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foundations of statistics
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foundations of statistics Didactical paper showing how one can maximize chance of getting significant results using inappropriate tricks, and giving recommendations such as that one should specify stopping rule beforehand. Something that is unverifiable (brings benefits to the dishonest people at the cost of the honest peope), and that works differently in the Bayesian approach … %} Simmons, Joseph P., Leif D. Nelson, & Uri Simonsohn (2011) “False-Positive Psychology: Undisclosed Flexibility in Data Collection and Analysis Allows Presenting Anything as Significant,” Psychological Science 22, 1359–1366. {% Together with his ’56 paper the classics that introduce bounded rationality. On informational and computational limits on rationality. %} Simon, Herbert A. (1955) “A Behavioral Model of Rational Choice,” Quarterly Journal of Economics 69, 99–118. {% %} Simon, Herbert A. (1956) “Rational Choice and the Structure of the Environment,” Psychological Review 63, 129–138. {% %} Simon, Herbert A. (1982) “Models of Bounded Rationality, Vols 1 and 2.” The MIT Press, London. {% %} Simon, Leo K. & Maxwell B. Stinchcombe (1995) “Equilibrium Refinement for Infinite Normal-Form Games,” Econometrica 63, 1421–1443. {% Imagine that journals only accept significant results (publication bias), and other than that all rules are satisfied (no p-value hacking for instance). What is the real value of a p-value? If for all studies a single (containing only one parameter) null hypothesis H0 is true, then there will be equally many p-values between 0.05 and 0.04 as between … 0.01 and 0.00. So their distribution is homogenous. The more the alternative hypothesis is true, the more skewed it will be. We can observe the distribution of p-values published in the journal, and then, making all kinds of distributional assumptions, can do simulations that reproduce that distribution of p-values, and then see what the real p-values are to correct for the publication bias. One problem is that this correction does not handle p-hacking and even may reinforce the distortions due to p-hacking. %} Simonsohn, Uri, Leif D. Nelson, & Joseph P. Simmons (2014) “p-Curve and Effect Size: Correcting for Publication Bias Using Only Significant Results,” Psychological Science 9, 666–681. {% %} Simonsohn, Uri, Joseph P. Simmons, & Leif D. Nelson (2014) “Anchoring is Not a False-Positive: Maniadis, Tufano, and List’s (2014) “Failure-to-Replicate” is Actually Entirely Consistent with the Original,” working paper. {% %} Simonson, Itamar & Amos Tversky (1992) “Choice in Context: Tradeoff Contrast and Extremeness Aversion,” Journal of Marketing Research 29, 281–295. {% PT, applications: nonadditive measures, portfolio inertia %} Simonsen, Mario H. & Sérgio R.C. Werlang (1991) “Subadditive Probabilities and Portfolio Inertia,” Revista de Econometria 11, 1–19. {% Use Mazur (1987) discounting function, use hypothetical questions, assume linear utility, and fitted data at an individual level, for N = 17 subjects. Did two measurements separated by one week, and found stable results. %} Simpson, Cathy A., & Rudy E. Vuchinich (2000) “Reliability of a Measure of Temporal Discounting,” Psychological Record 50, 3–16. {% Writes that EU is normative and nonEU may only be “shortcut,” so not just to be used for policy making. %} Sims, Christopher A. (2001) “Pitfalls of a Minimax Approach to Model Uncertainty,” American Economic Review, Papers and Proceedings 91, 51–54. {% %} Singh, Jagbir & William A. Thompson, Jr. (1968) “A Treatment of Ties in Paired Comparisons,” Annals of Mathematical Statistics 39, 2002–2015. {% Theoretical textbook on Bayesian statistics, with introductory chapters on decision foundation of Bayesian statistics. %} Singpurwalla, Nozer (2006) “Reliability and Risk: A Bayesian Perspective.” Wiley, New york. {% DOI: http://dx.doi.org/10.1111/j.1468-0068.2012.00864.x conservation of influence: discuss intentionality %} Sinhababu, Neil (2013) “The Desire-Belief Account of Intention Explains Everything,” Noûs 47, 680–696. {% Paper presented at FUR VII conference in Oslo, 1994 %} Siniscalchi, Marciano (1997) “Conditional Preferences, Ellsberg Paradoxes and the Sure Thing Principle.” In Pierpaolo