Algorithmical complexity of some statistical decision processes. I
Statistical theory of logical derivability
Algorithmical complexity of some statistical decision processes. II
On -estimators viewed as -estimators
Variational theorems in gnostical theory of uncertain data
Statistical deducibility testing with stochastic parameters
Infinite pseudo-random sequences of high algorithmic complexity
On pseudo-random sequences and their relation to a class of stochastical laws
Kolmogorov complexity and probability measures
Classes of strings (infinite sequences resp.) with a specific flow of Kolmogorov complexity are introduced. Namely, lower bounds of Kolmogorov complexity are prescribed to strings (initial segments of infinite sequences resp.) of specified lengths. Dependence of probabilities of the classes on lower bounds of Kolmogorov complexity is the main theme of the paper. Conditions are found under which the probabilities of the classes of the strings are close to one. Similarly, conditions are derived under...
On unequally spaced AR(1) process
Discrete autoregressive process of the first order is considered. The process is observed at unequally spaced time instants. Both least squares estimate and maximum likelihood estimate of the autocorrelation coefficient are analyzed. We show some dangers related with the estimates when the true value of the autocorrelation coefficient is small. Monte-Carlo method is used to illustrate the problems.
Kolmogorov complexity, pseudorandom generators and statistical models testing
An attempt to formalize heuristic concepts like strings (sequences resp.) “typical” for a probability measure is stated in the paper. Both generating and testing of such strings is considered. Kolmogorov complexity theory is used as a tool. Classes of strings “typical” for a given probability measure are introduced. It is shown that no pseudorandom generator can produce long strings from the classes. The time complexity of pseudorandom generators with oracles capable to recognize “typical” strings...
Stochastic control optimal in the Kullback sense
The paper solves the problem of minimization of the Kullback divergence between a partially known and a completely known probability distribution. It considers two probability distributions of a random vector on a sample space of dimensions. One of the distributions is known, the other is known only partially. Namely, only the conditional probability distributions of given are known for . Our objective is to determine the remaining conditional probability distributions of given such...
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