To estimate the unknown parameters in a linear model in which the observations are linear functions of the unknowns, one of the conventional methods is the least-square estimation. The best linear unbiased estimation (BLUE) is achieved when the inverse of the variance covariance matrix of the observables is considered as the weight matrix in the estimation process. Therefore having a realistic assessment of the precision of the observations is an important issue. One of the methods to reach this goal is the use of the least-square variance component estimation (LS-VCE). However, in this method, it is not impossible to estimate negative variances. But, they are not acceptable from the statistical point of view. In this paper, numerical methods such as genetic algorithm and also iterative methods based on LS-VCE are presented for non-negative estimation of variance components. By using non-negative variance components estimation methods not only one guarantees the non-negative variance components but also one can investigate to incorporate different noise components into the stochastic model. Those components that are not likely present are automatically estimated zeros. In this paper, using the above-mentioned methods, we assess the noise characteristics of time series of GPS permanent stations. The data used in this research are the coordinates of IGS stations located in Mehrabad-Tehran and also two other stations in Ahvaz and Mashhad (2005-2010). To deal with this amount of data, the iterative methods are superior over the numerical methods such as the genetic algorithm. The results indicate the noise of GPS position time series are a combination of white noise plus flicker noise, and in some cases combined with random walk noise.

When the parametric model does not hold, and we cannot fit a parametric model to the data, the true density may be estimated non-parametrically, as in the case of a kernel estimate. The purpose of this paper is to present a comparison between parametric and non-parametric models. The parametric investigation contains Vuong's test, and tracking interval based on the known maximum likelihood estimation theory. The presented non-parametric analysis involves kernel density estimation. Modified differences of Kullback-Leibler criteria between two rival models and Vuong's test, have been considered. In this circumstance, we address the problem of cross-validation estimation of variance for Kullback-Leibler divergence between the true but unknown density and its kernel estimator. A simulation study and data analysis have shown that the parametric density is a more realistic estimate of the data generating density.

Introduction: In some biological, environmental or ecological studies, there are situations in which obtaining exact measurements of sample units are much harder than ranking them in a set of small size without referring to their precise values. In these situations, ranked set sampling (RSS), proposed by McIntyre (1952), can be regarded as an alternative to the usual simple random sampling (SRS) to draw a more representative sample from the population of interest than what is possible in SRS. To draw a ranked set sample, one first draws n simple random samples, each of size n, from the population of interest and ranks them in an increasing magnitude. The ranking process is done without measuring sample units and therefore it need not to be accurate. One then identifies the ith sample unit from the ith sample for actual quantification (for i=1, … , n). Finally, he repeats this process m times (cycle) if he/she is required to obtain a sample of size mn. Since a ranked set sample contains information from both measured sample units and their corresponding ranks, one intuitively expects that statistical inference based on RSS to be more accurate than what is possible to obtain based on SRS. This paper is concerned with problem of estimating variance of the normal distribution in RSS. Several methods of estimation of variance of the normal distribution are described and compared via a Monte Carlo simulation study. Material and methods: All simulation studies in this paper have been done using R statistical software version R-3. 3. 1 Results and discussion: In this paper, we consider estimation of the normal variance based on a ranked set sample with single (multiple) cycle(s) and propose different unbiased estimators for each case. Our simulation results indicate that the mean square error (MSE) of each estimator is decreased as the values of n or m increases while the other parameters are kept fixed. It is also found that the estimator based on combining variance estimators of within and between ranking classes has typically better performance than the others. Conclusion: The following results can be obtained based on our simulation study: If there is a single cycle in RSS, then the proposed estimator in the case of single cycle beats Stokes-modified unbiased estimator. In the multiple cycle case in RSS, the estimator based on combining variance estimators of within and between ranking classes is the best one.

The noise in the GNSS position time series is mainly a combination of white noise and power law noise. Noise amplitudes are estimated using variance component estimation (VCE) procedures. These methods require repeated inversion of covariance matrix, which is a computational burden for analysis of long time series. This work proposes an algorithm to estimate the white noise amplitude, through the estimation of wavelet variance based upon the Maximal Overlap Discrete Wavelet Transform (MODWT). MODWT can be used for any sample size and number of wavelet and scaling coefficients does not decrease by factor 2 for each increase in the level of the transform, so it does not decrease our ability to perform statistical analysis. To test the performance of the proposed algorithm, we used 180 synthetic daily time series with different lengths (2000, 4000 and 8000) emulating real GNSS time series. They composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The results of proposed method were compared to those of REstricted Maximum Likelihood (REML) approach. Biases of white noise amplitudes for the proposed and REML method indicated that results given by the two methods are in good agreement. Moreover, the proposed algorithm has computational complexity of order O(N) where N is the number of observations. Also, the results demonstrated that this proposed algorithm can be about 450-10000 times faster than REML method depending on the length of time series. For further evaluation of the method, the time series of 19 real stations were used, and the results indicated the effectiveness of the proposed method. The low complexity of the proposed algorithm can considerably speed up the processing of GNSS time series.

We develop a two phase sampling procedure to determine the sample size necessary to estimate the population mean of a normally distributed random variable and show that the resulting estimator has preassigned variance and is unbiased under a regular condition. We present a necessary and sufficient condition under which the final sample mean is an unbiased estimator for the population mean.

