Order Statistics; Conditional Expectation

Transcription

1 Department of Mathematics Ma 3/3 KC Border Introduction to Probability and Statistics Winter 27 Lecture 4: Order Statistics; Conditional Expectation Relevant textbook passages: Pitman [5]: Section 4.6 Larsen Marx [2]: Section Order statistics Given a random vector X,..., X n ) on the probability space S, E, P ), for each s S, sort the components into a vector X ) s),..., X n) s) ) satisfying Pitman [5]: 4.6 X ) s) X 2) s) X n) s). The vector X ),..., X n) ) is called the vector of order statistics of X,..., X n ). Equivalently, { } X k) min max{x j : j J} : J {,..., n} & J k. Order statistics play an important role in the study of auctions, among other things. 4.2 Marginal Distribution of Order Statistics From here on, we shall assume that the original random variables X,..., X n are independent and identically distributed. Let X,..., X n be independent and identically distributed random variables with common cumulative distribution function F, and let X ),..., X n) ) be the vector of order statistics of X,..., X n. By breaking the event X k) x ) into simple disjoint subevents, we get Xk) x ) X n) x ) Xn) > x, X n ) x ). Xn) > x,..., X j+) > x, X j) x ). Xn) > x,..., X k+) > x, X k) x ). 4

2 KC Border Order Statistics; Conditional Expectation 4 2 Each of these subevents is disjoint from the ones above it, and each has a binomial probability: Xn) > x,..., X j+) > x, X j) x ) n j of the random variables are > x and j are x), so Thus: P X n) > x,..., X j+) > x, X j) x ) ) n ) n jf F x) x) j. j The cdf of the k th order statistic from a sample of n is: F k,n) x) P X k) x ) jk ) n ) n jf F x) x) j. ) j 4.3 Marginal Density of Order Statistics Pitman [5]: p. 326 If F has a density f F, then we can calculate the marginal density of X k) by differentiating the CDF F k,n) : d dx F k,n)x) d ) n F x) j F x) ) n j dx j jk ) n d j dx F x)j F x) ) n j jk ) n jf x) j F x) ) n j F x) n j)f x) j F x) ) n j F x)) j jk ) n jf x) j F x) ) n j n j)f x) j F x) ) ) n j fx) j jk jk n! j )!n j)! F x)j F x) ) n j fx) n n! ) n j F F x) x) j fx) j!n j )! jk n! ) n jf F x) x) k fx) k )!n k)! n! ) n jf + F x) x) j fx) j )!n j)! jk+ n n! ) n j F F x) x) j ft). j!n j )! jk v ::3.25 KC Border

3 KC Border Order Statistics; Conditional Expectation 4 3 The last two terms above cancel, since using the change of variables i j, jk+ n! ) n jf F x) x) j j )!n j)! n ik n! ) n if F x) x) i. i!n i )! So the density of the k th order statistic from a sample of size n is: f k,n) x) n! ) n kf F x) x) k fx) k )!n k)! ) n ) n ) k )F n F x) x) k fx). 2) k Note: I could have written n k instead of n ) k ), but then the binomial coefficient would have seemed more mysterious. 4.4 Some special order statistics The st order statistic X ) from a sample of size n is just the minimum X ) min{x,..., X n }. The event that min{x,..., X n } x is just the event that at least one X i x. The complement of this event is that all X i > x, which has probability F x) ) n, so the cdf is and its density is F,n) x) F x) ) n f,n) x) n F x) ) n ) fx). The n th order statistic X n) from a sample of size n is just the maximum X n) max{x,..., X n }. The event that max{x,..., X n } x is just the event that all X i x, so its cdf is and its density is F n,n) x) F x) n f n,n) x) nf x) n ) fx). The n ) st order statistic is the second-highest value. Its cdf is and its density is F n,n) x) n F x) ) F x) n 2 + F x) n nf x) n n )F x) n, nn ) F x) ) F x) n 2 fx). In the study of auctions, the second-highest bid is of special interest. Indeed a secondprice auction awards the item to the highest bidder, but the price is the second-highest bid. A standard auction provides incentives for bidders to bid less than their values, so the distribution of bids and the distribution of values is not the same. Nevertheless it can be shown see my online notes) that the expected revenue to a seller in an auction with n bidders with independent and identically distributed values is just the expectation of the second-highest order statistic for the distribution of values. KC Border v ::3.25

