How To Test For A Null Hypothesis

Transcription

1 TESTING THE ROULETTE WHEEL Mihael Perman University of Ljubljana Osijek, June 4th, 29

2 SOME QUOTATIONS The generation of random numbers is too important to be left to chance. Robert R. Coveyou, Oak Ridge National Laboratory, ZDA The only way to win at roulette is to steal chips when the croupier looks the other way. Albert Einstein With this roulette betting system, you will bet on the almost even money bets. You will bet on black or red or even or odd. You will not bet on the high-odds choices. All you need to win with this system is to win 7 out of 2 times. Internet, 1995 Roulette is the most glamorous of all the casino games. An air of elegance surrounds the roulette table, and its spinning wheel seems to be a perfect agent by which the goddess of fortune may intevene in the affairs of mortal men. How much superior is this unapproachable mechanistic device to those games like dice and cards, where human hands may tamper with fate. But more than glamour, the game presents to me a certain irresistible challenge. The roulette is intended

3 to be a symmetrical gambling device, the odds for which always favour the house. In the long run, it would appear that a player must inevitably lose. But due to a certain degree of asymmetry in the wheel s production, or due to its later wear, the odds may shift enough to favor a player on certain bets. The shrewd observer may spot such a case and actually be able to play a winning game. Herein lies the challenge. Allan N. Wilson, Tha Casino Gambler s Guide.

4 Unless we inconvenience ourselves by staying a long time in the casino to increase the sample of spins, how may we distinguish statistically a true weak biased number from the false random winners that eventually fluctuate all over the place and through which we lose? This questions I have put to two professors of mathematics, experts in roulette theory and play, whom I quote elsewhere in this book, and they both declare sadly that they have thus far no statistical method to offer as a practical solution. For the greater success of biased-wheel play, let s hope that some day a solution may be found. Russel T. Barnhart, Beating the Wheel

5 THE PROBLEM The roulette wheel in principle generates random numbers uniformly distributed on the set {, 1, 2,...,36}. Mechanical imperfections or wilful manipulation can lead to deviations from uniformity in various ways. Gambling houses are interested in statistics that would detect such deviations as soon as possible with the smallest probability of false alarms. The reasons why quality control is desirable are the following: The odds offered by the house should be those advertised. Skilled enough groups could take advantage of deviations if they notice them before the supervisors of the house. Relatively small deviations can nudge the expected gain into the positive. Quality control should include the human factor. Croupiers are human and could potentially cheat in collusion with gamblers.

6 MATHEMATICAL FORMULATION In statistical terms the problems is formulated as follows: We have observations X 1, X 2,... from the roullette wheel taking values in {, 1, 2,...,36}. We will assume that the observations are independent. There are two main objectives: We would like to test the hypothesis (possibly sequentially) H : X 1, X 2,... Uniform{, 36} against H 1 : X 1, X 2,... / Uniform{, 36}. The question is what test statistics to choose and how to decide whether to reject or accept the null hypothesis. We would like to detect a change point. The observations can start out as uniform but change to another distribution as we are collecting the observations. How does one detect such a change?

7 THE CLASSICAL χ 2 TEST The usual χ 2 test is the first idea to try. Notation: Nn k is the frequency of outcome k after n spins of the wheel. p is the probability of each outcome under the null hypothesis. m = 37. We compute χ 2 = m 1 k= (N k np ) 2 np.

8 .3 Probability distribution over cells Fig. 1 Probability distribution for a hanging wheel 7 Trajectory of CHI statistic Number of spins Fig. 1a Behaviour of χ 2 statistics for a hanging wheel

9 .4 Probability distribution over cells Fig. 2 Probability distribution for a dented wheel 14 Trajectories of CHI statistic and the likelihood ratio statistic Number of spins Fig. 2a Behaviour of χ 2 statistics for a dented wheel

10 .35 Probability distribution over cells Fig. 3 Probability distribution for a nicked wheel 6 Trajectories of CHI statistic and the likelihood ratio statistic Number of spins Fig. 3a Behaviour of χ 2 statistics for a nicked wheel

11 .3 Perfect probability distribution over cells Fig. 4 Probability distribution for a perfect wheel 6 Trajectories of CHI statistics and likelihood ratio statistic Number of spins x 1 4 Fig. 4a Behaviour of χ 2 statistics and the likelihood ratio statistic for a perfect wheel

12 CENTRAL LIMIT THEOREM The distributions of statistics to be used are all derived from a simple observations based on the central limit theorem. The vector 1 np (N 1 n np, N 2 n np,..., N m n np ) d N m (,Σ) where Σ = I 11 T /m. Remark: Multivariate normal vectors with the above distribution are easy to simulate on the computer. One only needs to simulate (Z 1 Z, Z 2 Z,..., Z m Z) where the Z 1, Z 2,..., Z m are independent standard normals.

