Том 234, № 6 (2018)
- Жылы: 2018
- Мақалалар: 12
- URL: https://journals.rcsi.science/1072-3374/issue/view/14965
Article
Statistical Analysis of Rounded Data: Measurement Errors vs Rounding Errors
Аннотация
Since data for statistical analysis are always given in a discretized form, observations contain not only measurement errors but also rounding errors which are determined by the discretization step. In this paper we consider situations where the rounding errors are considerable: they are comparable to or even greater (in average) than the measurement errors. It is shown that it can be reasonable to increase the measurement errors in order to reduce the error of the final result.
On Unreported Claims Models and Beta-Model for Randomized Probability of Reported Claims
Аннотация
We study nonlinear models of the expected number of unreported claims by year end with a randomized probability of claims to be reported. According to the developed simulation models the models with randomized probability more adequately describe the actual arrival process of insurance claims.
Nonlinear Regularization of Inverse Problems for Linear Homogeneous Transforms by Stabilized Hard Thresholding
Аннотация
In this paper we consider the problem of inverting linear homogeneous transforms by Vaguelette–Wavelet decomposition and stabilized hard thresholding of noisy wavelet coefficients. We also prove asymptotic normality and strong consistency of the mean-square risk estimate for this method.
Portfolio Analysis with General Commission
Аннотация
The well-known problem of finding explicit formulas for the expected return and risk of portfolios with general commission is completely solved. It is assumed that the commission depends on the asset and the asset position, and on whether the given position is opened or closed. For portfolios with only the budget constraint and initial commission, we prove that the function of expected portfolio return and portfolio variance function are bounded.
Multivariate Analogs of Classical Univariate Discrete Distributions and Their Properties
Аннотация
Some discrete distributions such as Bernoulli, binomial, geometric, negative binomial, Poisson, Polya–Aeppli, and others play an important role in applied problems of probability theory and mathematical statistics. We propose a variant of a multivariate distribution whose components have a given univariate discrete distribution. In fact we consider some very general variant of the so-called reduction method. We find the explicit form of the mass function and generating function of such distribution and study their properties. We prove that our construction is unique in natural exponential families of distributions. Our results are the generalization and unification of many results of other authors.
The Asymptotic Behavior of the Optimal Threshold Minimizing the Probability-of-Error Criterion
Аннотация
In this paper we consider the problem of estimation of a signal function from the noised observations via thresholding its wavelet coefficients. We find the asymptotic order of the optimal threshold that minimizes the probability of the maximum error between the estimates and the true wavelet coefficients exceeding a critical value.
On Large Deviations for Sums of i.i.d. Bernoulli Random Variables
Аннотация
Tail probabilities are studied for the binomial distribution. The Hoeffding inequality is sharpened in this particular case through estimating an integral factor in the Esscher transform, which is omitted in Hoeffding’s proof. This approach was already used by Talagrand (1995) in the general case. However, our results are much more precise. In particular, all involved constants are given in the explicit form.
The Berry–Esseen Bound for General Markov Chains
Аннотация
A rate of convergence in the CLT is studied for recurrent Markov chains. The Berry–Esseen type bound is obtained with explicit dependence on parameters of the chain. The method of the proof is purely analytical. It is based on the complex analysis of generating functions.
Esseen–Rozovskii Type Estimates for the Rate of Convergence in the Lindeberg Theorem
Аннотация
We present structural improvements of Esseen’s (1969) and Rozovskii’s (1974) estimates for the rate of convergence in the Lindeberg theorem and also compute the appearing absolute constants. We introduce the asymptotically exact constants in the constructed inequalities and obtain upper bounds for them. We analyze the values of Esseen’s, Rozovskii’s, and Lyapunov’s fractions, compare them pairwise, and provide some extremal distributions. As an auxiliary statement, we prove a sharp inequality for the quadratic tails of an arbitrary distribution (with finite second-order moment) and its convolutional symmetrization.
Scale Mixtures of Frechet Distributions as Asymptotic Approximations of Extreme Precipitation
Аннотация
This paper is a further development of the results of [20] where, based on the negative binomial model for the duration of wet periods measured in days [16], an asymptotic approximation was proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Fr´echet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor–Fisher distribution. Here we extend this result to the mth extremes, m ∈ ℕ, and sample quantiles. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution [17] and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions [10]. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution of the maximum precipitation is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions this limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. It is also shown that the limit distribution of sample quantiles is the well-known Student distribution. Several methods are proposed for the estimation of the parameters of the asymptotic distributions. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability (“Bayesian”) models.