4 KC Border Order Statistics; Conditional Expectation Uniform order statistics and the Beta function Pitman [5]: pp For a Uniform[,] distribution, F t) t and ft) on [, ]. In this case equation 2) tells us: The density f k,n) of the k th order statistic for n independent Uniform[, ] random variables is ) n f k,n) t) n t) n k t k. k Since f k,n) is a density, ) n f k,n) t) dt n t) n k t k, k or t) n k t k Now change variables by setting Then rewrite 3) as n n k ) k )!n k)!. 3) n! r k and s n r + so s n r and n r + s ). t) s t r r )!s )! s + r )! Γs)Γr) Γr + s). Recall that the Gamma function is a continuous version of the factorial, and has the property that Γs + ) sγs) for every s >, and Γm) m )! for every natural number m. See Definition 3... This fact suggests to at least some people) the following definition: 4.5. Definition The Beta function is defined for r, s > not necessarily integers), by Br, s) So for integers r and s, we see that Br +, s + ) Γs + )Γr + ) Γr + s + 2) t r t) s dt Γs)Γr) Γr + s). r+s r s)!r)! s)!r)! r + s) r + s )! r + s)! r + s) ) Definition The betar, s) distribution has the density fx) on the interval [, ] and zero elsewhere. Br, s) xr x) s v ::3.25 KC Border

5 KC Border Order Statistics; Conditional Expectation 4 5 Note that for integer rand s, the density of the betar +, s + ) distribution is fx) ) r + s x r x) s, r + s r which is /r + s) times the Binomial probability of r successes and s failures in r + s trials, where the probability of success is x. The mean of a betar, s) distribution is r r + s. Proof : xfx) dx Br, s) x x r x) s dx x r+ x) s Br, s) Br +, s) Br, s) Γs)Γr + ) Γr + s) Γr + + s) Γs)Γr) Γs)rΓr) r + s)γr + s) r r + s. Γr + s) Γs)Γr) Thus for a Uniform[,] distribution, the k, n) order statistic has a betak, n k + ) distribution and so has mean k n +. [Application to breaking a unit length bar into n + pieces by choosing n breaking points. The expectation of the k th breaking point is at k/n + ), so each piece has expected length /n.] 4.6 The war of attrition In the 97s, the ethologist John Maynard Smith [3] began to apply game theory to problems of animal behavior. One application was to the settlement of intraspecies conflict. In some species e.g., peafowl), conflicts are not settled by violent means, but by means of displays. The rivals will fan their tails, and eventually one will depart, leaving to the other whatever was the source of the conflict. Maynard Smith modeled this as a war of attrition. In the war of attrition game there are two rival contestants i, 2 for a prize of value v. Each chooses a length of time t i at random according to a common probability distribution with cumulative distribution function F. Waiting is costly, and the cost of waiting a length of time t is ct. The rivals continue their displays, until the lesser time elapses and that animal leaves. The distribution is an symmetric equilibrium distribution if it has the following properties. i.) KC Border v ::3.25

