|
|
| Line 1: |
Line 1: |
| In [[probability theory]] and [[statistics]], a '''stochastic order''' quantifies the concept of one [[random variable]] being "bigger" than another. These are usually [[partial order]]s, so that one random variable <math>A</math> may be neither stochastically greater than, less than nor equal to another random variable <math>B</math>. Many different orders exist, which have different applications.
| | A reverse cell phone directory will help you out of trouble when you start having the feeling that your partner is cheating. Being efficient at trace a call is really a way to find out information that your partner would think that have got. Find out how.<br><br>He puts an 800 number regarding press release, and a computerized system where it's all automated required handle 10,000 calls on the other hand. Calls can come directly you r from the press release.<br><br><br><br>Even still, I experienced more positive experiences with T-Mobile than by using my previous cell phone provider. After six years of horrible service with Sprint, I finally decided to cancel my account. Deals with straw was an agent changing my cell [http://paydaymobilityphonenumber.com/ payday mobility loan phone number] with out my permission or experience. It took over a week to have it returned. Even in me moving in order to a new cell phone provider Sprint did some foolish foods.<br><br>Further in this particular article we are going to discuss the functioning of services. Every just one of us must have received some unwanted calls. These calls end up being for several reasons. End up being someone really wants to be your friend, or he may wish to request for some ransom. Before taking any step further, you would like to always consult a cop. It is of utmost importance to do so. Just what exactly you do should be absolutely legal in your eyes of legislations. Otherwise, you might land up hurting all by yourself.<br><br>Try to keep your resume a minimum of one page. If one makes your resume longer, you lessen the odds of it being read. An employer has likely advertised widely and might have to scan dozens of resumes - added length may mean they won't read your business opportunity.<br><br>Instead for being "easy", think about your prospect as that guy or gal you have to beg and beg to go out out with you. And can then be after every thing finessing she only agrees to check out breakfast! You have to strategize november 23 her over. You bring flowers. You do apparent random acts of kindness just to get into a second date and maybe first make. Before Free Love, this was called Dating.<br><br>Finally, consider joining several different sites in order to increase your opportunities. Generally, the top survey sites only cost about $35 for an entire life membership. So provided may joined a very good site by using the above guidelines a lot more be poised to a few excellent extra revenue. |
| | |
| ==Usual stochastic order==
| |
| A real random variable <math>A</math> is less than a random variable <math>B</math> in the "usual stochastic order" if
| |
| | |
| :<math>\Pr(A>x) \le \Pr(B>x)\text{ for all }x \in (-\infty,\infty),</math>
| |
| | |
| where <math>\Pr(\cdot)</math> denotes the probability of an event.
| |
| This is sometimes denoted <math>A \preceq B</math> or <math>A \le_{st} B</math>. If additionally <math>\Pr(A>x) < \Pr(B>x)</math> for some <math>x</math>, then <math>A</math> is stochastically strictly less than <math>B</math>, sometimes denoted <math>A \prec B</math>.
| |
| | |
| ===Characterizations===
| |
| The following rules describe cases when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.
| |
| | |
| #<math>A\preceq B</math> if and only if for all non-decreasing functions <math>u</math>, <math>{\rm E}[u(A)] \le {\rm E}[u(B)]</math>.
| |
| #If <math>u</math> is non-decreasing and <math>A\preceq B</math> then <math>u(A) \preceq u(B)</math>
| |
| #If <math>u:\mathbb{R}^n\mapsto\mathbb{R}</math> is an increasing function and <math>A_i</math> and <math>B_i</math> are independent sets of random variables with <math>A_i \preceq B_i</math> for each <math>i</math>, then <math>u(A_1,\dots,A_n) \preceq u(B_1,\dots,B_n)</math> and in particular <math>\sum_{i=1}^n A_i \preceq \sum_{i=1}^n B_i</math> Moreover, the <math>i</math>th [[order statistic]]s satisfy <math>A_{(i)} \preceq B_{(i)}</math>.
| |
| #If two sequences of random variables <math>A_i</math> and <math>B_i</math>, with <math>A_i \preceq B_i</math> for all <math>i</math> each [[convergence in distribution|converge in distribution]], then their limits satisfy <math>A \preceq B</math>.