Battigalli, Aldo Montesano, & Fausto Panunzi (eds.) Decisions, Games and Markets. Studies in Risk and Uncertainty, 31–53, Kluwer Academic Publishers, Dordrecht. {% This paper assumes the Anscombe-Aumann model, where multiple priors was axiomatized by Gilboa & Schmeidler (1989). What this paper adds is a necessary and sufficient condition for a prior to be contained in the set of multiple priors. Such a prior is characterized by the existence of a convex subset of acts such that on this convex subset EU is satisfied w.r.t. the prior, and such that there is no other probability measure with respect to which this holds. In the main result, axiom 6 (no local hedging in the sense that for each sequence of acts converging to an act there is a subsequence of acts that, losely speaking, provide no hedge against each other) characterizes the existence of a finite coverage of acts such that within each coverage, EU holds. While this paper characterizes whether or not a single probability measure is contained in the set of priors, it does not provide a verifiable characterization of the set of priors. For the latter one would have to check for every single probability measure whether or not it is contained, which is an infinite task. The author formulates this point in Ghirardato & Marinacci (2012 p. 2832) as: “that plausible priors are identified individually, rather than as element of a set.” %} Siniscalchi, Marciano (2006) “A Behavioral Characterization of Plausible Priors,” Journal of Economic Theory 128, 91–135. {% Last para pleas for doing descriptive research into ambiguity: “Ultimately, however, I think NW’s critique can be interpreted constructively by proponents of ambiguity. NW’s paper does show that it is difficult to debate the appeal of different approaches to dynamic choice under ambiguity from a purely abstract (“normative”) point of view. New empirical and experimental evidence concerning how individuals actually behave in dynamic situations under ambiguity may provide more effective guidance for theoretical development in this exciting field.” %} Siniscalchi, Marciano (2008) “Two out of Three Ain't Bad: A Comment on “The Ambiguity Aversion Literature: A Critical Assessment”,” Economic Philosophy 25, 335–356. {% S is state space, f is act from S to outcomes. It is AA model with outcomes being probability distributions over prizes, which mathematically amounts to utility being linear in outcomes. P denotes the subjective probability measure on state space S used in EU and elsewhere. V(f) = EUP(f) + A((EP(i(s).u(f(s))))0in) where i is a random variable, density of a signed measure if you want, with P-expectation 0, and the dot following denotes inner product. Because has P expectation 0, the inner product gives the P covariance between i and u(f(s)). Can simplify some by taking and P together as just one signed measure with total measure 0. (Keeping absolute continuity w.r.t. P in the back of one’s mind, primarily to avoid violations of monotonicity.) The depend on the states and not only on their probabilities implying that we do not have probabilistic sophistication. A deviation from probabilistic sophistication is needed to accommodate Ellsberg. A(x) = A(x) for all xn. So, A is a generalized Absolute value function. The idea is that each i captures an informational interaction (ambiguity) between events. And that A is mostly negative and punishes for variance over ambiguous events. So in Ellsberg 3-color with red know color and black and yellow the unknown colors, P assigns 1/3 to all colors, (R) = 0, (B) = 1, (Y) = 1, and A punishes for nonzero covariance with . Big descriptive problem of the model is that A(x) = A(x) excludes inverse-S because, with outcomes in utils, for an unlikely event E the prospect 1E0 is undervalued as much as 1Ec0 is (turn 1E0 into 1E0 and then use weak certainty independence to add 1 util to all outcomes, which does not affect A), whereas inverse-S implies that the former is overvalued but the latter is undervalued. This makes the model descriptively problematic (in addition to the problems of the Anscombe-Aumann model). The s are not unique but become so if sharpness is imposed: then they are required to be orthonormal (linearly independence + orthogonality) and to assign value 0 to any crisp act (crisp means informally entailing no