6 KC Border Order Statistics; Conditional Expectation 4 6 Each rival, knowing that the opponent has drawn a time t i from the distribution specified by F, is also willing to choose a time specified by F. ii.) When the time t i has elapsed, and contestant i s opponent has not left, then i does not have an incentive to stay longer, and so will leave. Suppose contestant 2 chooses a waiting time s at random according to an exponential distribution with parameter λ. Now consider contestant s decision. Suppose contestant chooses to wait a length of time t. If s < t, which happens with probability e λt, hew ins the prize and receives V, but he also incurs a waiting cost cs. If s > t, which happens with probability e λt, then his cost is ct and he doe not get the prize. The expected total payoff φt) to is then [ t φt) V e λt ) cse λs ds + cte λt]. Rather than integrate this to find out its value, let s see how it depends on t by computing its derivative. Recalling the Fundamental Theorem of Calculus, we see that [ φ t) vλe λt ctλe λt + ce λt cλte λt)] vλ c)e λt. Now if λ is chosen so that the expected waiting cost is equal to the value of the prize, c v vλ c, λ then φ t). That is, contestant receives the same expected payoff regardless of when he leaves. As a result he is perfectly content to choose his waiting time at random according to the same exponential distribution. Thus an exponentially distributed waiting time with parameter λ c/v satisfies property i) of a symmetric equilibrium distribution. It is easy to see that φ), so the expected payoff is zero. This makes sense, as the expected payoff is the same for both players, and they both can t win.) To verify property ii) of a symmetric equilibrium distribution, suppose some length of time passes and neither contestant has dropped out. Since the exponential distribution is memoryless, each contestant can redo the calculation above, and conclude there is no advantage to choosing a different time to leave. Thus contestant i is happy to leave at time t i. If both contestants choose t i according to this λ, then we have an equilibrium. The length of the contest is min{t, T 2 }. Now min{t, T 2 } t if and only if it is not the case that T > t and T 2 > t. Thus P min{t, T 2 } t) P T > t & T 2 > t) P T > t) P t 2 > T ) e λt e λt e 2λt, which is an exponential survival function with parameter 2λ. Thus the expected length of the contest is /2λ). So the winner and loser both expect to wait /2λ) and the expected total cost incurred is equal to v, the value of the prize. Note that this model implies that the observed length of contest durations should be exponentially distributed, provided c and v are the same in each contest. The noted game theorist Robert Rosenthal once told me that in fact the duration of display contests among dung flies is exponentially distributed, but he didn t mention any citations. A little digging found a paper by Parker and Thompson [4] where they find qualified support for this distribution, but argue that an asymmetric model would fit better. 4.7 The Winner s Curse The Winner s Curse is a phenomenon that was first observed in oil leases. The Federal government claims all land on the continental shelf up to 2 miles offshore. It would auction off the v ::3.25 KC Border

7 KC Border Order Statistics; Conditional Expectation 4 7 right to drill for oil on tracts, and oil companies found that despite their best efforts to employ scientific methods to estimating the value of the lease, they systematically lost money on their leases. The explanation is straightforward. The value of a lease is an unknown number V, and each company i got an estimate X i of the value of V. Let s suppose that each estimate is an independent draw from the same distribution F. If the company bids based on the distribution F, they will subject to the winner s curse. Namely, the winner will be the company with the largest X i. But the distribution of the largest value of X i is the n th order statistic, the maximum, which has cdf F n, not F. In order to avoid the winner s curse you have to take into account the fact that you win only when your X i X n). 4.8 The σ-algebra of events generated by random variables Recall that the Borel sets of the real numbers are the members of the smallest σ-algebra that contains all the intervals. Given a set X,..., X n of random variables defined on the probability space S, E, P ) and intervals I,..., I n, X I, X 2 I 2,... X n I n ) is an event. The smallest σ-algebra that contains all these events, as the intervals I j range over all intervals, is called the σ-algebra of events generated by X,..., X n, and is denoted σx,..., X n ). A function g : S R is σx,..., X n )-measurable if for every interval I, the set g I) belongs to E. The following theorem is beyond the scope of this course, but may be found, for instance, in Aliprantis Border [, Theorem 4.4] or M. M. Rao [6, Theorem.2.3, p. 4] Theorem Let X X,..., X n ) be a random vector on S, E, P ) and let g : S R. Then the function g is σx,..., X n )-measurable if and only if there exists a Borel function h: R n R such that g h X. This means there is a one-to-one correspondence between X-measurable functions and functions that depend only on X. As a corollary we have: Corollary The set of X-measurable functions is a vector subspace of the space of random variables. 4.9 Conditioning on the value of a Random Variable: The discrete case Let X and Y be discrete random variables with joint pmf px, y), and let p X and p Y be the respective marginals. Then Y y) and X x) are events so the conditional probability of Y y) given X x) is Pitman [5]: Section 6. P Y y X x ) P Y y, X x) P X x) px, y) px, y) y px, y ) p X x). 4) This is a function of x, known as the conditional pmf of Y given X x and it defines the conditional distribution of Y given X x 4.9. Example Pitman [5, Exercise 6..5, p. 399]) Let X and Y be independent Poisson random variables with parameters µ and λ. What is the distribution of X given X +Y n? KC Border v ::3.25

