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8 votes
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Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?

Presumably here the null hypothesis is $H_0:$ You are not pregnant the alternative hypothesis is $H_1:$ You are pregnant so being pregnant would be the positive result. You take a pregnancy test ...
Henry's user avatar
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7 votes
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Is First Order Stochastic Dominance (FOSD) relation convex?

The first order stochastic dominance relation is convex. An easy way to prove this is to use the property that a cdf $F$ FOSD another cdf $G$ if and only if $F(x)\le G(x)$ for all $x$. That is, $F$ ...
Herr K.'s user avatar
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6 votes
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second order stochastic dominance without the same mean

Let $u(x) = x$ which is increasing and concave. Then the defining condition of SOSD reads $$\int x\mathrm dF(x)\ge \int x\mathrm dG(x) \implies E_F(X) \geq E_G(X) \tag{1}$$ ..which would contradict ...
Alecos Papadopoulos's user avatar
6 votes
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Difficulty in understanding the notation related from probability theory with game theory

We have that ${\cal I} = ((X^i)_i, \mu)$​ and ${\cal J} = ((Y^i)_i, \nu)$​ are two information structures. An Interpretation mapping for player $i$​​ is a mapping $\phi^i: X^i \to \Delta(Y^i)$​ so it ...
tdm's user avatar
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6 votes
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A question about conditional expectation involving independence

Well assuming the random variables are absolutely continuous, you can use densities: $$f(u|x,v)=\frac{f(u,v,x)}{f(x,v)}=\frac{f(u,v)f(x)}{f(x,v)}=\frac{f(u|v)f(v)f(x)}{f(x,v)}=f(u|v),$$ where the ...
Golden_Ratio's user avatar
6 votes

What does it mean when I say that CDF is bounded away from 1?

This would mean that $\sup\{F(\theta):\,\underline\theta\le\theta\le\bar\theta\}<1$, which makes no sense as $F(\bar\theta)=1$.
VARulle's user avatar
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5 votes
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understanding the proof of stochastic dominance.

That's just a question of notation. Note that $$\mathcal{d}F(t) = f(t)\mathcal{d}t$$ Then the proof should be easy to understand.
saguru's user avatar
  • 181
5 votes

How to average CDFs of one variable across years

Why don't you just take a weighted average? Suppose you have ten years $t \in \{1,...,10\}$ and year $t$ has $N_t$ observations such that in total you have $\sum_t N_t=N$ observations. Let the year-$t$...
Bayesian's user avatar
  • 5,291
5 votes
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Understanding the properties of extensive form games

So $h$ is just some history of the game. Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities ...
Walrasian Auctioneer's user avatar
5 votes
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What is the intuition behind Blackwell's Equivalence Theorem on Information Structures?

To answer the first part of your question, we do not need any more assumptions for the comparison of experiments (besides some measurability issues). Before going on, I'll fix some notations to ones ...
djsteve's user avatar
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5 votes
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What does it mean when I say that CDF is bounded away from 1?

Does it mean that F can never take the value 1 in this interval or that F never takes the value 1? In a sense yes, but it must be specified. Roughly speaking, it means that it is 'enough away' from $...
BakerStreet's user avatar
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5 votes
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Deriving an equation in Banerjee "A Simple Model of Herd Behavior" (1992)

To get the required probability, we just need to sum the probability of all the events with the initial history of the following type. The reason being that after any initial history of this type, it ...
Amit's user avatar
  • 9,011
4 votes
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Investment and probability

The simplest thing is to calculate the expected cost in the two scenarios: Scenario 1: Develop both at the same time $$E[cost]= X + Y$$ Scenario 2: Wait and see if task two is needed $$E[cost]= X + ...
snoram's user avatar
  • 1,007
4 votes
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Show that $W_t - \int_0^t \xi_s ds$ is forward-measure-Brownian

(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let $W$ be standard Brownian motion with respect to filtration $( \...
Michael's user avatar
  • 2,619
4 votes

How to average CDFs of one variable across years

The answer by @Baysiean proposed to compute a weighted average of the per-period empirical distribution functions $EDF_t(w)$ (where $w$ is the value in the support of a random variable $W$), a value ...
Alecos Papadopoulos's user avatar
4 votes
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How to compare investments with different risk and expected return?

