Estimation of a probability density function and its derivatives

We investigate statistical estimates of a probability density distribution function and its derivatives. As the starting point of the investigation we take a priori assumptions about the degree of smoothness of the probability density to be estimated. By using these assumptions we can construct estimates of the probability density function itself and its derivatives which are distinguished by the high rate of decrease of the error in the estimate as the sample size increases.Translated from Matematicheskie Zametki, Vol. 12, No. 5, 621–626, November, 1972.     

Asymptotically optimal choice of the smoothing parameter in a nonparametric kernel estimator for a probability density

This paper presents a method of estimation of an “optimal” smoothing parameter (window width) in kernel estimators for a probability density. The obtained estimator is calculated directly from observations. By “optimal” smoothing parameters we mean those parameters which minimize the mean integral square error (MISE) or the integral square error (ISE) of approximation of an unknown density by the kernel estimator. It is shown that the asymptotic “optimality” properties of the proposed estimator correspond (with respect to the order) to those of the well-known cross-validation procedure [1, 2]. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 67–80, Perm, 1990.     

Computing the probability density function of the stable Paretian distribution

The practical implementation of a stable Paretian model is a nontrivial task, because—with the exception of a few special cases—its probability density function cannot be expressed analytically. Here, we present an algorithm for calculating the probability density function of the asymmetric stable Paretian distribution. Due to the use of the Fast Fourier Transform, the algorithm is computationally efficient, easily implemented, and of similar accuracy as existing algorithms.     

Properties of the probability density function of the non-central chi-squared distribution

In this paper we consider the probability density function (pdf) of a non-central χ² distribution with arbitrary number of degrees of freedom. For this function we prove that can be represented as a finite sum and we deduce a partial derivative formula. Moreover, we show that the pdf is log-concave when the degrees of freedom is greater or equal than 2. At the end of this paper we present some Turán-type inequalities for this function and an elegant application of the monotone form of l'Hospital's rule in probability theory is given.     

Total probability of two‐parameter generalized transition function

In this paper, we research some properties of the total probability of a two‐parameter generalized transition function and obtain some important and interesting results.     

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.     

Asymptotic behavior of statistical estimates of the shift parameter for samples with unbounded density

The present paper is a continuation of the authors' work [1]. As in [1], we consider a sample from a general collection with distribution density f(x– ) depending on the shift parameter . In contrast to [1] it is assumed that the function f(x) is unbounded in a neighborhood of points x₁,h., x_N where it can be represented in the form (1.1). The main results assert that for the Bayes estimates t_n the normalized differences n^1/(1+)(t_n–) have a proper limit distribution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 175–184, 1976.     

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions. Research supported by NSERC of Canada.     

Markov shift in non-commutative probability

We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of ‘quantum detailed balance’ introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.     

Estimating a general function of a quadratic function

Let be an observation from a spherically symmetric distribution with unknown location parameter . For a general non-negative function c, we consider the problem of estimating c(||x − θ||²) under the usual quadratic loss. For p ≥ 5, we give sufficient conditions for improving on the unbiased estimator γ₀ of c(||x − θ||²) by competing estimators γ _s = γ₀ + s correcting γ₀ with a suitable function s. The main condition relies on a partial differential inequality of the form k Δs + s ² ≤ 0 for a certain constant k ≠ 0. Our approach unifies, in particular, the two problems of quadratic loss estimation and confidence statement estimation and allows to derive new results for these two specific cases. Note that we formally establish our domination results (that is, with no recourse to simulation).      

Estimation of heavy-tailed probability density function with application to Web data

Summary  Common non-parametric estimators of a probability density function (PDF) show bad performance for heavy-tailed PDFs. Using a parametric approximation of the true cumulative distribution function (CDF), the transformation-retransformation of the data is explored here as a useful tool for the reliable PDF prediction. The PDF estimators are compared by their capacity to solve a classification problem. Simulation results and an application to Web data analysis are presented, too.     

On sequential estimation of the shift parameter

The problem is considered of estimating the minimal sample size that guarantees the required accuracy, with confidence level fixed, of the estimate of the expectation. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 32–36, Perm, 1991.     

Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion

We consider a diffusion (ξ _t ) _t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ? of interest. We consider sequences of local models at ? corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ?. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ?, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n ^?3/2 at ?, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n ^?2 at ?, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.     

Asymptotic approximation for the probability density function of an arbitrary sequence of random variables

We establish a large deviation approximation for the density function of an arbitrary sequence of random variables. The results are analogous to those obtained by Chaganty and Sethuraman (1985). We apply our theorems to the sample variance and the Mann–Whitney two-sample statistic.     

Estimating the value of a linear functional of a distribution density

One considers the problem of the estimation of a continuous linear functional L(f) in L_p at an unknown point f from a sample X₁,..., X_n of size n from a distribution with density f.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 45–59, 1986.     

Inadmissibility of polynomial estimates of the shift parameter

In this note we demonstrate the inadmissibility of an extensive class of polynomial estimates of the shift parameter in the case of a quadratic loss function.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 885–894, December, 1973.     

Estimating the distance to a monotone function

In standard property testing, the task is to distinguish between objects that have a property 𝒫 and those that are ε‐far from 𝒫, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy 𝒫. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…,n} ↦ R is at distance ε_f from being monotone if it can (and must) be modified at ε_fn places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses ε_f. The running time of our algorithm is O(ε_f⁻¹ log log ε_f^{− 1} log n), which is optimal within a factor of loglog ε_f⁻¹ and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε_f⁻¹ log nlog log log n). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Estimating a density under pointwise constraints on the derivatives

Suppose we want to estimate a density at a point where we know the values of its first or higher order derivatives. In this case a given kernel estimator of the density can be modified by adding appropriately weighted kernel estimators of these derivatives. We give conditions under which the modified estimators are asymptotically normal. We also determine the optimal weights. When the highest derivative is known to vanish at a point, then the bias is asymptotically negligible at that point and the asymptotic variance of the kernel estimator can be made arbitrarily small by choosing a large bandwidth.     

Estimating the first zero of a characteristic function

For a characteristic function (Fourier transform of a probability distribution), the first zero encodes important information. We present a general lower bound estimation of the first zero in terms of a moment of any order. The result proves the complementary nature between the first zero and moments, and has interesting implications for quantum mechanical uncertainty relations. To cite this article: S. Luo, Z. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).