13 MAIN ALTERNATIVE HYPOTHESES The alternative hypotheses we will consider:.4 Probability distribution over cells Celica Fig. 5 Dented wheel. The payoff for betting on triplets is 1:11. In the case shown betting on the first three cells gives an expected payoff of.294.

14 .3 Probability distribution over cells Cells Fig. 6 Hanging wheel. The payoff for betting on triplets is 1:11. In the case shown betting on the best three cells gives an expected payoff of.44.

15 CHOICE OF STATISTICS We would like to devise statistics which would be better traps for the given types of faults. The χ 2 test for the multinomial are based on the expression χ 2 = (Observed i Expected i ) 2 Expected i where the index i refers to cell number i. Normally the cells would be non overlapping but that is because only in that case one can obtain analytical expressions for the limit distribution. We want to monitor sectors of three, five or seven. A plausible choice to monitor, say, triplets would be CHI3 = m 1 k= (N k n + N k+1 n + N k+2 n 3np ) 2 3np. Here we interpret k + 1 and k + 2 modulo m 1.

16 Another plausible statistic is the maximum deviation from the expected frequency of a cell which can also be a sector of three, five or seven adjacent pockets on the wheel. MAX3 = max k<m Nn k + Nk+1 n + Nn k+2 3np. np Yet another alternative is the likelihood ratio test. One looks at the quantity LRATIO = log( sup p P p (Observed values) ). P p (Observed values) As we observe more and more outcomes Wilks theorem asserts that 2 LRATIO converges in distribution to a χ 2 (36). REMARK: As we do everything sequentially we actually observe stochastic processes of various statistics and need to keep that in mind. So is there a process version of Wilks theorem?

17 DISTRIBUTIONS OF STATISTICS The distributions were obtained by simulation. Here are the distributions of a sample of test statistics. CHI3 is the χ 2 like statistic for triplets. M AX1 is a suitably standardised maximal positive deviation from the expected frequences of cells. M AX3 is a suitably standardised maximal positive deviation from the expected frequences for triplets. 18 Distribution of CHI Density CHI3 Fig. 7 Distribution of CHI3 statistic

18 18 Distribution of MAX Density MAX1 Fig. 8 Distribution of MAX1 statistics 16 Distribution of MAX Density MAX3 Fig. 9 Distribution of MAX3 statistics

19 TRAJECTORIES FOR THE CHOSEN STATISTICS The next few slides show various trajectories of CHIx and M AXx statistics: Trajectories for an honest wheel. Trajectories for a dented wheel. Trajectories for a hanging wheel. Trajectories in two cases of real data. LEGEND: CHI1 or MAX1 CHI3 or MAX3 CHI5 or MAX5 CHI7 or MAX7

20 Perfect wheel 5 Trajectories of CHIx statistics with 95% tresholds Number of spins x 1 4 Fig. 1 Trajectories of CHIx statistics for a prefect wheel. 8 Trajectories of MAXx statistics with 95% tresholds Number of spins x 1 4 Fig. 1a Trajectories of MAXx statistics for a prefect wheel.

21 Dented wheel 2 Trajectories of CHIx statistics with 95% tresholds Number of spins Fig. 11 Trajectories of CHIx statistics for a dented wheel. 18 Trajectories of MAXx statistics with 95% tresholds Number of spins Fig. 11a Trajectories of MAXx statistics for a dented wheel.

22 Hanging wheel 14 Trajectories of CHIx statistics with 95% tresholds Number of spins Fig. 12 Trajectories of CHIx statistics for a hanging wheel. 1 Trajectories of MAXx statistics with 95% tresholds Number of spins Fig. 12a Trajectories of MAXx statistics for a hanging wheel.

23 Wheel AR4 5 Trajectories of CHI statistics Number of spins Fig. 13 Trajectories of CHIx statistics for wheel AR4. 8 Trajectories of MAX statistics Number of spins Fig. 13a Trajectories of MAXx statistics for wheel AR4.

24 Wheel HISPAR2 16 Statistike CHIx za cilinder HISPAR2 s 95% pragom Fig. 14 Trajectories of CHIx statistics for wheel HISPAR2. 14 Statistike MAXx za cilinder HISPAR2 s 95% pragom Fig. 14a Trajectories of MAXx statistics for wheel HISPAR2.

