Exponential moving average versus moving exponential average

In this note we discuss the mathematical tools to define trend indicators which are used to describe market trends. We explain the relation between averages and moving averages on the one hand and the so called exponential moving average (EMA) on the other hand. We present a lot of examples and give the definition of the most frequently used trend indicator, the MACD.     

Complete q-order moment convergence of moving average processes under φ-mixing assumptions

Let {Y _i , ?∞ < i < ∞} be a doubly infinite sequence of identically distributed φ-mixing random variables, and {a _i , ?∞ < i < ∞} an absolutely summable sequence of real numbers. We prove the complete q-order moment convergence for the partial sums of moving average processes $\left\{ {X_n = \sum\limits_{i = - \infty }^\infty {a_i + Y_{i + n} ,n \geqslant 1} } \right\}$ based on the sequence {Y _i , ?∞ < i < ∞} of φ-mixing random variables under some suitable conditions. These results generalize and complement earlier results.     

Functional regular variation of Lévy-driven multivariate mixed moving average processes

We consider the functional regular variation in the space $\mathbb {D}$ of càdlàg functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int \int f(A, t - s) \Lambda (d A, d s)$ . We give sufficient conditions for an MMA process $(X_t)$ to have càdlàg sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb {D}$ if the driving Lévy basis is regularly varying and the kernel function f satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered in detail.     

Can adaption help on the average?

Summary We study adaptive information for approximation of linear problems in a separable Hilbert space equipped with a probability measure . It is known that adaption does not help in the worst case for linear problems. We prove that adaption also doesnot help on the average. That is, there exists nonadaptive information which is as powerful as adaptive information. This result holds for orthogonally invariant measures. We provide necessary and sufficient conditions for a measure to be orthogonally invariant. Examples of orthogonally invariant measures include Gaussian measures and, in the finite dimensional case, weighted Lebesgue measures.This research was supported in part by the National Science Foundation under Grant MCS-7823676     

Are unbounded linear operators computable on the average for Gaussian measures?

Traub and Werschulz [Complexity and Information, Cambridge University Press, New York, 1999] ask whether every linear operator S:⊆X→Y$S:\subseteq X\to Y$ is “computable on the average” w.r.t. a Gaussian measure on X. The question is inspired by an analogous result in information-based complexity on the average-case solvability of linear approximation problems. We give several interpretations of Traub and Werschulz’ question within the framework of type-2 theory of effectivity. We have negative answers to all of these interpretations but the one with minimal requirements on the algorithm's uniformness. On our way to these results, we give an effective version of the Mourier–Prokhorov characterization of Gaussian measures on separable Hilbert spaces.     

When can we improve on sample average approximation for stochastic optimization?

We explore the performance of sample average approximation in comparison with several other methods for stochastic optimization. The methods we evaluate are (a) bagging; (b) kernel density estimation; (c) maximum likelihood estimation; and (d) a Bayesian approach. We use two test sets: first a set of quadratic objective functions allowing different types of interaction between the random component and the univariate decision variable; and second a set of portfolio optimization problems. We make recommendations for effective approaches.     

Complete moment convergence of moving average processes under <Emphasis Type="Italic">ρ</Emphasis>-mixing assumption

Let {Y _i : ?∞ < i < ∞} be a doubly infinite sequence of identically distributed ρ-mixing random variables, and {a _i : ?∞ < i < ∞} an absolutely summable sequence of real numbers. In this paper we prove the complete moment convergence for the partial sums of moving average processes \(\{ X_n = \sum\limits_{i = - \infty }^\infty {a_i Y_{i + n,} n \geqslant 1} \} \) based on the sequence {Y _i : ?∞ < i < ∞} of ρ-mixing random variables under some suitable conditions.     

The average distance on the Sierpiński gasket

Summary The canonical distance of points on the Sierpiski gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysis of graphs connected with the Tower of Hanoi problem.     

Quasi-likelihood estimation of average treatment effects based on model information

In this paper, the estimation of average treatment effects is considered when we have the model information of the conditional mean and conditional variance for the responses given the covariates. The quasi-likelihood method adapted to treatment effects data is developed to estimate the parameters in the conditional mean and conditional variance models. Based on the model information, we define three estimators by imputation, regression and inverse probability weighted methods. All the estimators are shown asymptotically normal. Our simulation results show that by using the model information, the substantial efficiency gains are obtained which are comparable with the existing estimators.     

Cohen’s linearly weighted kappa is a weighted average

An n × n agreement table F?=?{f _ij } with n ?? 3 ordered categories can for fixed m?(2??? m??? n ? 1) be collapsed into ${\binom{n-1}{m-1}}$ distinct m × m tables by combining adjacent categories. It is shown that the components (observed and expected agreement) of Cohen??s weighted kappa with linear weights can be obtained from the m × m subtables. A consequence is that weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to the m?× m tables, where the weights are the denominators of the kappas. Moreover, weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to all nontrivial subtables.     

Nonstationary denumerable state Markov decision processes – with average variance criterion

In this paper, we consider the nonstationary Markov decision processes (MDP, for short) with average variance criterion on a countable state space, finite action spaces and bounded one-step rewards. From the optimality equations which are provided in this paper, we translate the average variance criterion into a new average expected cost criterion. Then we prove that there exists a Markov policy, which is optimal in an original average expected reward criterion, that minimizies the average variance in the class of optimal policies for the original average expected reward criterion.     

On the average crosscap number II: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less thanβ(G)-1/2β(G)-1β(G) and not larger thanβ(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

Little’s law when the average waiting time is infinite

One version of Little’s law, written as $L = \lambda w$ , is a relation between averages along a sample path. There are two others in a stochastic setting; they readily extend to the case where the average waiting time $w$ is infinite. We investigate conditions for the sample-path version of this case to hold. Published proofs assume (our) Eq. (3) holds. It is only sufficient. We present examples of what may happen when (3) does not hold, including one that may be new where $w$ is infinite and $L$ is finite. We obtain a sufficient condition called “weakly FIFO” that is weaker than (3), and through truncation, a necessary and sufficient condition. We show that (3) is sufficient but not necessary for the departure rate to be equal to the arrival rate.     

Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

The average errors for Hermite-Fejr interpolation on the Wiener space

For 1≤ p ∞, firstly we prove that for an arbitrary set of distinct nodes in [-1, 1], it is impossible that the errors of the Hermite-Fejr interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on [-1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejr interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1≤ p ∞ and 2≤ q ∞, the p-average errors of Hermite-Fejr interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejr interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.     

Null controllability of the viscous Camassa–Holm equation with moving control

In this paper, we study the null controllability of the viscous Camassa–Holm equation on the one-dimensional torus. By using a moving distributed control, we obtain that the system is null controllable for a given data with certain regularity.     

An average of special values of Dirichlet series of Rankin–Selberg type

The Ramanujan Journal - For an integer n and a Dirichlet character $$\xi $$ modulo N, we denote by $$\mathcal {S}_n(N,\xi )$$ the space of cusp forms of weight n with respect to $$\varGamma _0(N)$$...