ambiguity or hedge against it, formalized by being replacable in any mixture by its certainty equivalent). The model can be related to anchoring and adjustment à la Einhorn & Hogarth (properly cited by the author on p. 802). The model chosen here with interaction captured through inner product with complementarity between positive and negative part of s primarily captures n “binary” complementarities in a natural way. If the urn contains k exchangeable ambiguous colors with k > 2, then I don’t see an easy way to model this. Maybe many ’s must be defined (for each color one?) and A must capture the k-interactions? Not clear. The axioms characterizing the model are some usual ones: weak ordering (A1), monotonicity (A2), continuity in outcomes (A3), nondegeneracy (A4), weak certainty independence (A5: only mixing with sure prospects to give independence under translations but not under rescalings), monotone continuity (A6) to give countable additivity of P, a probably redundant Complementary translation axiom (A8; only needed to handle two-sided bounded utility), and the crucial axiom of Complementary independence (A7), which I reformulate: Assume that f and f* are complete hedges (their sum is constant as is their 50-50 mixture; the author calls it complementary), and so are g and g*. Assume that f ~ f* and g ~ g*. Then for all mixture weights , f + (1)g ~ f* + (1)g*. Key in this model is pairs of acts that are perfect hedges (complementary) for each other, meaning that they sum to a constant act. Particularly useful are such pairs if they are indifferent (obtainable by adding constant utility to the worst of a pair of perfect hedges). Then their sum gives a constant act equal to the value of the two acts if EU were to hold; i.e., if A were 0. How much this constant act exceeds the certainty equivalent of the acts is how big A is. Thus we can measure the EU functional and also A. Being able to measure EU means that we can also measure P. Complementary independence will ensure, I expect, that the P measured this way is additive. The model holds together with CEU (Choquet expected utility) if and only if there is a probability measure P such that, with W the weighting function, W underweights each event as much as its complement: W(E) P(E) = W(Ec) P(Ec) for all events E. This property obviously contradicts inverse-S. More ambiguity averse results are derived implying same subjective probability P and utility u, characterized by one A function always dominating the other. biseparable utility violated: the model is not biseparable utility, although it does intersect with the latter (see above intersection with CEU). The main reason is that the function A can be too general and nonlinear. For example, take S = {s1,s2}, payment in vNM utility (for instance prizes are [0,100], u is the identity on prizes, and for known probabilities we have EV). p1 = P(s1) = p2 = P(s2) = 0.5, and only one 0 = , defined by (s1) = 1/3 = (s2). A() = || if || 37/3, and A() = |37/3|/2 37/3 if || > 37/3. It means that, as long as outcomes within an act differ by no more than 37, then we have RDU with linear utility and (sj)b (the decision weight of state sj when having the best outcome) = 1/3 and (sj)w (the decision weight of state sj when having the worst outcome) = 2/3. In other words, W(s1) = W(s2) = 1/3. If the difference in outcomes exceeds 37, then whatever the best outcome has more than the worst + 37, is weighted only half as much. Then (using stimuli of Wakker 2010, §4.1) we have, with (x1,x2) denoting the act that yields vNM utility x1 under s1 and x2 under s2, (38,1) ~ (24,8) and (24,1) ~ (10,8) implying, in Wakker’s (2010, Eq. 10.5.2) notation, 38 24 ~tc 24 10. However, (39,0) ~ (24,7) and (24,0) ~ (10,7) imply, 39 24 ~tc 24 10. We have a violation of rank-tradeoff consistency (Wakker 2010 Def. 10.5.5), and RDU is violated by Wakker (2010, Theorem 10.5.6). %} Siniscalchi, Marciano (2009) “Vector Expected Utility and Attitudes toward Variation,” Econometrica 77, 801–855. {% dynamic consistency: favors abandoning time consistency, so, favors sophisticated choice; %} Siniscalchi, Marciano (2011) “Dynamic Choice under Ambiguity,” Theoretical Economics 6, 379–421. {% %} Sinn, Hans-Werner (1983) “Economic Decisions under Uncertainty,” North Holland, Amsterdam. {% Here is the ASCII spelling of the author’s name, for searching purposes: Sipos. Already proposes, in §3, a variation of the symmetrical Choquet integral à la prospect theory. Here the 0 outcome plays a central role, with an integral symmetrical about it. The negative part is integrated with respect to the dual capacity; i.e., it is the PT functional with reflection which also appeared in Starmer & Sugden (1989). Lemma 6.