8 KC Border Order Statistics; Conditional Expectation 4 8 You may or may not recall what the distribution of X + Y is, so let s just roll out the old convolution formula recalling that X and Y are always nonnegative): P X + Y n) pk, n k) k e k µ µk e µ+λ) n! k! k which is a Poissonµ + λ) distribution. So e λ λn k n k)! n! k!n k)! µk λ n k convolution & nonnegativity independence arithmetic e µ+λ) µ + λ) n Binomial Theorem n! P X k X + Y n ) P X k, X + Y n) P X + Y n) defn. P X k, Y n k) P X + Y n) arithmetic P X k) P Y n k) independence P X + Y n) ) ) µ µk λ λn k e k! e n k)! Poisson e µ+λ) n! µ + λ) n ) n µ k λ n k k n k µ + λ) n ) µ µ + λ ) k λ µ + λ ) n k. This is just the probability that a Binomialn, p) random variable is equal to k, when p µ/µ + λ) Application to the Poisson arrival process In the Poisson process we discussed last time, N t is the number of arrivals in the interval [, t] and it has Poissonλt) distribution where λ is the arrival rate the rate in the Exponentialλ) waiting time distribution. Now let < s < t. What can we say about P N s k N t n)? Well N s is Poissonλs) random variable and it is independent of N t N s, which is distributed according to the Poisson λt s) ) pmf. Moreover N t N s + N t N s ), so according to Example 4.9. that we just worked out, P N s k N t n) is the Binomial v ::3.25 KC Border

9 KC Border Order Statistics; Conditional Expectation 4 9 probability P N s k N t n) ) n k n k ) s λs λs + λt s) t ) ) k n k t s. t ) k λt s) ) n k λs + λt s) When you think about it, one interpretation of the Poisson process, that arrivals are uniformly scattered in an interval, so the probability of hitting [, s] given that [, t] has been hit is just s/t. Remember s < t.) So the probability of getting k hits on [, s] given n hits on [, t] is given by the Binomial with probability of success s/t. 4. Conditional Expectation In one way, conditional expectation is quite simple. It is the expectation of a random variable with respect to a conditional distribution. For instance, E Y X x ) y yp Y y X x) y px, y) y px). Note that we have used the common convention not to subscript the probability mass functions, but to use the names of the arguments, x or y or x, y), to indicate whether we are talking about the marginal distribution X or Y, or the joint distribution of the vector X, Y ). 4.. Example Continuing with the previous example, Example 4.9.: Let X and Y be independent Poisson random variables with parameters µ and λ. Then we saw that the distribution of X given X + Y n was a Binomial n, µ/µ + λ) ) distribution: P X k X + Y n ) n k ) ) k ) n k µ λ. µ + λ µ + λ Now a Binomial n, µ/µ + λ) ) has expectation nµ/µ + λ), so Similarly E X X + Y n ) nµ µ + λ E Y X + Y n ) which implies the comforting conclusion that nλ µ + λ E X X + Y n ) + E Y X + Y n ) n. 4. Conditional Expectation, Part 2 So far we have defined EY X x) for discrete random variables. This quantity depends on x, so we can write it as a function of x. Let s use the name υ for this function, because y is the Latin equivalent of the Greek υ ypsilon). Pitman [5]: p. 42 υx) E Y X x ). KC Border v ::3.25

