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1
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Approximate sums of squares in analysis of variance

100%
Stępniak C.
Mathematica Applicanda
|
1977
|
tom 5
|
nr 10
PL
.
EN
Consider the two-way crossed classification model, in which there are a levels of the factor A, b levels of the factor B and nij observations y(i,j,k), k=1,⋯,n(i,j), for the (i,j)th cell, i=1,⋯,a, j=1,⋯,b. The sum of squares for testing interactions in this model can be written as Q=∑(i,j)n(i,j)(y(i,j,⋅)/n(i,j)−y(i,⋅,⋅)/n(i,⋅)−y(⋅,j,⋅)/n(j,⋅)+y(⋅,⋅,⋅)/n(⋅,⋅))^2, where y(i,j,⋅)=∑(k)y(i,j,k), y(i,⋅,⋅)=∑(j)y(i,j,⋅), y(⋅,j,⋅)=∑(i)y(i,j,⋅), y(⋅,⋅,⋅)=∑(i)y(i,⋅,⋅), n(i,⋅)=∑(j)n(i,j), n(⋅,j)=∑(i)n(i,j) and n(⋅,⋅)=∑(i)n(i,⋅). It is well known that if the numbers of observations are proportional, i.e., if (1) n(i,j)=n(i,⋅)n(⋅,j)/n(⋅,⋅) for all i=1,⋯,a and j=1,⋯,b, then the quadratic form Q(0)=∑(i,j)y(i,j,⋅)^2/n(i,j)−∑(i)y(i,⋅,⋅)^2/n(i,⋅)−∑(j)y^2(⋅,j,⋅)/n(⋅,j)+y^2(⋅,⋅,⋅)/n(⋅,⋅) is nonnegative definite, being then identical with Q. The author proves the converse of this implication; he shows that the nonnegative definiteness of Q0 implies the proportionality condition (1). He considers a similar problem also for the case of the three-way crossed classification model.
2
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On distribution of waiting time for the first failure followed by a limited length success run

100%
Stępniak C.
Applicationes Mathematicae
|
2013
|
tom 40
|
nr 4
421-430
EN
Many doctors believe that a patient will survive a heart attack unless a succeeding attack occurs in a week. Treating heart attacks as failures in Bernoulli trials we reduce the lifetime after a heart attack to the waiting time for the first failure followed by a success run shorter than a given k. In order to test the "true" critical period of the lifetime we need its distribution. The probability mass function and cumulative distribution function of the waiting time are expressed in explicit and concise form by binomial coefficients.
3
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Inverting covariance matrices

100%
Stępniak C.
Discussiones Mathematicae Probability and Statistics
|
2006
|
tom 26
|
nr 2
163-177
EN
Some useful tools in modelling linear experiments with general multi-way classification of the random effects and some convenient forms of the covariance matrix and its inverse are presented. Moreover, the Sherman-Morrison-Woodbury formula is applied for inverting the covariance matrix in such experiments.
4
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Selective lack-of-memory and its application

100%
Stępniak C.
Discussiones Mathematicae Probability and Statistics
|
2009
|
tom 29
|
nr 1
31-39
EN
We say that a random variable X taking nonnegative integers has selective lack-of-memory (SLM) property with selector s if P(X ≥ n + s/X ≥ n) = P(X ≥ s) for n = 0,1,.... This property is characterized in an elementary manner by probabilities pₙ = P(X=n). An application in car insurance is presented.
5
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Characterizing experimental designs by properties of the standard quadratic forms of observations

100%
Stępniak C.
Applicationes Mathematicae
|
2007
|
tom 34
|
nr 1
39-45
EN
For any orthogonal multi-way classification, the sums of squares appearing in the analysis of variance may be expressed by the standard quadratic forms involving only squares of the marginal and total sums of observations. In this case the forms are independent and nonnegative definite. We characterize all two-way classifications preserving these properties for some and for all of the standard quadratic forms.
6
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On useful schema in survival analysis after heart attack

100%
Stępniak C.
Discussiones Mathematicae Probability and Statistics
|
2014
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tom 34
|
nr 1-2
63-69
EN
Recent model of lifetime after a heart attack involves some integer coefficients. Our goal is to get these coefficients in simple way and transparent form. To this aim we construct a schema according to a rule which combines the ideas used in the Pascal triangle and the generalized Fibonacci and Lucas numbers
7
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A note on correlation coefficient between random events

100%
Stępniak C.
Discussiones Mathematicae Probability and Statistics
|
2015
|
tom 35
|
nr 1-2
57-60
EN
Correlation coefficient is a well known measure of (linear) dependence between random variables. In his textbook published in 1980 L.T. Kubik introduced an analogue of such measure for random events A and B and studied its basic properties. We reveal that this measure reduces to the usual correlation coefficient between the indicator functions of A and B. In consequence the resuts by Kubik are obtained and strenghted directly. This is essential because the textbook is recommended by many universities in Poland.
8
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Quotient of information matrices in comparison of linear experiments for quadratic estimation

100%
Stępniak C.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1599-1605
EN
The ordering of normal linear experiments with respect to quadratic estimation, introduced by Stępniak in [Ann. Inst. Statist. Math. A 49 (1997), 569-584], is extended here to the experiments involving the nuisance parameters. Typical experiments of this kind are induced by allocations of treatments in the blocks. Our main tool, called quotient of information matrices, may be interesting itself. It is known that any orthogonal allocation of treatments in blocks is optimal with respect to linear estimation of all treatment contrasts. We show that such allocation is, however, not optimal for quadratic estimation.
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