IFSM representation of Brownian motion with applications to simulation

Several methods are currently available to simulate paths of the Brownian motion. In particular, paths of the BM can be simulated using the properties of the increments of the process like in the Euler scheme, or as the limit of a random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve series. In this paper we first propose a IFSM (Iterated Function Systems with Maps) operator whose fixed point is the trajectory of the BM. We then use this representation of the process to simulate its trajectories. The resulting simulated trajectories are self-affine, continuous and fractal by construction. This fact produces more realistic trajectories than other schemes in the sense that their geometry is closer to the one of the true BM's trajectories. The IFSM trajectory of the BM can then be used to generate more realistic solutions of stochastic differential equations

Least squares volatility change point estimation for partially observed diffusion processes

A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$, with drift $b(x)$ and diffusion coefficient $\sigma(\theta, x)=\sqrt{\theta} \sigma(x)$ known up to $\theta>0$, is supposed to switch volatility regime at some point $t^*\in (0,T)$. On the basis of discrete time observations from $X$, the problem is the one of estimating the instant of change in the volatility structure $t^*$ as well as the two values of $\theta$, say $\theta_1$ and $\theta_2$, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length $\Delta_n$ with $n\Delta_n=T$. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant

On Rényi information for ergodic diffusion processes

In this paper we derive explicit formulas of the R\'enyi information, Shannon entropy and Song measure for the invariant density of one dimensional ergodic diffusion processes. In particular, the diffusion models considered include the hyperbolic, the generalized inverse Gaussian, the Pearson, the exponential familiy and a new class of skew-t diffusion

Controlling for Selection Bias in Social Media Indicators through Official Statistics: a Proposal

With the increase of social media usage, a huge new source of data has become available. Despite the enthusiasm linked to this revolution, one of the main outstanding criticisms in using these data is selection bias. Indeed, the reference population is unknown. Nevertheless, many studies show evidence that these data constitute a valuable source because they are more timely and possess higher space granularity. We propose to adjust statistics based on Twitter data by anchoring them to reliable official statistics through a weighted, space-time, small area estimation model. As a by-product, the proposed method also stabilizes the social media indicators, which is a welcome property required for official statistics. The method can be adapted anytime official statistics exists at the proper level of granularity and for which social media usage within the population is known. As an example, we adjust a subjective wellbeing indicator of \u201cworking conditions\u201d in Italy, and combine it with relevant official statistics. The weights depend on broadband coverage and the Twitter rate at province level, while the analysis is performed at regional level. The resulting statistics are then compared with survey statistics on the \u201cquality of job\u201d at macro-economic regional level, showing evidence of similar paths

On the Goodness-of-Fit Tests for Some Continuous Time Processes

We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size $\alpha$ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.Comment: 22 pages, 2 figure

Estimating nonresponse bias and mode effects in a mixed mode survey

In mixed-mode surveys, it is difficult to separate sample selection differences from mode-effects that can occur when respondents respond in different interview settings. This paper provides a framework for separating mode-effects from selection effects by matching very similar respondents from different survey modes using propensity score matching. The answer patterns of the matched respondents are subsequently compared. We show that matching can explain differences in nonresponse and coverage in two Internet-samples. When we repeat this procedure for a telephone and Internet-sample however, differences persist between the samples after matching. This indicates the occurrence of mode-effects in telephone and Internet surveys. Mode-effects can be problematic; hence we conclude with a discussion of designs that can be used to explicitly study mode-effects

An Italian Subjective Well-being Index: the Voice of Twitter Users from 2012 to 2017

Since 2012, driven by the desire to propose a subjective well-being index complementary to the traditional measures, with high time and space frequency, our team evaluates, analysing Twitter data, a composite index that captures various aspects and dimensions of individual and collective life. The Twitter data analysis is carried on with a human supervised sentiment analysis method, the Integrated Sentiment Analysis (iSA) algorithm, where a (even non random) sample of texts is first classified by human coders, in order to create a training set, and then the rest of the corpus is classified by the machine learning method. The Subjective Well-being Index (SWBI) is a multidimensional indicator whose components were inspired by the dimensions adopted for the Happy Planet Index provided by the New Economic Foundation. In detail, it consists of eight dimensions that describe three different areas: personal well-being, social well-being and well-being at work. The Italian Subjective Well-being Index (SWBIITA), that we propose here, audits the Italian subjective well-being revealed by tweets acquired via the public Twitter API, written in the Italian language, and posted from Italy from January 2012 to December 2017. Around 1 to 5% of the data includes geo-referenced information, which allows us to provide an index at local level. It should be noted, as a feature of this index, that SWBIITA is not the result of the aggregation of individual well-being measurement, but it directly estimates the aggregate composition of sentiment within the Italian society. In this work, after a weighting procedure adopted to partially overcome the selection bias caused by the use of data from social network, we describe the SWBIITA dimensions in the considered period at the regional level. Moreover we compare our results with the currently available data provided by Italian official statistics, emphasizing novelties, similarities and differences

Measuring Social Well Being in The Big Data Era : Asking or Listening?

The literature on well being measurement seems to suggest that "asking" for a self-evaluation is the only way to estimate a complete and reliable measure of well being. At the same time "not asking" is the only way to avoid biased evaluations due to self-reporting. Here we propose a method for estimating the welfare perception of a community simply "listening" to the conversations on Social Network Sites. The Social Well Being Index (SWBI) and its components are proposed through to an innovative technique of supervised sentiment analysis called iSA which scales to any language and big data. As main methodological advantages, this approach can estimate several aspects of social well being directly from self-declared perceptions, instead of approximating it through objective (but partial) quantitative variables like GDP; moreover self-perceptions of welfare are spontaneous and not obtained as answers to explicit questions that are proved to bias the result. As an application we evaluate the SWBI in Italy through the period 2012-2015 through the analysis of more than 143 millions of tweets