Size biased sampling from the Dickman subordinator

Size-biased sampling arises when a positive-valued outcome variable is sampled with selection probability proportional to its size. In this article, we propose a semiparametric linear regression model to analyze size-biased outcomes. In our proposed model, the regression parameters of covariates are of major interest, while the distribution of random errors is unspecified. Under the proposed model, we discover that regression parameters are invariant regardless of size-biased sampling. Following this invariance property, we develop a simple estimation procedure for inferences. Our proposed methods are evaluated in simulation studies and applied to two real data analyses.

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.

On some inequalities in the probabilistic number theory

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h < |\chi|$, then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

Sums of multiplicative functions and limit theorems in probabilistic number theory. I

ABSTRACTThe aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.

Let y = {ii, ... ,4t} be a finite set of coprime integers and let {l, 2,...} denote the mutiplicative semigroup generated by y, and arranged in increasing order.Let Dn(w) denote the discrepancy of the sequence {nkco}%=i mod 1, w e [0, 1).In this paper we solve a problem posed by R.C. Baker [3], by proving that for all w except on a set of Lebesgue measure 0 l<limsuP /*(ft,) <C.Here the constant C only depends on the total number of primes involved in the prime factorization of q ,... , qx.The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence {cos2nnk<}kx>=l by sums of independent standard normal random variables.P{co : t]k((o) < x} = x, 0<JC<1, *>1.If Fff denotes the empirical distribution function of the sequence {nk} at stage N, i.e., FN(s) = FN(s ; to) := N-l*{k < N : nk(w) <s}, 0 < s < 1, then DN(co):= sup \Fjf(s) -s\ 0<s<X denotes the discrepancy (or the Kolmogorov-Smirnov statistic) of the sequence {jt<y}f=1 mod 1. (For the basic definition and facts on uniform distribution

The paper presents an overview of main methods and results of the value distribution problem of additive arithmetical functions.

Let (nk)k≥1 be an increasing sequence of positive integers, and let f (x) be a real function satisfyingIf the distribution ofconverges to a Gaussian distribution. In the casethere is a complex interplay between the analytic properties of the function f , the number-theoretic properties of (nk)k≥1, and the limit distribution of (2).In this paper we prove that any sequence (nk)k≥1 satisfying lim contains a nontrivial subsequence (mk)k≥1 such that for any function satisfying (1) the distribution ofconverges to a Gaussian distribution. This result is best possible: for any ε &gt; 0 there exists a sequence (nk)k≥1 satisfying lim such that for every nontrivial subsequence (mk)k≥1 of (nk)k≥1 the distribution of (2) does not converge to a Gaussian distribution for some f.Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

1. The new Theory of Local or Geometrical Probability, so far as it is known, seems to present, in a remarkable degree, the same distinguishing features which characterize those portions of the general Theory of Probability which we owe to the great philosophers of the past generation. The rigorous precision, as well as the extreme beauty of the methods and results, the extent of the demands made on our mathematical resources, even by cases apparently of the simplest kind, the subtlety and delicacy of the reasoning, which seem peculiar to that wonderful application of modern analysis—ce calcul delicat, as it has been aptly described by Laplace—reappear, under new forms, in this, its latest development. The first trace which we can discover of the Theory of Local Probability seems to be the celebrated problem of Buffon, the great naturalist— a given rod being placed at random on a space ruled with equidistant parallel lines, to find the chance of its crossing one of the lines. Although the subject was noticed so early, and though Buffon’s and one or two similar questions have been considered by Laplace, no real attention seems to have been bestowed upon it till within the last few years, when this new field of research has been entered upon by several English mathematicians, among whom the names of Sylvester and Woolhouse are particularly distinguished. It is true that in a few cases differences of opinion have arisen as to the principles, and discordant results have been arrived at, as in the now celebrated three-point problem, by Mr. Woolhouse, and the four-point problem of Professor Sylvester; but all feel that this arises, not from any inherent ambiguity in the subject matter, but from the weakness of the instrument employed; our undisciplined conceptions of a novel subject requiring to be repeatedly and patiently reviewed, tested, and corrected by the light of experience and comparison, before they are purged from all latent error. The object of the present paper is, principally, the application of the Theory of Probability to straight lines drawn at random in a plane; a branch of the subject which has not yet been investigated. It will be necessary to begin by some remarks on the general principles of Local Probability. Some portion of what follows I have already given elsewhere.

Bimodal size distributions have been commonly observed in Arctic char (Salvelinus alpinus). We document an example of such bimodality caused solely by biased gillnet sampling. The observed bimodality was a direct artefact of the sampling method resulting from an abrupt increase in gillnet catchability of fish larger in total length than between 25 and 30 cm. Mean gillnet selectivity (catchability) of char in the upper mode of the observed bimodal size distribution was about 40 times higher than the corresponding value for char in the observed lower mode. Fish of intermediate size, lacking in the gillnet samples, were present in the population and readily obtainable by electrofishing. The observed size difference in gillnet vulnerability is likely to result from behavioural changes following ontogenetic niche shifts.

