A stochastic process describes evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples - Gaussian process, Markov Process, etc.

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How to calibrate REGARCH model? (Finding MLE)

I have a model whose specification is $$R_{t+1} = \sigma_{t+1} \varepsilon_{t+1} \text{ with } \ \ \ \ \varepsilon_{t+1} \sim N(0,1)$$ $$r_{t+1} \sim N(0.43 + \log \sigma_{t+1}, 0.29^2)$$ $$\log\...
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Testing for Stochastic Dominance - Mann Whitney with Unequal Variance

I've been looking for methods like Mann Whitney, but without homogeneity of variance. So far, I found that I suppose to test for stochastic dominance instead - i.e. modifying the null hypothesis. ...
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13 views

KL Divergence between an i.i.d sequence and non i.i.d sequence

I have two sequences of probability density functions $Q=\prod_{i=1}^{n} q(x_i)$ (independent and identically distributed, i.i.d )$P=p(x_1,\cdots,x_n)$ (non i.i.d). How can I calculate the KL ...
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AR(2) process: are leptokurtic residuals OK?

I have a time series of logarithmic returns. After inspection of the ACF and PACF plots, I tried to fit AR(2), MA(2) and ARMA(1,1) models and eventually found out that the AR(2) version can possibly ...
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49 views

Adam: stochastic gradient descent?

I would like to get a better idea of stochastic gradient descent algorithms, especially and most important Adam, since I've expierenced reasonable results with Adam and refuse to use something "just ...
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17 views

Predicting a Chi-Square Process

Assume that $W(t)$ is a one-parameter stochastic process given by $W(t) := X_1^2(t) + X_2^2(t)$ where $X_i(t)$ are independent copies of a stationary gaussian process with known covariance function. ...
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25 views

Is weakly stationary equivalent to $I(0)$?

I'm currently reading some time series lecture notes. It says that: Weakly stationary (or wide-sense stationary) processes are said to be $I(0)$ (integrated of order $0$). Let's call the above ...
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21 views

Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
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133 views

Wavelet-domain gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be ...
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56 views

What's the variance of the following stochastic integral and is it weakly stationary?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
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27 views

the intuition behind that the variance of increment for Brownian Motion is time interval

Could anybody help me to understand that why is that for Brownian motion, the variance of the increment $Z(t+s)-Z(t)$ is the time interval $s$? I understand the math, but what is the intuition ...
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34 views

predicting the future in a stationary stochastic process

Let's say I have a strictly-stationary stochastic process with known PSD (power spectral density). The process has been running, and I have all the data from time $t=-\infty$ to $t=0$. I want to ...
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15 views

Is it reasonable to have a zero mean Gaussian prior for the coefficients of an AR(p) process, assuming it is stable?

I wanna perform parameter estimation of an underlying AR(p) process given some data. Let's say it's stable. For example an AR(2) process is stable when the conditions $a_2 - a_1 < 1,$ $a_2 + a_1 &...
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30 views

Stochastic vs statistical models

While there are plenty of discussion of the difference between statistics and stochastic, I didn't find a nice explanation for the difference between statistical and stochastic models. Could anyone ...
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26 views

Computing the number of renewals

My question is from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. Page 463, $\S$7.8. There is a paragraph in the book talking about the computation of renewal function: ...
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19 views

Esscher transform (extension)

The Esscher-transform is a well know tool in the financial section. Given a Levy-process $(X_t)_{t\geq 0}$ under $P$. Let be $u$ be real such that $\phi_{t}(u)=\log\left(E\left[\exp(uX_t)\right]\...
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What are the differences between stochastic v.s. fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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27 views

Checking whether a given formula is correct for a homogeneous Markov chain

I am new to cross validated so I hope my question belongs here. I saw in a paper where I study someone claiming the following: Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain (...
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Interpreting graphs from an Audjusted Dickey Fuller in R: library(plm)

I performed an Augmented Dickey Fuller test in R. However I do not understand how to interpret the results of the graphs. Can someone help me with that? ...
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How the process parameters changes with the length of data aggregation?

Is there any general relationship for a process(e.g. ARMA, O-U process) applied to financial data over different time intervals. e.g.In this question there is an answer telling the O.P. to aggregate ...
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34 views

How to fit a discrete distribution that can only be sampled from to count data?

My question is similar to this one. Assume we have a distribution from which we can only sample, but have no information on its pmf and consider further some count data: ...
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1answer
39 views

Predicting the maximum of a function given a set of samples

The main aspects of the question are highlighted in bold Let $f: \mathbb{R}^n \mapsto \mathbb{R}$ be a function. Supposing that we have access to a set of samples $(X,Y)$ obtained by sampling the ...
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39 views

Estimate transition matrix from many short Markov chains

I have a situation where data from the following process is observed: For $i = 1, \dots, n$ let $(X_{i,1}, \dots, X_{i,m_i})$ be a sequence of $m_i$ random variables coming from a discrete-space ...
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$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
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25 views

Expected time between two events

I'm having trouble with the following problem: Consider a game between two players A and B. Player A must complete three tasks each of which take an exponentially distributed amount of time with ...
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What does fixed regressor say about our linearity condition?