The trade-off between risk and expected returns depends on your own preferences. Assume that you are expected utility maximizer and let the return of the investment be given by the random variable $X$....
tdm's user avatar
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3 votes
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Average ability conditioning on having accepted an offer

If $\alpha \sim U$, then how come there is no expectation in your profit function? The $\alpha$ is unknown and which $\alpha$-types the firm gets depends on salary $v$. This should be reflected in the ...
Bayesian's user avatar
  • 5,291
3 votes

Risk neutral probability for each of 3 states

I cannot really follow your formulas, what is the logic behind them? Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation $$ ...
Giskard's user avatar
  • 29.3k
3 votes

Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?

I'm sorry, this is probably better a comment than an answer, but I don't have sufficient points: In the diagram you've included, Type I and Type II errors are more properly conditional probabilities. ...
Paul Harrison's user avatar
3 votes
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Higher order beliefs and coherency in game theory

The way hierarchies of beliefs are specified, beliefs on the same events are encoded in different places. The basic idea is actually quite simple. You have two players, Ann and Bob, say. Ann's first ...
Michael Greinecker's user avatar
3 votes

Integration by parts with CDF

Hint: Simply apply integration by parts to the integral on the LHS. Simplify and you should arrive at the following expression: \begin{equation} (1-R)-\int_{R-k}^1G(\theta)\mathrm d\theta. \end{...
Herr K.'s user avatar
  • 15.5k
3 votes
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What is the meaning of the support set in game theory?

From Wikipedia: Suppose that $f : X \to \mathbb{R}$ is a real-valued function whose domain is an arbitrary set $X.$ The set-theoretic support of $f$, written $\operatorname{supp}(f),$ is the set of ...
Giskard's user avatar
  • 29.3k
3 votes

Negative certainty equivalent

I mispoke in the comments, this certainty equivalent should indeed not be negative. The certainty equivalent in your example is $w_0+c$, this certain payoff's utility is equivalent with the lottery's....
Giskard's user avatar
  • 29.3k
3 votes
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Bayes’ rule in "The sources of capital misallocation"

It follows immediately from Bayesian Updating using Bayes Rule for normal random variables. You can find derivations in e.g. Baley/Veldkamp (2021): Bayesian Learning See also the related answer at ...
jpfeifer's user avatar
  • 529
3 votes

Resampling for a Probabilistic Model to Balance Outcomes

Imbalanced classes present minimal problem to proper statistical methods. A standard criticism of class imbalance is that it can result in models always or often classifying as the majority class. ...
Dave's user avatar
  • 410
3 votes
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How realistic is the conclusion that players do not change their mixing proportions in response to changes in their own payoffs?

I don't think the major lesson is quite that. If an expected payoff maximizing player mixes between several pure strategies, they must be indifferent between playing all of them. This has the curious ...
Michael Greinecker's user avatar
3 votes
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Slight Uncertainty of Continuation in Repeated Prisoner's Dilemma

Imagine you have played the first 19 rounds. Now a chance event decides on whether there will be another, final, round. What's your optimal action in this last round, in case it actually occurs? ...
VARulle's user avatar
  • 6,994
3 votes

Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?

Formally, a NE is indeed a different mathematical object than a CE. But, roughly speaking, the probability distribution on outcomes generated by a NE is a CE. This is what is really meant when saying ...
VARulle's user avatar
  • 6,994
3 votes
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Third price auction from Auction Theory by Krishna, Order statistics

For questions like this, it can help to first consider the CDF. We can then take the derivative and have our density. I'll give you some hints and you need to fill in the gaps. Ask yourself: What is ...
Bayesian's user avatar
  • 5,291
2 votes

Investment and probability

Regardless of period if we know the cost 100%, it is simply cost. If there is a degree of uncertainty on what the cost will be (such as in the future) we regard it as expected cost. Regarding your ...
Lee Sin's user avatar
  • 661

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