Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 6, N° 1 2012 Articles 8 Effective Measurand Estimators for Samples of Trapezoidal PDFs Zygmunt Lech Warsza Submitted 5 th December 2010; accepted 22 nd March 2011 Abstract: This paper is final overview of investigations on the ac- curacy of basic the estimators of trapezoidal probability distribution samples of the measured data. For symmetri- cal trapezoidal PDF of straight as well concaved sides, using Monte-Carlo method of simulation, the standard deviation (SD) of linear 1- and 2-component estimators are evaluated. Approaches for theirs evaluation are pro- posed. It is established that in the ratio of upper and bot- tom bases of trapezoidal PDF in the range from 1 to 0,35 the mid-range value has smaller standard deviation (SD) than the mean value and median. It is find then for the whole family of the symmetric linear trapezoidal PDF more accurate than above single element estimators are two-component (2C) estimators as the linear form of the mean and mid-range values of the sample. Their coeffi- cients are found, properties discussed and formulas of SD are given. The new simplified 2C-estimator of equal coef- ficients is also proposed. These estimators successfully extend estimation of the measurand value as the sample mean and description of its accuracy by the uncertainty type A recommended by the international guides of un- certainty evaluation in measurement GUM-2008 [1], EA- 4/02 [2] and by Handbook NASA [3]. Approaches of de- scribed below investigations could be effectively applied also for other models of convoluted PDF-s. Keywords: estimators of probability density function, trapezoidal PDF, mid-range, uncertainty evaluation 1. Introduction Random components of measurement data can be in many cases more accurately modelled by non-Gaussian probability density distribution function (PDF) than by Normal distribution as the range of data random disper- sion is commonly limited in reality. The mean value as the most effective measurand estimator of the n-element sample of Normal distribution is also used for other dis- tributions. Its standard deviation (SD) is defined in GUM [1] as the uncertainty type A. For data processing it is very important to choose an effective estimator of the centre coordinate of PDF, i.e. estimator of the smallest SD, as not proper evaluation en- tails incorrect assessment of the measurement accuracy. For samples modelled by Normal, Uniform and La- place (double-exponential) PDF distributions, it is pre- sented in the paper [4] of 15 th IMEKO TC4 Symposium in Iasi Romania, how to regard the data autocorrelation and which estimator has the smallest standard deviation (SD) to be chosen as the better accurate for any of them. E. g. more effective estimator then mean value of meas- urand of Uniform samples is mid-range and for Laplace sample – median, respectively. Using one of goodness- of-fit tests (Kolmogorov–Smirnov, Chi-Square and other tests) we make decision about the estimation choice. The main purpose of this work is the expansion of op- portunities for choosing the best single or a few com- ponent estimators of empirical data modelled by more complex non-Gaussian distributions than the above models. It is assumed that treated measurement data do not contain unknown systematic errors and are not self- correlated. The estimator of the distribution parameter should meet also requirements of solvency, sufficiency, efficiency and be unbiased. First of all, efficiency of esti- mators is researched. 2. Single component estimators Let’s check up which one of single-component estima- tors of PDF of particular samples: mean X , mid-range 2 V q / or median med X , satisfies the requirement of effi- ciency, i.e. has the least-possible sum of the square dis- persion, denotes a minimum standard deviation in com- parison with other estimators. Similarly, it is possible to receive results for other basic non-Gaussian distributions. In columns 3–5 of Tab. 1 values of standard deviations of three estimators of a few basic distribution models of em- pirical data (for demonstration of difference order only) are presented. Standard deviation of the best single component esti- mator of the particular non-Gaussian distribution is sig- nificantly less then of other estimators even if difference between their values, e.g. between midrange and mean, is small. This is the cause to search for estimators better then the sample mean. 3. The best single component estimators of trapeze distributions 3.1. Linear trapeze It is important to consider the problem of choice of an effective estimator for composition of simple dis- tributions. In the measurement systems practically all analogue signals now are digitalised, and then uniform distributions are very common in these systems. So, with convolution of two different uniform distributions we get PDF as a symmetrical trapezoid of linear sides, from triangular to the uniform distribution as its bound- ary cases. The effective single component estimators of the centre of the triangular and uniform distributions are the sample mean and the mid-range respectively – see again Table 1.