(i) explains that this integral is a sum of the positive and negative part. Does not refer to Choquet, apparently did not know it? %} Šipoš, Ján (1979) “Integral with Respect to a Pre-Measure,” Mathematica Slovaca 29, 141–155. {% Here is the ASCII spelling of the author’s name, for searching purposes: Sipos. %} Šipoš, Ján (1979) “Non Linear Integrals,” Math. Slovaca 29, 257–270. {% probability communication & ratio bias: Reconsider Pighin et al. (2011), who argued that 1 in X is a bad way to communicate risk. This paper does a more extensive study and finds that the effect is weaker than in Pighin et al., but is existing. %} Sirota, Miroslav, Marie Juanchich, Olga Kostopoulou, & Robert Hanak (2014) “Decisive Evidence on a Smaller-than-You-Think Phenomenon: Revisiting the “1-in-X” Effect on Subjective Medical Probabilities,” Medical Decision Making 34, 419–429. {% Considers nonarchimedean EU %} Skala, Heinz J. (1975) “Non-Archimedean Utility Theory.” Wiley, New York. {% Considers Choquet integrals on Riesz spaces. %} Skala, Heinz J. (1999) “Comonotonic Additive Operators and Their Representations,” Glasgow Mathematical Journal 41, 191–196. {% fuzzy set theory %} Skala, Heinz J., Settimo Termini, & Enric Trillas (1984) “Aspects of Vagueness.” Reidel, Dordrecht. {% Takes acts and events as primitive, consequences are act-event pairs. In beginning of paper, value of consequence can depend on counterfactual consequence and context, leading to a general model that can explain regret, disappointment, and most other things. §4 considers additive aggregation that in itself does not yet seem restrictive but in presence of “separability” (which does not relate solely to global prefs so might better be called something like forgone-branch independence (often called consequentialism)) it becomes restrictive. It results from making the structure preferentially isomorphic to Debreu (1960). The appendix extends to infinitely many events. Because the model is in fact state-dependent utility, the probability measure, that is indeed used, is pointed out to identify only null events (Example 1, (a), in the appendix) The technique is as follows. A general model is assumed for DUU. A substructure is assumed, however, that satisfies the SEU assumptions (say; in fact, the paper does it for state-dependent SEU). Say the substructure concerns all acts with monetary outcomes and here SEU is satisfied. Let us call this substructure the canonical structure. Next, for a general act where all interactions whatsoever between outcomes are permitted, we make a corresponding canonical act that is such that for each state of nature it yields the monetary amount that is equally good for that state of nature as the outcome resulting there for the general act. In this manner, the SEU representation from the canonical structure is extended to all acts, while permitting for all interactions thinkable. %} Skiadas, Costis (1997) “Conditioning and Aggregation of Preferences,” Econometrica 65, 347–367. {% Tradeoff method: it builds on his 1997 Econometrica paper but restricts the additive (state-dependent) functional there further by means of an indifference tradeoff consistency condition (Axiom A10, p. 257), to obtain an SEU model. %} Skiadas, Costis (1997) “Subjective Probability under Additive Aggregation of Conditional Preferences,” Journal of Economic Theory 76, 242–271. {% %} Skiadas, Costis (1998) “Recursive Utility and Preferences for Information,” Economic Theory 12, 293–312. {% event/utility driven ambiguity model Has a nice variation of the Anscombe-Aumann (AA) model with a finite roulette-event space and a finite horse-event space, and uncertainty joint. The resulting product structure of the state space can nicely be used. One can better discuss the order of resolution of uncertainty (done in final para of main text, p. 73). Assumes quasi-convexity/uncertainty aversion. Weak certainty independence now more clearly