The linearity condition states that $\mathbb{E}[y_i]=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for ...
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26 views

Generating Brownian motion on a manifold using charts

Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in $x\left(U\right)\...
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11 views

What methods can be used for tractable computation of probabilities for evolutionary model of non-independent entities?

I'm trying to extend a simple model which works as follows. We have n 'original' entities which each have a colour. This population evolves by the following events, which occur at exponential rates: ...
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26 views

How do I choose the initial features vectors for a Stochastic Gradient Descent trained SVD++ algorithm?

I'm reading the SVD++ Netflix Recommender Systems paper because I want to be able to properly assess this approach to building a recommender system. How should I choose the initial values of $q_i$ ...
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Kolmogorov Forward and Backward Equation Intepretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
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Hypothesis testing to discriminate between two renewal processes

We have time [0,T] to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of inter-...
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83 views

Simple question about Ornstein-Uhlenbeck process

My question comes from this paper. The picture bellow provides a summary of the equations. Suppose prices of two stocks satisfy (2.1) SDE. Then X(t) is expressed as (2.2) and can be modeled with as ...
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28 views

Bayesian Optimization for a Stochastic Target that changes over time

Let's say there is a single slot machine that: costs zero to play can only be played once per day has a payout that is conditionally normal and is a function of the date and time. I want to use ...
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72 views

Best random variable for infinite trials of a true/false event?

if you were to toss a fair coin a finite amount of times, and a success = heads, then the best random variable to represent it would be a binomial random variable. However, if you were to toss it an ...
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What is the difference between a truncated normal distribution and a half normal distribtion in a Stochastic Frontier Analysis?

I am trying to replicate a SFA where the error term u is assumed to have a cumulative normal distribution function truncated from below at zero. In my opinion, that refers to a truncated normal ...
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How to properly show the efficiency of a process?

I'm no statistician but my background is in computer science. At work, we are trying to improve the efficiency of a system where 5 people (A-E) each produce one part of a report and send it to 2 key ...
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Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
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Estimating correlation of two HW1F processes

I have been thinking of an efficient way to estimate 2 HW1F processes efficiently. I assume two processes to be separate short rates (for Libor & Euribor). I was just planning to use the ...
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57 views

Optimization of stochastic computer models

This is a tough topic for myself to google since having the words optimization and stochastic in a search almost automatically defaults to searches for stochastic optimization. But what I really want ...
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73 views

monte carlo simulation using exponential distributions

I'm trying to simulate a stochastic model of deterministic exponential population growth, where $dN/dt = rN$ where $N$ is population size and $r$ is rate ($t$ time). I'm assuming there's no carrying ...
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1answer
83 views

The distinction between stochastic independent variable and measurement error in independent OLS variable

Assume that OLS regression of the form: $$Y_t = X_t'\beta + u_t$$ Suppose $X_t$ are stochastic, thus standard Gauss-Markov assumptions need to be accommodated. Given that: $$\text{E} {(\hat\beta)} ...
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Expressing a non-unique stationary distribution for a markov chain?

I am working ahead of my stochastic processes class, so some of what is written below may be inaccurate. I am working a problem that asks me to compute a stationary distribution for a Markov Chain, ...
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Ergodicity and percentiles

The mean ergodicity of a homogeneous/stationary random field/process is such a property under which given a sufficiently long realization, the mean of the realization converges to the mean of the ...
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Will there ever be an unhappy Tribble in Oz?

Here's an amusing problem brought to me by a student. Although it was originally phrased in terms of mutually annihilating bullets fired at regular intervals by a gun, I thought you might enjoy a ...
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How can I model a process which follows a pdf that changes w.r.t time?

I'm interested in modeling the probability of a gym having had k number of arrivals at time = t. Clearly this should be modeled by some type of time cont. stochastic process but a poisson process will ...
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31 views

Gradient descent for a noisy system

I have a system with tuning parameter $w$. To evaluate this system I use cost function $f(w)$. I try finding the optimum value for $w$ using Gradient Descent starting from $w_0$. The problem with ...
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Why is the periodogram of differenced white noise not flat?

I'm a final year undergrad who was doing a project that involved the implementation of a frequency-domain volatility estimator. I haven't a lot of stats background so not understanding a point that ...
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22 views

Kolmogorov Equations for a 3 state model. CTMC with a 3x3 generator matrix. Solving for $p_{11}$

I have a matrix $Q= \left[ \begin{array}{ccc} -3&3&0\\ 2&-5&3\\ 0&4&-4 \end{array} \right] $ where the state space is $S=[0,1,2]$ I need to solve the ...
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52 views

Joint Density for Renewal Processes

I'm trying to derive the joint density for the time-average age Z and time-average residual life Y for a renewal process, and use that result to determine if Z and Y are independent. If we call $X(t)...
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Deriving the age process CDF from a renewal process

My question is whether I correctly derived the CDF of the steady-state age process. For context on why I' am asking this question is problem 6.1(a) from Introduction to Stochastic Processes by Lawler ...