**W**hen looking at questions on X validated, I came across this seemingly obvious request for an unbiased estimator of **P**(X=k), when X~**B**(n,p). Except that X is not observed but only Y~**B**(s,p) with s<n. Since **P**(X=k) is a polynomial in p, I was expecting such an unbiased estimator to exist. But it does not, for the reasons that Y only takes s+1 values and that any function of Y, including the MLE of **P**(X=k), has an expectation involving monomials in p of power s at most. It is actually straightforward to establish properly that the unbiased estimator does not exist. But this remains an interesting additional example of the rarity of the existence of unbiased estimators, to be saved until a future mathematical statistics exam!

## Archive for binomial distribution

## unbiased estimators that do not exist

Posted in Statistics with tags binomial distribution, counterexample, cross validated, exam, existence of unbiased estimators, monomial, teaching, unbiasedness on January 21, 2019 by xi'an## an accurate variance approximation

Posted in Books, Kids, pictures, R, Statistics with tags binomial distribution, cross validated, Monte Carlo Statistical Methods, Poisson distribution, R, simulation, variance estimation on February 7, 2017 by xi'an**I**n answering a simple question on X validated about producing Monte Carlo estimates of the variance of estimators of exp(-θ) in a Poisson model, I wanted to illustrate the accuracy of these estimates against the theoretical values. While one case was easy, since the estimator was a Binomial B(n,exp(-θ)) variate [in yellow on the graph], the other one being the exponential of the negative of the Poisson sample average did not enjoy a closed-form variance and I instead used a first order (δ-method) approximation for this variance which ended up working surprisingly well [in brown] given that the experiment is based on an n=20 sample size.

Thanks to the comments of George Henry, I stand corrected: the variance of the exponential version is easily manageable with two lines of summation! As

which allows for a comparison with its second order Taylor approximation:

## simulating correlated Binomials [another Bernoulli factory]

Posted in Books, Kids, pictures, R, Running, Statistics, University life with tags binomial distribution, cross validated, inverse cdf, Jacob Bernoulli, Parc de Sceaux, R, random simulation, stackoverflow on April 21, 2015 by xi'an**T**his early morning, just before going out for my daily run around The Parc, I checked X validated for new questions and came upon that one. Namely, how to simulate X a Bin(8,2/3) variate and Y a Bin(18,2/3) such that corr(X,Y)=0.5. (No reason or motivation provided for this constraint.) And I thought the following (presumably well-known) resolution, namely to break the two binomials as sums of 8 and 18 Bernoulli variates, respectively, and to use some of those Bernoulli variates as being common to both sums. For this specific set of values (8,18,0.5), since 8×18=12², the solution is 0.5×12=6 common variates. (The probability of success does not matter.) While running, I first thought this was a very artificial problem because of this occurrence of 8×18 being a perfect square, 12², and cor(X,Y)x12 an integer. A wee bit later I realised that all *positive* values of cor(X,Y) could be achieved by randomisation, i.e., by deciding the identity of a Bernoulli variate in X with a Bernoulli variate in Y with a certain probability ϖ. For *negative* correlations, one can use the (U,1-U) trick, namely to write both Bernoulli variates as

in order to minimise the probability they coincide.

I also checked this result with an R simulation

> z=rbinom(10^8,6,.66) > y=z+rbinom(10^8,12,.66) > x=z+rbinom(10^8,2,.66) cor(x,y) > cor(x,y) [1] 0.5000539

Searching on Google gave me immediately a link to Stack Overflow with an earlier solution with the same idea. And a smarter R code.

## prayers and chi-square

Posted in Books, Kids, Statistics, University life with tags binomial distribution, chi-square test, exercises, medical school, prayer, Richard Dawkins, The God Delusion on November 25, 2014 by xi'an**O**ne study I spotted in Richard Dawkins’ *The God delusion* this summer by the lake is a study of the (im)possible impact of prayer over patient’s recovery. As a coincidence, my daughter got this problem in her statistics class of last week (my translation):

1802 patients in 6 US hospitals have been divided into three groups. Members in group A was told that unspecified religious communities would pray for them nominally, while patients in groups B and C did not know if anyone prayed for them. Those in group B had communities praying for them while those in group C did not. After 14 days of prayer, the conditions of the patients were as follows:

out of 604 patients in group A, the condition of 249 had significantly worsened;out of 601 patients in group B, the condition of 289 had significantly worsened;out of 597 patients in group C, the condition of 293 had significantly worsened.

Use a chi-square procedure to test the homogeneity between the three groups, a significant impact of prayers, and a placebo effect of prayer.