10 KC Border Order Statistics; Conditional Expectation 4 The random variable υx) is known as the conditional expectation of Y given X, which is written E Y X ) υx). The thing to see is that this is a random variable since it is a function of the random variable X. By Theorem 4.8. E Y X ) is σx)-measurable. Another way to say this is that E Y X ) is a random variable that equals E Y X x ) when X x. In terms of the probability space S, E, P ) on which X is defined, we have Xs) x E Y X ) s) υ Xs) ) E Y X x ). 4.. Example Let S be the six-point sample space consisting of x, y) pairs S {, ),, ),, 2),, ),, ),, 2)} with probability measure given in the following diagram where the numbers in each box represent the probability of the sample point: 3 y 2 5 y 4 y 3 8 x x Let X and Y be random variables on S defined by Xx, y) x and Y x, y) y. Then so p X ) p X ) 2, E X p X ) + p X ) 2, and so The event X is highlighted below. p Y ) 2, p Y ) 3, p Y 2) 6 E Y 2p Y 2) + p Y ) + p Y ) v ::3.25 KC Border

11 KC Border Order Statistics; Conditional Expectation 4 So EY X ) 2P Y 2 X ) + P Y X ) + P Y X ) P Y 2, X ) P Y, X ) P Y, X ) P X ) P X ) P X ) Similarly EY X ) 5 2. Thus the random variable EY X) is defined on S by EY X)x, y) { 2 x, 5 2 x, which can be represented in the diagram, where now the numbers in each box represent the value of the random variable EY X): The expectation of the random variable EY X) is From the calculation above, E EY ) X) E EY ) X) 2 3 E Y. 4.2 Conditional Expectation is a Positive Linear Operator Too Ordinary expectation is a positive linear operator that assigns random variables a real number. Conditional expectation assigns random variables another random variable, but it is also linear and positive: Pitman [5]: p. 42 E ay + bz X ) a E Y X ) + b E Z X ) Y E Y X ). KC Border v ::3.25

12 KC Border Order Statistics; Conditional Expectation Iterated Conditional Expectation Since E Y X ) is a random variable, we can take its expectation. Pitman [5]: p. 43 E E Y X )) E Y. More remarkable is the following generalization Theorem For a Borel) function φ, E φx) E Y )) ) X E φx)y. Proof : Let υx) E Y X x ). Then υx) y ypx, y), px) and so E φx) E Y X )) E φx)υx) ) φx)υx)px) x ) ypx, y) φx) px) px) x y ) φx) ypx, y) x y φx)ypx, y) x,y) E φx)y ). In light of Theorem 4.8. we have the following corollary Corollary If Z is σx)-measurable, that is, if we can write Z φx) for some Borel) function φ, then E Z E Y X )) EZY ). 4.4 Conditional Expectation is an Orthogonal Projection Recall Corollary 4.8.2, which states that the space of σx)-measurable random variables is a vector subspace. If we further restrict attention to L 2, the space of square-integrable random variables, we have the inner product defined by X, Y ) EXY ). See Section 9.. Recall from your linear algebra class that in an inner product space, the orthogonal projection of a vector y on a subspace M is the unique vector y M such that y M M, and y y M ) M. It turns out that conditional expectation with respect to X is the orthogonal projection on to the subspace of σx)-measurable random variables. v ::3.25 KC Border

