Anton Cedilnik

Comparison of a two-stage sampling design and its composite sample alternative: An application to soil studies

The objective of a long-term soil survey is to determine the mean concentrations of several chemical parameters for the pre-defined soil layers and to compare them with the corresponding values in the past. A two-stage random sampling procedure is used to achieve this goal. In the first step, n subplots are selected from N subplots by simple random sampling without replacement; in the second step, m sampling sites are chosen within each of the n selected subplots. Thus n · m soil samples are collected for each soil layer. The idea of the composite sample design comes from the challenge of reducing very expensive laboratory analyses: m laboratory samples from one subplot and one soil layer are physically mixed to form a composite sample. From each of the n selected subplots, one composite sample per soil layer is analyzed in the laboratory, thus n per soil layer in total. In this paper we show that the cost is reduced by the factor m — 1 when instead of the two-stage sampling its composite sample alternative is used; however, the variance of the composite sample mean is increased. In the case of positive intraclass correlation the increase is less than 12.5%; in the case of negative intraclass correlation the increase depends on the properties of the variable as well. For the univariate case we derive the optimal number of subplots and sampling sites. A case study is discussed at the end.

Continuity of homomorphisms into complete normed algebraic algebras

We prove that, if A and B are complete normed non-associative algebras, and if B is strongly semisimple and algebraic, then dense range homomorphisms from A to B are continuous. As a consequence, we obtain that homomorphisms from complete normed algebras into a complete normed power-associative algebraic algebra B are continuous if and only if B has no nonzero elements with zero square. We also show that homomorphisms from complete normed algebras into a finite-dimensional algebra B are continuous if and only if B has no nonzero elements with zero square.

A new bivariate distribution

We propose a new bivariate distribution with five shape parameters and two scale parameters. It allows great flexibility for the bivariate situation and successfully replaces the assumption of the bivariate normal distribution when the ratio of two positive bounded variables is in question

Modification of the Fisher's information measure to optimize a sampling design

In statistics, Fisher was the first to introduce the measure of the amount of information supplied by the data about the unknown parameter. We analyze the disadvantages of Fishers information measure for optimization of sampling designs. To overcome this problem, we modify Fishers information measure and we upgrade it to the multivariate setting. On a case study of soil we demonstrate the evaluation of different information measures derived from Fishers information measure. The variables under study were the concentrations of several chemical compounds in soil (such as pH, N, C, Zn, etc.).

Estimation Procedure for a New Bivariate Distribution

We introduced a new bivariate distribution for a random vector Z = [X, Y]T, where X and Y are positive continuous variables. The distribution is a generalization of the Weibull distribution, it has 7 parameters: r and s are power parameters, m and n are moment parameters, a and b are scaling parameters, and p is the linking parameter. The ML estimation turns out to be a very difficult task. In this paper we present the estimation procedure. It was tested on simulated data, which we generated using the acceptance-rejection method. The results are very satisfactory.

Quantifying the efficiency of soil sampling designs: A multivariate approach

The objective of this paper is to quantify and compare the loss functions of the standard two-stage design and its composite sample alternative in the context of multivariate soil sampling. The loss function is defined (conceptually) as the ratio of cost over information and measures design inefficiency. The efficiency of the design is the reciprocal of the loss function. The focus of this paper is twofold: (a) we define a measure of multivariate information using the Kullback–Leibler distance, and (b) we derive the variance-covariance structure for two soil sampling designs: a standard two-stage design and its composite sample counterpart. Randomness in the mass of soil samples is taken into account in both designs. A pilot study in Slovenia is used to demonstrate the calculations of the loss function and to compare the efficiency of the two designs. The results show that the composite sample design is more efficient than the two-stage design. The efficiency ratio is 1.3 for pH, 2.0 for C, 2.1 for N, and 2.5 for CEC. The multivariate efficiency ratio is 2.3. These ratios primarily reflect cost ratios; influence of the information is small.

Ranges of Elements of a Nonassociative Algebra

We generalize the notion of the numerical range in such a way that the ranges of elements of a given normed algebra are sets of elements of some other normed algebra, and that the states are linear operators. We show that many properties of numerical range rest unchanged, including the connection between range and spectrum. For Hermitian elements we deduce Vidav’s lemma and associator identities [x, x, x] = [x, x 2, x] = 0.