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24 views

Applying Markov Decision Processes to the Selling House Problem with waiting times

I'd like to apply the Markov Decision Process theory to this problem. We have a house to sell. Each day an offer of $X_n$ comes for the house. Each offer costs an amount $c$ to observe. You may think ...
3
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1answer
60 views

Prove $Z_n = X_n1_{n \le T} + Y_n1_{n+1\ge T}$ is a martingale

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}, \mathbb P)$, let $X = (X_n)_{n \in \mathbb N}$ and $Y = (Y_n)_{n \in \mathbb N}$ be $(\{\mathscr F_n\}, \mathbb ...
0
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1answer
36 views

Prove Doob's using a certain Lemma

I am to prove Doob's (d) in the red box below: What I tried: Since $T < \infty$ a.s., we have $$E[X_T] = E[\lim X_{T \wedge n}].$$ By Fatou's Lemma, we have $$E\left[\lim X_{T \wedge ...
2
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1answer
76 views

How to show that $S \wedge T, S \vee T, S + T$ are stopping times?

From Williams (1991) Probability with Martingales: $$(S \wedge T \le n) = (S \le n) \cup (T \le n) \in \mathscr F_n$$ $$\because (S \le n), (T \le n) \in \mathscr F_n$$ $$(S \ \vee \ T \le n) ...
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0answers
57 views

Optimal stopping rule for sequential subsampling?

I have a set of ranking models added sequential one at a time. Where each model generate ranks for the sample. Finally the expected rank is done by taking the average rank from all models. scenario I ...
2
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1answer
151 views

Secretary Problem (Optimal Stopping) When Interviews Are Costly

The Secretary problem is an optimal stopping problem. Imagine hiring one secretary out of $n$ applicants, who are interviewed in random order and either rejected or hired on the spot (as soon as one ...
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37 views

How to determine the optimal time averaging window

I have two large time series datasets with some background noises. I would assume the two datasets are either lag correlated or lead correlated. I tried to use time averaging to smooth out the dataset ...
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0answers
160 views

Optimal Stopping for Bernoulli One-Armed Bandit with a Fixed, Known Payout

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. Suppose I have two arms $k = {1, ...
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0answers
38 views

How important is the quality of solutions to NP-hard problems arising in machine learning problems?

Machine learning and inference problems give rise to intractable problems. For instance the exact inference in Bayes nets is an NP-hard problem. At the same time there are polynomially tractable ...
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0answers
58 views

bias of an estimator when using stopping rules

Consider the setting where $X_1,X_2,...$ are i.i.d. real-valued random variables with $\mathbb{E}[X_i] = \theta$ and let the random variable $\tau$ be an associated stopping time. I'm wondering what ...
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0answers
111 views

How are optional stopping rules based on e.g. sample confidence (width of confidence interval) biased?

Inspired by this: http://pss.sagepub.com/content/22/11/1359 In the context of open-ended data collection where the necessary sample size cannot be properly estimated, for the purpose of a ...
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0answers
420 views

Understanding Sequential Probability Ratio Test (SPRT) Likelihood Ratio

I am a software developer looking to develop an alternative for the simple hypothesis testing scheme described here. In short, the test works as follows: Two URLs are compared for their ability to ...
13
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3answers
532 views

How could one develop a stopping rule in a power analysis of two independent proportions?

I am a software developer working on A/B testing systems. I don't have a solid stats background but have been picking up knowledge over the past few months. A typical test scenario involves comparing ...
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0answers
652 views

Determining Optimal Number of Cluster in Hierarchical Clustering in Consideration of Variance of Data

I'm applying a Hierarchical Agglomerative Clustering (HAC) for grouping my data and I need to determine the number of the cluster automatically. To determine the optimal number of cluster, I obtain ...
3
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1answer
118 views

Thoughts on model self-penalization amidst difficult parameter estimation

It is well accepted that one should account for model complexity when performing model comparisons, and the general procedure is to penalize more complex models more strongly. While this makes sense ...
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2answers
77 views

Difference between the Stopping Criteria [closed]

I would like to know the difference between below mentioned stopping criteria used in various gradient descent algorithm $\frac{Prev\_fun\_value - curr\_fun\_value}{Pre\_fun\_value} \le tol$ ...
4
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0answers
140 views

Stopping rule for chi-squared discretization algorithm

I developed an algorithm that uses the chi-squared test to perform supervised discretization of a continuous variable. I described it in the paper "ChiD-A Chi-Squared Discretization Algorithm" ...
2
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0answers
76 views

Optimal stopping under partially observable state

This problem is basically the classic asset selling problem but with imperfect state information. In the classical problem, we have an asset that we wish to sell, we receive offers w(0) to w(N-1). ...
3
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2answers
366 views

Optimal stopping from an unknown distribution

The Secretary problem has an algorithm for fixed N and immediate accept/reject (that is, reject reject ... accept one, stop). There are several variants; in mine, secretaries or samples come from a ...