13 KC Border Order Statistics; Conditional Expectation Theorem Let X, Y, and Z have finite variances, and assume that Z is σx)-measurable. Then E Y E Y X )) Z ). Proof : Expand Y E Y X )) Z to get E Y Z E Y X )) Z ) EY Z) E E Y X ) Z ), since expectation is a linear operator. By Corollary 4.3.2, E E Y X ) Z ) EY Z). Substituting this in the previous equation proves the theorem. What this says is that the random variable Y E Y X ) is orthogonal to Z for every σx)-measurable random variable Z. Since E Y X ) is itself σx)-measurable, we have E Y X ) is the orthogonal projection of Y onto the vector space of σx)-measurable random variables, where the inner product X, Y ) is given by EXY ). 4.5 Conditional Expectation and Densities When X and Y have a joint density the definition of P Y y X x ) seems ill-defined: Since when X has a density, P X x) for every x, we cannot define P Y y X x ) as P Y y, X x) /P X x), since that would entail division by zero. It is beyond the scope of this course to prove it, but the following approach works. Given an interval B, a real number x, and ε >, consider P Y B X x ε, x + ε) ) P Y B, X x ε, x + ε)). P X x ε, x + ε)) If the marginal density f X of X is positive and continuous at x, then the denominator is positive, so we are no longer dividing by zero. What we want is for this to tend to a limit as ε tends to zero, and in fact it does, a result known as the Radon Nikodym Theorem. Define so f Y y X x ) fx, y) f X x) fx, y) f Y y X x ) f X x). For a function h: R R, E hy ) ) X x hy)f Y y X x ) dy fx, y) hy) f X x) dy KC Border v ::3.25

14 KC Border Order Statistics; Conditional Expectation Conditioning with Several Variables Let Y, X,, X n have joint density fy, x,..., x n ). The conditional density of Y given X X,..., X n x n is then f Y y x,..., x n ) fy, x,..., x n ) f X,...,X n x,..., x n ), and we may speak of E Y ) X,..., X n, etc. Similarly, f X,...,X n X,..., X n Y y) fy, x,..., x n ), f Y y) and we may speak of E X,..., X n Y ), etc. 4.7 Conditional Independence If f X,Y ) x, y Z z) f X x Z z)f Y y Z z), we say that X and Y are conditionally independent given Z z. If this is true for all z, we say that X and Y are conditionally independent given Z. Here is a transparent but artificial example Example The outcome space has eight points, each equally probable. The values of the random variables X, Y, and Z are given in the following table. s s 2 s 3 s 4 s 5 s 6 s 7 s 8 Z X 2 2 Y 2 2 You can see that X and Y are not independent. For example, P X 2, Y ) P X 2) P Y ) 6. But X and Y are conditionally independent given Z. conditional on values of z: Here is the distribution of X and Y Z Z X X 2 X X Y Y 4 4 Y 4 4 Y 4 4 It is easy to verify conditional independence. A common way that dependent, but conditionally independent random variables can arise is like this. Let U, V, Z be independent random variables, and let X Z + U and Y Z + V. Then usually X and Y are not independent, but they are conditionally independent given Z. Bibliography [] C. D. Aliprantis and K. C. Border. 26. Infinite dimensional analysis: A hitchhiker s guide, 3d. ed. Berlin: Springer Verlag. v ::3.25 KC Border

15 KC Border Order Statistics; Conditional Expectation 4 5 [2] R. J. Larsen and M. L. Marx. 22. An introduction to mathematical statistics and its applications, fifth ed. Boston: Prentice Hall. [3] J. Maynard Smith The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology 47): DOI:.6/ )9-6 [4] G. A. Parker and E. A. Thompson. 98. Dung fly struggles: a test of the war of attrition. Behavioral Ecology and Sociobiology 7): DOI:.7/BF3256 [5] J. Pitman Probability. Springer Texts in Statistics. New York, Berlin, and Heidelberg: Springer. [6] M. M. Rao. 98. Foundations of stochastic analysis. Mineola, NY: Dover. Reprint of the 98 Academic Press edition. KC Border v ::3.25

16