This may sound a wee bit weird for a school test, but she is in medical school after all so it is a good way to enforce rational thinking while learning about the chi-square test! (*Answers:* [even though the data is too sparse to clearly support a decision, esp. when using the chi-square test!] homogeneity and placebo effect are acceptable assumptions at level 5%, while the prayer effect is not [if barely].)

## differences between Bayes factors and normalised maximum likelihood

Posted in Books, Kids, Statistics, University life with tags Bayes factors, binomial distribution, model complexity, normalised maximum likelihood, Rissanen on November 19, 2014 by xi'an**A** recent arXival by Heck, Wagenmaker and Morey attracted my attention: Three Qualitative Differences Between Bayes Factors and Normalized Maximum Likelihood, as it provides an analysis of the differences between Bayesian analysis and Rissanen’s Optimal Estimation of Parameters that I reviewed a while ago. As detailed in this review, I had difficulties with considering the normalised likelihood

as the relevant quantity. One reason being that the distribution does not make experimental sense: for instance, how can one simulate from this distribution? [I mean, when considering only the original distribution.] Working with the simple binomial B(n,θ) model, the authors show the quantity corresponding to the posterior probability may be constant for most of the data values, produces a different upper bound and hence a different penalty of model complexity, and may differ in conclusion for some observations. Which means that the apparent proximity to using a Jeffreys prior and Rissanen’s alternative does not go all the way. While it is a short note and only focussed on producing an illustration in the Binomial case, I find it interesting that researchers investigate the Bayesian nature (vs. artifice!) of this approach…

## [weak] information paradox

Posted in pictures, Running, Statistics, University life with tags binomial coefficient, binomial distribution, combinatorics, cross validated, elm tree, MAP estimators, William Feller on December 2, 2011 by xi'an**W**hile (still!) looking at questions on ** Cross Validated** on Saturday morning, just before going out for a chilly run in the park, I noticed an interesting question about a light bulb problem. Once you get the story out of the way, it boils down to the fact that, when comparing two binomial probabilities,

*p*and

_{1}*p*

_{2}, based on a Bernoulli sample of size n, and when selecting the MAP probability, having either n=2k-1 or n=2k observations lead to the same (frequentist) probability of making the right choice. The details are provided in my answers here and there. It is a rather simple combinatoric proof, once you have the starting identity [W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 1968, [II.8], eqn (8.6)]

but I wonder if there exists a more statistical explanation to this weak information paradox…

## another lottery coincidence

Posted in R, Statistics with tags binomial distribution, coincidence, Le Loto, lottery on August 30, 2011 by xi'an**O**nce again, meaningless figures are published about a man who won the French lottery (*Le Loto*) for the second time. The reported probability of the event is indeed one chance out of 363 (US) trillions (i.e., billions in the metric system. or 10^{12})… This number is simply the square of

which is the number of possible loto grids. Thus, the probability applies to the event “*Mr so-&-so plays a winning grid of Le Loto on May 6, 1995 and **a winning grid of Le Loto on July 27, 2011*“. But this is not the event that occured: one of the bi-weekly winners of *Le Loto* won a second time and this was spotted by *Le Loto* spokepersons. If we take the *specific* winner for today’s draw, Mrs such-&-such, who played bi-weekly one single grid since the creation of *Le Loto* in 1976, i.e. about 3640 times, the probability that she won earlier is of the order of

.

There are thus two chances in 10 thousands to win again for a given (unigrid) winner, not much indeed, but no billion involved either. Now, this is also the probability that, for a given draw (like today’s draw), one of the 3640 previous winners wins again (assuming they all play only one grid, play independently from each other, &tc.). Over a given year, i.e. over 104 draws, the probability that there is no second-time winner is thus approximately

showing that within a year there is a 2% chance to find an earlier winner. Not so extreme, isn’t it?! Therefore, less bound to make the headlines…

**N**ow, the above are rough and conservative calculations. The newspaper articles about the double winner report that the man is playing about 1000 euros a month (this is roughly the minimum wage!), representing the equivalent of 62 grids per draw (again I am simplifying to get the correct order of magnitude). If we repeat the above computations, assuming this man has played 62 grids per draw from the beginning of the game in 1976 till now, the probability that he wins again conditional on the fact that he won once is

,

a small but not impossible event. (And again, we consider the probability only for Mr so-&-so, while the event of interest does not.) (*I wrote this post before Alex pointed out the four-time lottery winner in Texas, whose “luck” seems more related with the imperfections of the lottery process…)*

**I** also stumbled on this bogus site providing the “probabilities” (based on the binomial distribution, nothing less!) for each digit in *Le Loto*, no need for further comments. (Even the society that runs *Le Loto* hints at such practices, by providing the number of consecutive draws a given number has not appeared, with the sole warning “*N’oubliez jamais que le hasard ne se contrôle pas*“, i.e. *“Always keep in mind that chance cannot be controlled*“…!)