Random Sums Research Articles

Improved face recognition with accelerated robust features improved by means of mean shift k-means clustering

To improve the precision of heterogeneous face recognition model, a heterogeneous face recognition model method based on binary multilayer Gabor Extreme Learning Machine (GELM) is proposed in this paper. Firstly, a random weighted Gabor feature extraction scheme is proposed based on pixel weight. It propagates the locally geometric input image sub-block to the hidden node, and embeds the extracted Gabor feature to the hidden layer. Moreover, it conducts random weighting and sum using a group of Gabor kernels so as to realise convolution operation of non-linear activation function of the propagated pixel; then, it estimates the output layer by means of linear weighting that is similar to Extreme Learning Machine (ELM). At last, the performance of heterogeneous face recognition method of the proposed algorithm is verified through BERC VIS-TIR database and CASIA NIR-VIS 2.0 database.

On the rates of convergence to symmetric stable laws for distributions of normalized geometric random sums

Let X1,X2,... be a sequence of independent, identically distributed random variables. Let ?p be a geometric random variable with parameter p?(0,1), independent of all Xj, j ? 1: Assume that ? : N ? R+ is a positive normalized function such that ?(n) = o(1) when n ? +?. The paper deals with the rate of convergence for distributions of randomly normalized geometric random sums ?(?p) ??p,j=1 Xj to symmetric stable laws in term of Zolotarev?s probability metric.

Efficient expressions for moments of dependent random sums using copulas

Mao and Zhao (2014) derived the mean and variance of a dependent random sum, where the dependence was specified by a copula. Here, we derive expressions for the general moment of the dependent random sum. We also extend Mao and Zhao’s results to the case that the components of the sum are not identically distributed. The practical usefulness of the results (in terms of computational time and computational accuracy) is examined by simulation.

On the law of the iterated logarithm for random exponential sums

The asymptotic behavior of exponential sums ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp ( 2\pi i n_k \alpha ) for Hadamard lacunary ( n k ) (n_k) is well known, but for general ( n k ) (n_k) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random ( n k ) (n_k) , and in this paper we prove the law of the iterated logarithm for ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp (2\pi i n_k \alpha ) if the gaps n k + 1 − n k n_{k+1}-n_k are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of { n k α } \{n_k \alpha \} under the same random model, exhibiting a completely different behavior.

On the accuracy of approximation of the distribution of negative-binomial random sums by the gamma distribution

On the accuracy of approximation of the distribution of negative-binomial random sums by the gamma distribution

Stochastic modeling of global risk severity

The paper formulates a stochastic model by incorporating a discrete random variable, two sequences of discrete random variables, and a sequence of positive random variables. The paper also provides an interpretation of the formulated stochastic model in investigating the severity of a global risk. Sufficient conditions for the representation of the formulated stochastic model as a random sum are also established by the paper.

Spectral expansion of random sum complexes

Let [Formula: see text] be a finite abelian group of order [Formula: see text] and let [Formula: see text] denote the [Formula: see text]-simplex on the vertex set [Formula: see text]. The sum complex [Formula: see text] associated to a subset [Formula: see text] and [Formula: see text], is the [Formula: see text]-dimensional simplicial complex obtained by taking the full [Formula: see text]-skeleton of [Formula: see text] together with all [Formula: see text]-subsets [Formula: see text] that satisfy [Formula: see text]. Let [Formula: see text] denote the space of complex-valued [Formula: see text]-cochains of [Formula: see text]. Let [Formula: see text] denote the reduced [Formula: see text]th Laplacian of [Formula: see text], and let [Formula: see text] be the minimal eigenvalue of [Formula: see text]. It is shown that if [Formula: see text] and [Formula: see text] are fixed, and [Formula: see text] is a random subset of [Formula: see text] of size [Formula: see text], then [Formula: see text]

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein\u2019s Method I

The variance-gamma (VG) distributions form a four-parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein’s method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper (Gaunt 2018), which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.

Bounds of decryption failure probability in NTRUEncrypt encryption scheme for a fixed key

The asymmetric encryption scheme NTRUEncrypt is one of the fastest post-quantum encryption schemes. To date, there are several versions of this encryption scheme but all of them have an unwanted feature that assumes decryption failure. Besides the inconvenience for authorized users, this feature leads to specific attacks on the encryption scheme and consequently reduces its security. The traditional approach to estimating the decryption failure probability assumes that this probability is determined by random selection of all elements used to form the encrypted message: the plain text, the key and so-called randomizing polynomial. At the same time, from a practical point of view, a more adequate indicator of the failure frequency is the set of probabilities calculated for each fixed value of the secret key. In this article, we get upper bounds for the decryption failure probability for a fixed key for one of the most extensive versions of the NTRUEncrypt encryption scheme. The first of two obtained bounds is approximate in the sense that, when it is proved, the replacement of the probability distribution of certain independent random variables sum by the limit (normal) distribution is carried out. The second obtained bound is due to Hoeffding's inequality and it is not based on any heuristic assumptions. In general, the obtained results provide more adequate information about the frequency of decryption failure for the considered version of NTRUEncrypt and can be used later in choosing the parameters of this encryption scheme to optimize it for security or practicality.