| |
| #If <math>A</math>, <math>B</math> and <math>C</math> are random variables such that <math>\sum_c\Pr(C=c)=1</math> and <math>\Pr(A>u|C=c)\le \Pr(B>u|C=c)</math> for all <math>u</math> and <math>c</math> such that <math>\Pr(C=c)>0</math>, then <math>A\preceq B</math>.
| |
| | |
| ==Other properties==
| |
| If <math>A\preceq B</math> and <math>{\rm E}[A]={\rm E}[B]</math> then <math>A=B</math> in distribution.
| |
| | |
| ==Stochastic dominance==
| |
| [[Stochastic dominance]]<ref>http://www.mcgill.ca/files/economics/stochasticdominance.pdf</ref> is a stochastic ordering used in [[decision theory]]. Several "orders" of stochastic dominance are defined.
| |
| *Zeroth order stochastic dominance consists of simple inequality: <math>A \preceq_{(0)} B</math> if <math>A \le B</math> for all [[state of nature|states of nature]].
| |
| *First order stochastic dominance is equivalent to the usual stochastic order above.
| |
| *Higher order stochastic dominance is defined in terms of integrals of the [[distribution function]].
| |
| *Lower order stochastic dominance implies higher order stochastic dominance.
| |
| | |
| ==Multivariate stochastic order==
| |
| An <math>\mathbb R^d</math>-valued random variable <math>A</math> is less than an <math>\mathbb R^d</math>-valued random variable <math>B</math> in the "usual stochastic order" if
| |
| | |
| :<math>{\rm E}[f(A)] \le {\rm E}[f(B)]\text{ for all bounded, increasing functions } f:\mathbb R^d\longrightarrow\mathbb R </math>
| |
| | |
| Other types of multivariate stochastic orders exist. For instance the upper and lower orthant order which are similar to the usual one-dimensional stochastic order. <math>A</math> is said to be smaller than <math>B</math> in upper orthant order if
| |
| | |
| :<math>\Pr(A>\mathbf x) \le \Pr(B>\mathbf x)\text{ for all } \mathbf x \in \mathbb R^d </math>
| |
| | |
| and <math>A</math> is smaller than <math>B</math> in lower orthant order if
| |
| | |
| :<math>\Pr(A\le\mathbf x) \ge \Pr(B\le\mathbf x)\text{ for all } \mathbf x \in \mathbb R^d </math>
| |
| | |
| All three order types also have integral representations, that is for a particular order <math>A</math> is smaller than <math>B</math> if and only if <math>{\rm E}[f(A)] \le {\rm E}[f(B)] </math> for all <math>f:\mathbb R^d\longrightarrow \mathbb R</math> in a class of functions <math>\mathcal G</math>.<ref>Alfred Müller, Dietrich Stoyan: ''Comparison methods for stochastic models and risks.'' Wiley, Chichester 2002, ISBN 0-471-49446-1, S. 2.</ref> <math>\mathcal G</math> is then called generator of the respective order.
| |
| | |
| ==Other stochastic orders==
| |
| | |
| ===Hazard rate order===
| |
| The ''[[hazard rate]]'' of a non-negative random variable <math>X</math> with absolutely continuous distribution function <math>F</math> and density function <math>f</math> is defined as
| |
| :<math>r(t) = \frac{d}{dt}(-\log(1-F(t))) = \frac{f(t)}{1-F(t)}.</math>
| |
| | |
| Given two non-negative variables <math>X</math> and <math>Y</math>
| |
| with absolutely continuous distribution <math>F</math> and <math>G</math>,
| |
| and with hazard rate functions
| |
| <math>r</math> and <math>q</math>, respectively,
| |
| <math>X</math> is said to be smaller than <math>Y</math> in the hazard rate order
| |
| (denoted as <math>X \le_{hr}Y</math>) if
| |
| :<math>r(t)\ge q(t)</math> for all <math>t\ge 0</math>,
| |
| or equivalently if | |
| :<math>\frac{1-F(t)}{1-G(t)}</math> is decreasing in <math>t</math>.
| |
| | |
| ===Likelihood ratio order===
| |
| Let <math>X</math> and <math>Y</math> two continuous (or discrete) random variables with densities (or discrete densities) <math>f \left( t \right)</math> and <math>g \left( t \right)</math>, respectively, so that <math>\frac{g \left( t \right)}{f \left( t \right)}</math> increases in <math>t</math> over the union of the supports of <math>X</math> and <math>Y</math>; in this case, <math>X</math> is smaller than <math>Y</math> in the ''likelihood ratio order'' (<math>X \le _{lr} Y</math>).
