I have heard that people often handle uncertainty using point estimates not probability distributions. However, I have been unable to find any evidence for this in the heuristics and biases literature. I was wondering if anyone could point me to some relevant papers?

To clarify, there is no question whether some people have some beliefs about the higher moments of distributions. However, it would not shock to me to learn that when (for example) considering by how much the stock market will rise tomorrow, people form a point estimate (perhaps using historic averages) but do really not think about the uncertainty surrounding their estimate.


Perhaps there is some evidence toward your claim, but I would argue that in most situations, people do not use point estimates (although some "smoothing" likely occurs). In particular, there is no doubt that people care about the second movement (variance). The commonly employed, and robustly empirically documented, notion of risk aversion captures exactly this. Given two risky prospects with the same expectation (say different assets) it seems that most people prefer the one with lower variance (i.e., the less risky prospect). Think: which would you chose when given the option between \$50 for sure or a 50/50 toss up between \$0 and \$100? The difference between these two options is entirely in their second moment.

The evidence for this is so vast and varied it is hard to even give an encompassing reference; but since you ask about the stock market, it is worth pointing out that one of the key explanations for why stocks have a higher average return than bonds is that they are more risky. I imagine a google scholar search of 'risk aversion in the stock market" will be fruitful.

It is worth noting that there is also empirical evidence that people care about the third moment (skewness). This working paper by Yusufcan Masatlioglu, A Yesim Orhun, and Collin Raymond is an experiment in which subjects

reveal a strong preference for positive skew over negative skew; in other words for ruling out more uncertainty about the desired outcome (and tolerating uncertainty about the undesired outcome) compared to ruling out more uncertainty about the undesired outcome (and tolerating uncertainty about the desired outcome).

As for even higher moments, the jury is out, but I would imagine their importance diminishes rather quickly. The complexity of the distributions needed to generate differential higher moments increases and I would think heuristics would start to kick in.

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    $\begingroup$ (+1) I am fairly sure most SE visitors (I do not write most people) would be indifferent between \$50 sure and a 50/50 toss up between \$0 and \$100. At least this is what I find when I talk about it with my (perhaps wealthier than average) students. How about upping the stackes to \$50 million and a toss up between \$0 and \$100 million? $\endgroup$ – Giskard Apr 23 '18 at 21:04

I think there is a missing concept in your question. If you are discussing simple gambles and not things like the stock market, then the distributions that are involved usually have sufficient statistics.

A statistic is sufficient for a parameter if the point estimate could be substituted for the data itself with no loss in information. A statistic, $t$, is said to be sufficient for $\theta$ if $\Pr(\mathbf{x}|\theta)=\Pr(t|\theta)$, where $\mathbf{x}$ is a vector of data. What this implies is that the implied distribution from an observed set of data has a perfect substitute in the vector of point statistics when sufficiency holds. They are indistinguishable concepts mathematically.

If you are talking about the stock market, then those distributions lack both moments and sufficient statistics and people need to use the distribution or fully process the uncertainty of the parameters using Bayesian methods as Frequentist solutions do not exist that are admissible in the general case. The reason is relatively simple.

Consider a stock traded on the NYSE. It is sold in a double auction with many potential buyers and many potential sellers. Since the stock is sold in a double auction it follows that the winner's curse does not obtain. Since the winner's curse does not obtain, it follows that the rational behavior is to bid your expectation regarding the price. Since there are many potential buyers and sellers, the limiting distribution of those prices is the normal distribution from the central limit theorem.

If we assume we are in equilibrium, then the purchase price and the selling price are both normally distributed ignoring bankruptcies, mergers and liquidity costs. Those do not change the general principle here, but do radically alter the mixture distribution. Nonetheless, if we center our graph around $(p_t^*,p_{t+1}^*)$ then we could also think of it as being $(0,0)$ in the error space. The return can be operationally defined as $$r_t=\frac{p_{t+1}}{p_t}-1,$$ so it follows that returns are the ratio of two normal distributions. In error space, the distribution of two normals centered around zero is the Cauchy distribution. If you then translate the distribution back to price space and truncate due to the limit of liability your distribution of returns must be $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$

Although you could manually verify the absence of statistics by the Neyman Fisher Factorization Theorem, it is well known that the statistics from this distribution are not sufficient. This can also be seen from the Pitman–Koopman–Darmois theorem which basically states that only distributions in the exponential family of distributions, such as the normal, have sufficient statistics. Joint sufficiency does exist for purposes of inference, but not for projective purposes which are what matter here. As such, any point estimator not constructed from the Bayesian posterior predictive density will lose information, this is amplified by the truncation.

The Bayesian posterior predictive distribution can be used because it is defined as $\Pr(\tilde{x}|\mathbf{x})$. Notice that there are no parameters at all in that probability statement.

The problem with this distribution is that neither skew nor kurtosis are even defined for it.

Consequently, higher moment discussions with stocks are deeply flawed as the above distribution has no moments at all.

Old articles such as

SCOTT, R. C. and HORVATH, P. A. (1980), On The Direction of Preference for Moments of Higher Order Than The Variance. The Journal of Finance, 35: 915-919.

depend upon the existence of moments in the distributions at all.

Unfortunately, I couldn't find specific literature to what you are talking about outside the finance literature and most of it is built on flawed assumptions. It would be interesting to see how people respond to skewed distributions like waiting time models that possess sufficiency. I couldn't find an example.

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