (BLUEs) as well as best linear invariant estimators of the location and scale parameters of the presumed parametric families of life distributions. It is observed that the BLUEs based on the pooled sample are overall more efficient than those based on one sample of the same size and also than those based on independent samples. Furthermore, the best linear unbiased predictor and the best linear invariant predictor of a future repair time from an independent system are also obtained. A real data set of Boeing air conditioners, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes, is used to illustrate the inferential results developed here.

Pitman closeness, monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.

Use of Quasi-Ranges in the Estimation of Scale Parameter of a Symmetric Distribution

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the scale parameter of a symmetric distribution using quasi-ranges is considered. We also prove a sufficient condition for the non negativity of the scale estimator obtained by the above method. Further, we obtain necessary and sufficient conditions for the derived estimators to be constant multiple of the sample range.

Modified BLUEs and BLIEs of the location and scale parameters and the population mean using ranked set sampling

Ranked set sampling (RSS) is a sampling procedure that can be used to improve the cost efficiency of selecting sample units of an experiment or a study. In this paper, RSS is considered for estimating the location and scale parameters a and b>0, as well as the population mean from the family F((x−a)/b). Modified best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) are considered. Numerical computations with different location-scale distributions and different sample sizes are conducted to assess the efficiency of the suggested estimators. It is found that the modified BLIEs are uniformly higher than that of BLUEs for all distributions considered in this study. The modified BLUE and BLIE are more efficient when the underlying distribution is symmetric.

Estimation of Parameters of Uniform Distribution Based on K-Record Values

Abtsrcat There are many situations where experimental outcomes are a sequence of record-breaking observations. In this paper, an extension of record models, well known as k-records, is considered. The best linear unbiased estimators (BLUEs) for the location and scale parameters of the uniform distribution are derived using k-record values. In addition, best linear invariant estimators of two parameter uniform distribution are obtained on the basis of k-record values. We have also determined the e�ciency of BLUEs relative to maximum likelihood estimators (adjusted for bias). The best linear unbiased predictor of future k-record values is also determined.

Use of Spacings in the Estimation of Common Scale Parameter of Several Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several distributions using spacing of the pooled sample of all observations of individual samples. We derived conditions for the non negativity of the scale estimator obtained by the above methods. Further, we obtained necessary and sufficient conditions for the derived estimators to be constant multiples of the pooled sample range.

Linear invariant estimation of clock phase and trend from noisy phase data

We show how to solve two problems of optimal linear estimation from a set of noisy measurements of clock phase. The phase is modelled as a process with stationary dth increments, e.g. a sum of processes with power-law spectra. The additive measurement noise can be any mean-zero process with a known autocovariance function. The estimation targets are the phase at a given time and the coefficient of the overall trend, e.g. the frequency drift. A feasibility condition called ‘invariance’ is imposed according to the target and the degree of non-stationarity of the phase model. The solution of a set of linear equations gives the regression coefficients and the mean squared error of the best linear invariant estimate. Proofs of these results are available from the online version of this journal.

Estimation of location and scale parameters for the Burr XII distribution using generalized order statistics

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution.

Estimation of the parameters of Gumbel and Burr distributions in terms of kth record values

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on $m$ selected $k$th record values are presented for the parameters of the Gumbel and Burr distribution

Best linear invariant estimators using ranked set sampling procedure: comparative study

The best linear invariant estimators of the mean, location and scale parameters of the normal, Weibull, power, gamma, Gumbel, and log-logistic distributions are obtained by using ranked-set sampling procedure. The efficiency of the best linear invariant estimator of the mean relative to the simple random sample mean ( X ̄ ) is obtained and tabulated. Also, the efficiencies of these estimators of the location and scale parameters relative to the ordered best linear unbiased estimators for the given distributions are obtained and tabulated for perfect ordering case.

The use of spacings in the estimation of a scale parameter

Linear functions on spacings—instead of linear functions on order statistics—are considered, in order to simplify the form of best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the scale parameter in the classical location-scale family. Also, a sufficient condition for the non-negativity of the scale estimator is presented and, moreover, necessary and sufficient conditions for the BLUE (and the BLIE) to be a constant multiple of the sample range are derived. Finally, a modification of this approach is applied in order to simplify the derivations of both the location and the scale estimators in the Uniform Type-II Censored model.

Approximate inference for location and scale parameters with application to failure-time data

A new method (SP/CI) for obtaining approximate confidence intervals for the location, scale, and quantiles based on sample data parameters is presented. The basis of SP/CI is a saddlepoint expansion; the requirements are the maximum likelihood estimate (MLE), a few cross derivatives of the log-likelihood evaluated at the MLE, and quantiles of the log(F) distribution. SP/CI is illustrated numerically on several type-II censored reliability data sets using the extreme value and log-normal models. The accuracy is investigated by numerical comparison with the exact results, with best linear invariant estimator (BLIE), as well as with the results from other approximate methods discussed in the literature, including the asymptotic normality of the MLE, the likelihood ratio (LR) statistic, and corrections to the LR statistic recently developed. >

Approximate MLEs for the location and scale parameters of the extreme value distribution with censoring

For the extreme value distribution, the maximum likelihood method does not provide explicit estimators for the location and scale parameters. A method of deriving explicit estimators by approximating the likelihood function is provided. The authors derive the asymptotic variances, covariance, and conditional bias of these estimators, and show that they are almost as efficient as the maximum likelihood estimators and just as efficient as the best linear unbiased and the best linear invariant estimators. Two examples illustrate this method of estimation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>