25 .35 Empirical probability distribution over cells Fig. 14c Empirical probability distribution over cells for wheel HISPAR2.

26 SEQUENTIAL TESTS It is a luxury to assume a fixed number of observations. There are several reasons for that: We never now why data collection has stopped. Was that independent of teh outcomes? Usually that is not the case. The house wants to stop a table as soon as there is enough evidence that something is wrong.

27 MARTINGALES The idea of a sequential test is that we reject the nullhyposthesis as soon as possible given the significance level α. But when is as soon as possible? One possible solution is to observe that the transforms ˆχ 2 k = k(χ2 k mx(1 x/m)), where x is the width of the sector are MARTIN- GALES under the null-hypothesis. For martingales one has MAXIMAL INEQUALITIES. Under the null-hypothesis we can say k P( max 1 k n n (ˆχ2 k mx(1 x/m)) + a) E [ (ˆχ 2 n) q ] +, a q where x + is the positive part, and q 1.

28 TESTS The test now does the following: choose an appropriate q 1 and a >, and reject the null as soon as the maximal inequality is violated. One needs to calibrate the tests. As an example one gets when α =.1 - For sectors of width x = 1 choose q = 8 and a = For sectors of width x = 3 choose q = 6 and a = For sectors of width x = 5 choose q = 6 and a = For sectors of width x = 7 choose q = 6 and a = The constant q is choosen in such a way that it minimizes a.

29 EXAMPLES Hanging wheel 8 Trajektorije transformiranih statistik TCHIx za cilinder, ki visi Stevilo iger 35 Wheel AR4 Potek transformiranih statistik TCHIx v Stevilo iger

30 THE KLOTZ STRATEGY If the wheel is biassed there may be winning strategies. One possible way is to maximize the expected logarithm of your winnings. This idea from economics produces an interesting strategy called the Klotz strategy. The Klotz strategy is then combined with a Baysian estimate of probabilities of certain outcomes in the sense that ˆp i = n i + α n + nα. The parameter α may be interpreted as caution. The higher it is, the less we are inclined to get exited by seemingly more probable outcomes.

31 EXAMPLES Here are some simulated and some real examples. In all cases we take α = 1 and α = 2. Slightly biassed wheel 4 Potek kapitala pri previdnosti 1 4 Potek kapitala pri previdnosti Kapital 2 Kapital Igra Igra More seriously biassed wheel 2 x 15 Potek kapitala pri previdnosti 1 2 x 15 Potek kapitala pri previdnosti Kapital 1 Kapital Igra Igra

32 Real wheel AR4 1 Potek kapitala na cilindru, previdnost=1 1 Potek kapitala na cilindru, previdnost= Kapital 5 Kapital Igra Igra 1 x 14 Potek kapitala na cilindru 1, previdnost=1 Real wheel HISPAR2 1 x 14 Potek kapitala na cilindru 1, previdnost= Kapital 5 Kapital Igra Igra

33 Real wheel Cammegh 1 x 14 Potek kapitala na cilindru 2, previdnost=1 1 x 14 Potek kapitala na cilindru 2, previdnost= Kapital 5 Kapital Igra Igra Real wheel HISPAR4 1 x Potek kapitala na cilindru 4, previdnost= x Potek kapitala na cilindru 4, previdnost= Kapital 5 Kapital Igra Igra

34 TESTING One possible idea is to use the Klotz strategy as a test statistics. If the optimal player starts winning too much we reject the null-hypothesis. But what is too much? Again we observe a few facts: Under the null-hypothesis the current capital of the player is a non-negative supermartingale so it converges to a finite limit. The supremum of the entire capital trajectory is a finite random variable. One can either try to find an analytic estimate of the distribution of the maximum or simulate. Here is the simulated distribution. The advantage is that the p-values have the meaning in terms of money. It is not easy to get across simple statistical ideas to the end-user.

35 35 The distribution of the maximum Maximum

36 Main points? CONCLUDING REMARKS One has to focus on certain types of alternative hypotheses. The entire space is just to big. The classical χ 2 test does a poor job. If one takes marginal distributions of trajectories as approximations to the right critical values one has to proceed by simulation. Remaining questions? Are the statistics chosen the right ones? What are the rules for deciding? In particular, what are the right critical values for individual statistics? Is it correct to just look at the marginal distribution? Or does one have to consider the entire trajectory? If one were to test sequentially what is the right decision rule? Is there a process version of Wilks theorem? What can one say about the asymptotic behaviour of the test statistics? Do they converge under the null-hypothesis?