| |
| | |
| ===Mean residual life order===
| |
| | |
| ===Variability orders===
| |
| If two variables have the same mean, they can still be compared by how "spread out" their distributions are. This is captured to a limited extent by the [[variance]], but more fully by a range of stochastic orders.{{citation needed|date=February 2012}}
| |
| | |
| ====Convex order====
| |
| Convex order is a special kind of variability order. Under the convex ordering, <math>A</math> is less than <math>B</math> if and only if for all convex <math>u</math>, <math>{\rm E}[u(A)] < {\rm E}[u(B)]</math>.
| |
| | |
| ===Laplace transform order===
| |
| Laplace transform order compares both size and variability of two random variables. Similar to convex order, Laplace transform order is established by comparing the expectation of a function of the random variable where the function is from a special class: <math>u(x) = -\exp(-\alpha x)</math>. This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with <math> \alpha </math> a positive real number.
| |
| | |
| ===Realizable monotonicity===
| |
| Considering a family of probability distributions <math> ({P}_{\alpha})_{\alpha \in F} </math> on partially ordered space <math> (E,\preceq) </math>
| |
| indexed with <math> \alpha \in F </math> (where <math> (F,\preceq) </math> is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables <math> (X_\alpha)_{\alpha} </math> on the same probability space, such that the distribution of <math> X_\alpha </math> is <math> {P}_\alpha </math> and <math> X_\alpha \preceq X_\beta </math> almost surely whenever <math> \alpha \preceq \beta </math>. It means the existence of a monotone [[coupling (mathematics)|coupling]]. See<ref>Stochastic Monotonicity and Realizable Monotonicity
| |
| James Allen Fill and Motoya Machida, The Annals of Probability, Vol. 29, No. 2 (Apr., 2001), pp. 938-978, Published by: Institute of Mathematical Statistics, Stable URL: http://www.jstor.org/stable/2691998</ref>
| |
| | |
| ==See also==
| |
| *[[Stochastic dominance]]
| |
| | |
| ==References==
| |
| {{refimprove|date=February 2012}}
| |
| #M. Shaked and J. G. Shanthikumar, ''Stochastic Orders and their Applications'', Associated Press, 1994.
| |
| #E. L. Lehmann. Ordered families of distributions. ''The Annals of Mathematical Statistics'', 26:399–419, 1955.
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:Stochastic Ordering}}
| |
| [[Category:Theory of probability distributions]]
| |
A reverse cell phone directory will help you out of trouble when you start having the feeling that your partner is cheating. Being efficient at trace a call is really a way to find out information that your partner would think that have got. Find out how.
He puts an 800 number regarding press release, and a computerized system where it's all automated required handle 10,000 calls on the other hand. Calls can come directly you r from the press release.
Even still, I experienced more positive experiences with T-Mobile than by using my previous cell phone provider. After six years of horrible service with Sprint, I finally decided to cancel my account. Deals with straw was an agent changing my cell payday mobility loan phone number with out my permission or experience. It took over a week to have it returned. Even in me moving in order to a new cell phone provider Sprint did some foolish foods.
Further in this particular article we are going to discuss the functioning of services. Every just one of us must have received some unwanted calls. These calls end up being for several reasons. End up being someone really wants to be your friend, or he may wish to request for some ransom. Before taking any step further, you would like to always consult a cop. It is of utmost importance to do so. Just what exactly you do should be absolutely legal in your eyes of legislations. Otherwise, you might land up hurting all by yourself.
Try to keep your resume a minimum of one page. If one makes your resume longer, you lessen the odds of it being read. An employer has likely advertised widely and might have to scan dozens of resumes - added length may mean they won't read your business opportunity.
Instead for being "easy", think about your prospect as that guy or gal you have to beg and beg to go out out with you. And can then be after every thing finessing she only agrees to check out breakfast! You have to strategize november 23 her over. You bring flowers. You do apparent random acts of kindness just to get into a second date and maybe first make. Before Free Love, this was called Dating.
Finally, consider joining several different sites in order to increase your opportunities. Generally, the top survey sites only cost about $35 for an entire life membership. So provided may joined a very good site by using the above guidelines a lot more be poised to a few excellent extra revenue.