amounts to constant relative risk aversion. The paper examines the role of weak certainty independence in detail. The sure-thing principle together with weak certainty independence imply SEU with log-power utility. This is proved in Appendix B.1, but it had been known before (Blackorby & Donaldson 1982, International Economic Review; Corollary 1.1; Wakker 1989, Theorem VII.7.5). The main Theorem 5 (p. 65) embeds this in a multiple priors maxmin EU framework. Theorem 11 has an SEU representation with power utility both for horses and for roulette, but they are only linked through an ordinal monotonicity and CE substitution so they can have different powers, leading to source-dependent SEU (event/utility driven ambiguity model: utility-driven). Can refer to this as Skiadas’ source-dependence CRRA model. SDEU had been described before, briefly, by Chew, Li, Chark, & Zhong (2008). criticism of monotonicity in Anscombe-Aumann (1963) for ambiguity: P. 63 penultimate para, l. 6 writes, appropriately on monotonicity in the AA model: “This is not an innocuous assumption” %} Skiadas, Costis (2013) “Scale-Invariant Uncertainty-Averse Preferences and Source-Dependent Constant Relative Risk Aversion,” Theoretical Economics 8, 59–93. {% DOI 10.1007/s00199-015-0920-9 The paper considers two sources of uncertainty, one concerning a horse race and the other concerning a roulette wheel. Neither have objective probabilities, and for each subjective probabilities are derived from prefs. There is also time, with repeated resolutions of the uncertainty about the two sources. Time separability is assumed, and risk separability (so EU) within each source, but no overall separability. This is like Abdellaoui et al.’s (2011 AER) source method, with local within-source but no global between-source probabilistic sophistication. The author assumes SEU within each source and captures source preference (my term) through source-dependent utility, as in the smooth model. In spirit it is like Chew et al.’s (2008) source-dependent EU. (event/utility driven ambiguity model: utility-driven) I regret that, if there is source-dependence of preference, the author calls it different risk attitude. If a first source has more concave utility than a second (so, lower certainty equivalents), the author says that the first source has more risk aversion. This same unfortunate terminology was used by Chew et al. (2008) and Kilka & Weber (2001). It may be easier to sell to noninitiated audiences at first acquaintance, but this terminology cannot survive. Risk attitude should only concern known OBJECTIVE probabilities. The difference between the unknown and the known Ellsberg urns is due to ambiguity attitude, and not due to changed risk attitude. In the axiomatization, SEU within each source comes from separability giving state-dependent SEU, and then constant relative risk aversion which is known to then imply SEU (Wakker 1989-book Theorem VII.7.5), and give CRRA (logpower) utility. %} Skiadas, Costis (2015) “Dynamic Choice with Constant Source-Dependent Relative Risk Aversion,” Economic Theory 60, 393–422. {% This book follows Keynes (1937) (more than Keynes (1936) general theory, which is what Paul Krugman seems to prefer). Most of economics assumes that uncertainty can be reduced to risk, so that we can calculate expectations, correlations, and so on with certainty, and can use Lucas’ rational expectations. The efficient market hypothesis is based on it. A spokesman of Goldman Sachs’ (chief financial officer David Viniar ?), seems to have declared August 17 2007, at the beginning of the financial crisis, that events were occurring that according to the best models around should happen once in 10140 times. It shows that uncertainty isn’t like risk, a point raised forcefully before by Keynes (1921) (better than Knight 1921), and reiterated by Keynes (1937). The author even argues that macroeconomics should be dedicated to the study of uncertainties that cannot be reduced to risks. %} Skidelsky, Robert (2009) “Keynes: The Return of the Master.” Penguin, London. {% conservation of influence: deviates from Watson’s behaviorism, who took living beings as no more than mechanistacally reacting to stimuli, and added to that “operant gedrag” where the living being has influence. That is, Skinner added decision maker’s