The purpose of the present study was to estimation of (co)variance components and genetic parameters of the growth curve in crossbred population. For this purpose, body weight records were fitted through the Gompertz curve and curve parameters were estimated. The final weight (Wf), hatch weight (W0), mature rate index (K), age (ti) and weight (Wi) in inflection point and growth rate in different days for all of the birds were predicted by Gompertz model. The estimated of (co)variance and genetic parameters were performed using a multi-trait animal model through Gibbs sampling. Heritability of the growth cure parameters including final weight, hatch weight, mature index and weight and age in inflection point was 0. 335, 0. 269, 0. 273, 0. 291 and 0. 397, respectively. Also, the heritability of growth rate traits from hatch to 45 days was estimated high and varied between 0. 311 to 0. 424. The genetic correlation between hatch weight with mature index and final weight was positive (0. 14 and 0. 24), but the genetic correlation between hatch weight with age and weight at inflection point was negative (-0. 24 and-0. 12). The genetic correlation of the mature index with hatch weight, mature weight, age and weight at inflection point was negative. Genetic correlation of the mature index with growth rate in the early ages of bird was positive and it was estimated medium and negative for late. The weight at inflection point had the positive genetic correlation with most of growth rate traits and the change trend from beginning to end of growth period was ascending.

Geodetic data processing usually is performed using the least-squares method. To achieve the best linear unbiased estimation, it is necessary to use the proper and realistic stochastic model of the observables. The estimation of the unknown (co) variance components of the observables is referred to as variance component estimation (VCE). In geodetic applications, VCE is also known as the observables weights estimation. In this paper, least-squares variance component estimation is applied in a straightforward manner to GPS observables for determination of the realistic stochastic model. For this purpose, the functional model used in the analysis is the GPS geometry-based observation model (GFOM). The numerical results for two receivers, namely Trimble 4000 SSi and Trimble R7, are presented. The results indicate that the correlation between observation types is significant. A positive correlation of 0.55 is observed between the code observations on CA and P2 for Trimble 4000 SSi. Also, a significant positive correlation of 0.64 is observed between the phase observations on L1 and L2 for Trimble R7.

When a change occurs in a process, one expects to receive a signal from a control chart as quickly as possible. Upon the receipt of signal from the control chart a search for identifying the source of disturbance begins. However, searching for assignable cause around the signal time, due to the fact that the disturbance may have manifested itself into the process sometimes back, may not always lead to successful identification of assignable cause (s). If process engineers could identify the change point, i.e. the time when the disturbance first manifested itself into the process, then corrective actions could be directed towards effective elimination of the source of disturbance. In this paper we develop a maximum likelihood estimator (MLE) for process change point designed to detect changes in process variance of a normal quality characteristic when the change follows a linear trend. We describe how this estimator can be used to identify the change point when a Shewhart S-control chart signals a change in the process variance. Numerical results reveal that the proposed estimator outperforms the MLE designed for step change when a linear trend disturbance is present.

Measuring the 3D displacement fields provide essential information regarding the Earth crust interaction and the mantle rheology. The interferometric synthetic aperture radar (InSAR) has an appropriate capability in revealing the displacements of the Earth’ s crust. Although, it measures the real 3D displacements in the line of sight (LOS) direction. The 3D displacement vectors can be retrieved through multiple InSAR measurements acquired from at least three independent imaging geometries in a theoretical manner. However, this is a physically ill-posed inverse problem and consequently, the retrieving process of the components in 3D displacements become sensitive to observation errors, especially in the northern component due to the near-polar orbiting of SAR missions. Combining different datasets regarding this issue requires proper treatment of the weight of observations, which otherwise will have a negative effect on both the precision and accuracy of the estimated 3D displacement field. In retrieving the 3D displacement fields through InSAR technique, we deal with two major issues, integration of inhomogeneous precision of observations and instability of the estimation problem. These facts constitute the motivations to address the Tikhonov regularization (TR) and least squares variance component estimation (LS-VCE). In this article, to overcome these drawbacks, the regularized least squares variance component estimation (RLS-VCE) is proposed for retrieving the 3D displacement vectors. Usually, the number of InSAR observations in relation to the three unknowns of 3D displacements for each pixel is not enough to apply VCE. Therefore, observations of some neighborhood cells are taken into account to increase the redundancy of stochastic model. In this context, a moving frame including a window of 3 × 3 pixels is considered to increase the number of observations and consequently, the degree of freedom of stochastic model. To assess the efficiency of the proposed method, the RADAR dataset of the Envisat and ALOS missions for the 17 June 2007 eruption of Kilauea volcano on Hawaiian island are applied. To validate the results of the proposed method, co-event displacement vectors of 19 GNSS stations around the Kilauea volcano are used. Furthermore, the 3D displacements of GNSS stations are applied for detrending the displacements of InSAR from systematic or random disturbing effects (e. g. orbit errors, curvature and topography of the Earth, atmosphere, etc. ) through fitting a two variates linear or quadratic polynomial. Comparing the co-event retrieved 3D displacement vectors through RLS-VCE method and GNSS measurements indicates that the componential RMSE of northern displacements decreases drastically to 2. 2 cm from 11. 7 cm (for range displacements and primary weights). This is approximately equivalent to 80% improvement in the accuracy of estimating the northern component of displacement. The overall RMSE of retrieving 3D displacement vectors decrease from 7. 8 cm to 2. 6 cm, which is equal to 66% improvement. Achieving to this overall accuracy and for northern component is of major interest for all disciplines of geoscience dealing with 3D surface deformation analysis. Results indicate that retrieving the 3D displacement vectors through applying the RLS-VCE method has a meaningful improvement on the precision and accuracy of the results, the northern-southern component in special.