New Bounds on Poisson Approximation for Random Sums of Independent Binomial Random Variables}

In this paper, we use the Stein-Chen method to obtain new bounds on Poisson approximation for random sums of independent binomial random variables. Some results related to sums of independent binomial distributed random variables are also investigated. The received results in the present study are more general and sharper than some known results.

Collective Risk Models with Hierarchical Archimedean Copulas

In actuarial science, collective risk models, in which the aggregate claim amount of a portfolio is defined in terms of random sums, play a crucial role. In these models, it is common to assume that the number of claims and their amounts are independent, even if this might not always be the case. We consider collective risk models with different dependence structures. Due to the importance of such distributions in an actuarial setting, we first investigate a collective risk model with dependence involving the family of multivariate mixed Erlang distributions. Other models based on mixtures involving bivariate and multivariate copulas in a more general setting are then presented. These different structures allow to link the number of claims to each claim amount, and to quantify the aggregate claim loss. Then, we use Archimedean and hierarchical Archimedean copulas in collective risk models, to model the dependence between the claim number random variable and the claim amount random variables involved in the random sum. Such dependence structures allow us to derive a computational methodology for the assessment of the aggregate claim amount. While being very flexible, this methodology is easy to implement, and can easily fit more complicated hierarchical structures.

Some Results on Approximation for Distributions of Random Sums by Poisson Distribution

The main purpose of this paper is to investigate the bounds in Poisson approximation of the distributions for random sums by Poisson distribution via a probability distance based on linear operator introduced by Renyi. Some Le Cam type bounds for distributions of random sums of row-wise independent Poisson-binomial distributed and negative-binomial distributed random variables are obtained.

Forecasting aggregate claims using score‐driven time series models

In the insurance industry, premium estimation and ruin probability valuation depend fundamentally on the aggregate claims distribution. From the mathematical point of view, the aggregated claims variable is a random sum of random variables. Obtaining the analytical expression for its probability distribution is a hard task. In this paper, a new approach is proposed for the modeling of the aggregated claims predictive distribution. The newly proposed generalized autoregressive score models are combined to specify non‐Gaussian distributions for both the number of claims and the claims severity. In all models, appropriate parameters were made time varying according to a score‐driving mechanism. By the use of the fast Fourier transform, we are then able to numerically obtain the aggregated claims distribution. The proposed method is applied to real data, provided by a Brazilian motor insurer.

An improvement of a non-uniform bound for combinatorial central limit theorem

ABSTRACTIn this article, we develop a new Rosenthal Inequality for uniform random permutation sums of random variables with finite third moments and apply it to obtain a sharp non-uniform bound for the combinatorial central limit theorem using the Stein's method and the exchangeable pair techniques. The obtained bound is shown to be sharper than other existing bounds.

On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables

We present upper bounds for supx ∈ ℝ|P{Z N < x} − Φ(x)|, where Φ(x) is the standard normal distribution function, for random sums $$ {Z}_N={S}_N/\sqrt{\mathbf{V}{S}_N} $$ with variances VS N > 0 (S N = X1 + ⋯ + X N ) of centered strongly mixing or uniformly strongly mixing random variables X1, X2, . . . . Here the number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .

A Markov-switching generalized additive model for compound Poisson processes, with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.

Convergence Rate Estimates in the Global CLT for Compound Mixed Poisson Distributions

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278--294], we obtain mom...

Bounds for the Concentration Functions of Random Sums under Relaxed Moment Conditions

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The results obtained are extended to Poisson-binomial, binomial, and Poisson random sums. Under the same assumptions, bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are put forward for the accuracy of approximation of the concentration functions of geometric, negative binomial, and Sichel random sums by exponential, folded variance gamma, and folded Student distributions. Numerical estimates of all the constants involved are written down explicitly.

New bounds on poisson approximation for random sums of independent negative-binomial randomvariables

The aim of this paper is to establish new bounds on Poisson approximation for random sums of independent negative-binomial random variables. The bounds showed in current paper are a uniform bound and a non-uniform bound. The received results in this paper are extensions and generalizations of known results.

A Moment Inequality with Application to Convergence Rate Estimates in the Global CLT for Poisson-Binomial Random Sums

A moment inequality between the central and noncentral third-order absolute moments is proved, which is optimal for every value of the recentering parameter. By use of this inequality there are con...