influence! %} Skinner, Burrhus F. (1971) “Beyond Freedom and Dignity.” Knopf, New York. {% Seems to be last text he wrote, knowing he would die. It is a very opiniated text, arguing against the cognitive approach and favoring behaviorism. So he wants to keep things simple at the level of directly observable phenomena and predictions directly in terms of them and their (cor)relations. Wants no abstractions such as cognitive concepts. I did not understand several parts, conjecturing that they are not clearly written. In several parts he puts up straw men. His expectations of neurology is naïve. %} Skinner, Burrhus F. (1985) “Cognitive Science and Behavourism,” British Journal of Psychology 76, 291–301. {% probability intervals: pp. 192-193 mentions the difference between multiple priors and interval probabilities. Unfortunately, it takes combinations of Dempster-Shafer belief/plausibility functions, and of convex-concave capacities, as an example of interval probabilities. This is not formally incorrect, but can be confusing because, if the concave capacity is to be taken as the dual of the convex one (similarly as a plausibility function is the dual of the belief function), then the convex capacity alone captures all the info, and this capcity can in turn be related uniquely to a set of priors. So this is a case where the interval probabilities can be uniquely related to multiple priors, and the two models are not fundamentally different. Essential differences do arise if we relax some assumptions, such as allowing for nonconvex-nonconcave capacities. Full generality is achieved if we further allow the lower capacity not to be the dual of the upper capacity. %} Škulj, Damjan (2006) “Jeffrey’s Conditioning Rule in Neighbourhood Models,” International Journal of Approximate Reasoning 42, 192–211. {% %} Skyrms, Brian F. (1980) Book Review of: Arthur W. Burks (1977) “Cause, Chance, and Reason,” University of Chicago Press, Chicago; Theory and Decision 12, 299–309. {% second-order probabilities to model ambiguity %} Skyrms, Brian F. (1980) “Higher Order Degrees of Belief.” In David H. Mellor (1980, ed.) Prospects for Pragmatism. Essays in Memory of F.P. Ramsey, 109–137, Cambridge University Press, Cambridge. {% It seems that he lets states of nature be mappings from acts to outcomes. %} Skyrms, Brian F. (1980) “Causal Necessity.” Yale University Press, New Haven. {% foundations of probability %} Skyrms, Brian F. (1988) “Probability and Causation,” Journal of Econometrics 39, 53–68. {% Ch. 4 discusses Ramsey (1926). %} Skyrms, Brian F. (1990) “The Dynamics of Rational Deliberation.” Harvard University Press, Cambridge, MA. {% Dutch books; interpretations of sigma-additivity %} Skyrms, Brian F. (1995) “Strict Coherence, Sigma Coherence, and the Metaphysics of Quantity,” Philosophical Studies 77, 39–55. {% %} Slater, Patrick (1961) “Inconsistencies in a Schedule of Paired Comparisons,” Biometrika 48, 303–312. {% Patients will accept more risks to choose for chemotherapy than doctors/nurses will recommend. (Explanation I suggest: doctors & Nurses care more about costs/time which means, indirectly, interests of other patients.) %} Slevin, Maurice L., Linda Stubbs, Hilary J. Plant, et al. (1990) “Attitudes to Chemotherapy: Comparing Views of Patients with Cancer and Those of Doctors, Nurses, and General Public,” British Medical Journal 300, 1458–1460. {% %} Sloman, S., Yuval Rottenstreich, Edward Wisniewski, Constantinides Hadjichristidis, & Craig R. Fox (2004) “Typical versus Atypical Unpacking and Superadditive Probability Judgment,” Journal of Experimental Psychology: Learning, Memory & Cognition 30, 573–582. {% Elementair boek over statistiek, speciaal geschikt voor psychologen; het legt allerlei termen uit !zonder! formules. %} Slotboom, Anke M. (1987) “Statistiek in Woorden.” Wolters-Noordhof, Groningen. {% real incentives/hypothetical choice: for gain-loss gambles, more risk aversion for real payment. Gives nice early references. Feather (1959), for one, preceded this study. All gambles have one gain and one loss. Participants are more risk seeking for hypothetical lotteries than for real-payment lotteries. Not clear if this is caused by loss aversion or by other factors of risk aversion. Download 7.23 Mb. Share with your friends:
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