Riecan [12] and Chovanec [1] investigated states in MV-algebras. Earlier, Riecan [11] had dealt with analogous ideas in D-posets. In the monograph of Riecan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on MV-algebras. We remark that a different definition of a state in an MV-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term "state-homomorphism" (instead of "state"). The author is indebted to the referee for his suggestion concerning terminology. Let $$\mathcal{A}$$ be an MV-algebra which is defined on a set A with card A>1. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $$\mathcal{A}$$ and the system of all σ-closed maximal ideals of $$\mathcal{A}$$ . For MV-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between MV-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.

A NUMBER of the conceptual premises upon which statistical science and practice have long been based have recently been attacked and discarded as harmful to statistical development in the Union of Soviet Socialist Republics. It seems incumbent upon statisticians in other countries to study the Soviet theses, to determine the extent to which differences are real and not merely semantic, and to take issue with those appear to undermine the structure of upon which their own work is founded. The present paper is intended as a preliminary analysis of this kind and its conclusions should be regarded as tentative. The Soviet attack, in the writer's opinion, can best be interpreted as a new phase of the ancient conflict between dogma and science. The dogmas upon which it leans have the essential characteristics of those in a revealed religion. Pretending to be scientific, these dogmas are actually anti-scientific in spirit and in the consequences which would follow their general acceptance. They are part of a new orthodoxy which seeks to impose revelation and arbitrary theological dictates upon the reason and the scientific judgments of men. Freedom of and expression are incompatible with theocracy and they are debarred from Soviet statistical doctrine. In order to support this interpretation I must describe the recent promulgation of Soviet statistical doctrine in some detail. This occurred during a two day Conference on Methodology at the Central Statistical Administration of the USSR in Moscow on February 20-2I, I950. A summary record has been published and appears in translation in the official journals of some other countries.' Soviet leaders have previously pronounced orthodox doctrines in art and literature, economics, biology, and in other scientific fields. I am among those who felt there was reason to hope tllat as an important tool for scientific analysis, statistics could escape such a doctrinal imposition. This hope has been unfounded. The conference was opened by the Chief of the Central Statistical Administration, Mr. V. N. Starovskiy. He explained that harmful bourgeois influences and anti-Marxist distortion in Soviet statistical science and literature hamper its development. He identified the main obstacle to the development of statistical science as the formal mathematics school of thought which considers statistics (to be) a universal science for the study of nature and society based ultimately on the mathematical law of large numbers and not on MarxistLeninist theory. A report on the correct theoretical basis of statistics was then given by Mr. V. A. Sobol of Mr. Starovskiy's staff. Statistics, said Mr. Sobol, is a social science, whose tasks and theoretical foundation are formulated in the works of Lenin and Stalin. Its tasks are to aid in the building of a communist society. Its theoretical foundation rests upon historical materialism and (communist) political economy. Incorrect views, which insidiously lean upon the theory of probability, have been expressed by such writers as Academician V. S. Nemchinov, who supported the chromosome theory of heredity at the 6 August I948 session of the All-Union Academy of Agricultural Sciences. Among others tainted with heresy were Mr. I. Yu. Pisarev (also of Mr. Starovskiy's staff) who was the author of the article on Statistics in the Great Soviet Encyclopedia. Following Sobol's report, according to the record. there was lively discussion. Most of *A paper contributed to the Twenty-Seventh Session of the International Statistical Institute and published here with permission of the Institute and the Indian National Committee for the International Statistical Conferences of 195'. See, e.g., Vestnik Statistiki, c. I, 1950 (publication of the Central Statistical Office of the Ukrainian S.S.R.); and Statisticky Zpravodaj, Rocnik XIII, I5 rijna 1950, Cislo 809, pp. 253-69 (Statistical Bulletin published by the State Statistical Office of Czechoslovakia).

In recent years an increasing interest in the studies on outlier can be observed, however, for the time being there exists no general definition of outlier. In the present paper we introduced a generic descriptive definition of outlier. We observed that the outlier problems had so far been treated in statistical way without paying proper attention to probabilistic-theoretic backgrounds. In view of this gap, we made an attempt to establish a probabilistic background theory. Within this framework, the large deviations are considered as probabilistic-theoretic model of outlier, and the interrelationship of the laws of large numbers, the central limit theorems and the large deviations are clarified. These considerations are specialized for the case of statistical sample, which is important from the point of view of the assessment of data quality. Some methodological and historical aspects of geodesy, geophysics and astronomy are mentioned, too. We revealed that the data analysis carried out by Kepler in the process of discovery of his famous elliptic law of planetary motion has relevance to the outlier problem. This methodologically interesting fact is a new result in the history of geosciences. We established that the accuracy of Chebyshev inequality increases as the deviation of the random variable involved from its expectation, increases. The possibility of application of Chebyshev inequality to the outlier problem is pointed out.