Random weighting estimation of parameters in generalized Gaussian distribution
Introduction
Random weighting is an emerging computing method in statistics [7]. It has received considerable attention in the recent years, and has been widely studied for different problems [1], [2], [3], [4], [5], [6]. However, there is little research to investigate random weighting estimation of parameters in generalized Gaussian distribution (GGD) (see Appendix A.1) and the related convergence properties.
Suppose that X1, X2, … , Xn are the random variables of an independent and identical distribution with an unknown distributed function F. Let x1, … , xn be the corresponding observed realizations. Further, we shall denote and . Then, the random weighting process can be described as follows:
- (i)
Construct the sample (empirical) distribution function Fn from , i.e.
- (ii)
The random weighting estimation of the sample mean Fn iswhere is the characteristic function, and random vector (V1, … , Vn) obeys the Dirichlet distribution D(1, … , 1), i.e. A uniformly distributed density function of (V1, … , Vn) can be defined aswhere (V1, V2, … , Vn) ∈ S n−1, and .
This paper investigates the estimation of GGD parameters by using the random weighting method. An expression is established to describe the relationship between moments and parameters. The strong convergence for random weighting estimation of GGD parameters is also rigorously proved. Computational simulations and practical experiments have been conducted to comprehensively evaluate the performance for random weighting estimation of GGD parameters.
Section snippets
K-Order absolute moment of GGD parameters
The probability density function of GGD can be given aswhere μ, α and β are the mean, shape and scale parameters, respectively, and α > 0. is Γ function (see Appendix A.2). , where σ is the standard deviation. The shape parameter α determines the decay speed for the density function of GGD:
- •
If α → 0, the limitation of the density function is the δ function (see Appendix A.3);
- •
If α = 1, GGD corresponds to the Laplacian
Convergence analysis
We begin with the main result in our previous work [2] to analyze the strong convergence properties for random weighting estimation of parameters α and β. Theorem 1 Suppose E∣X1∣1+δ < ∞,(0 < δ < 1). Then, for any ε > 0, when n → ∞, we have[2]
Performance evaluations
In this section, random weighting estimation of GGD parameters is comprehensively evaluated by computational simulations and practical experiments.
Conclusions
This paper presents a study on random weighting estimation of GGD parameters. This study demonstrates that random weighting is an efficient method to estimate GGD parameters. The results of the study have extensive applications in many fields such as signal processing and image processing.
References (7)
- et al.
Large numbers law for sample mean of random weighting estimation
Information Sciences
(2003) - et al.
The random weighting estimate of quantile process
Information Sciences
(2004) - et al.
Convergence rate of empirical Bayes estimation for two-dimensional truncation parameters under linex loss
Information Sciences
(2005)
Cited by (33)
Random weighting method for estimation of error characteristics in SINS/GPS/SAR integrated navigation system
2015, Aerospace Science and TechnologyNonlinear weighted measurement fusion Unscented Kalman Filter with asymptotic optimality
2015, Information SciencesCitation Excerpt :For instance, the infrared sensor provides accurate angle but poor range while radar provides accurate range but poor angle [22]. There are two ways to achieve the fusion: state fusion methods (distributed convex combination of local estimators) [3,5,10,20,27] and measurement fusion methods [11,23]. For linear systems, there are perfect theories of state fusion methods, such as the federated filter [5], the unified optimal linear fusion [20] and three kinds of distributed state fusion methods [27].
Windowing-based random weighting fitting of systematic model errors for dynamic vehicle navigation
2014, Information SciencesWeak convergence for random weighting estimation of smoothed quantile processes
2014, Information SciencesCitation Excerpt :Random weighting is a computational method in statistics. It has received great attention in the recent years, and has been widely applied to solve different problems [2,7–16,18,19,24,25]. However, there has been very limited research to use the random weighting method for estimation of quantile processes, which is an important research topic in the areas such as computational statistics and information technology [1,13,20,21].
An innovation based random weighting estimation mechanism for denoising fiber optic gyro drift signal
2014, OptikCitation Excerpt :To the best of authors knowledge, there has been very limited research regarding the use of random weighting method for adjusting the Kalman filter parameters and its application to gyro signal processing. The random weighting estimation is regarded to be a promising method for improving the accuracy of the gyroscopes [27–29]. In this paper, random weighting of innovation sequence with gain correction adaptive Kalman filter is developed for denoising the IFOG signal.
Energy-saving estimation model for hypermarket HVAC systems applications
2011, Energy and Buildings