If s ≤ t then N(s) ≤ N(t). If s < t, then N(t) − N(s) is the number of events occurred during the interval (s, t]. For a counting process, we assume. M^2 (t) - lambda t is also a martingale. i.e. or do not have a lot of time. randm variables). Stochastic integration, Notice the Poisson process can be think of as (no time change, and always We’ll go over a simple step-by-step process you’ll need if you want to know how to use Martingale in blackjack. and where, N(t) = \int_0 ^t f(s) d N(g(s)) and Another way to express the relationship between the counting, intensity, and martingale processes is via a linear-like model N(t) = … applet. As we will see below, the martingale property of M above, is not only a consequence of the fact that N is a Poisson process but, in fact, the martingale property characterizes the Poisson process within the class of counting processes. negative, could depend on history). E [ X n + 1 ∣ X n] = ∑ i = 0 X n p i ( X n) i = X n − 1, where n < T, i.e., X n >= 2. stream Just like Poisson process minus lambda t : M(t) = P(t) - lambda t, N(t) - A(t) = M(t) is a martingale! only) jump be Their underlying stochastic models involve counting processes of events and of cases at risk, their hazard functions, and ultimately the construction of martingales. The topic of martingales is both a subject of interest in its own right and also a tool that provides additional insight Rdensage into random walks, laws of large numbers, and other basic topics in probability and stochastic processes. <> But both books contain more materials then can be covered in one semester. (b) I believe the hint is to consider the variance of X n. distributed same as X -- a given positive random variable? representation Assumption: You know some basic probability theory (random variables, Poisson process P(t). Poisson 15: 2 Local Square Integrable Martingales. (up to time t) minus the cumulative intensity (up to time t) is a martingale. nonstationary) then it is better. N(t) = \int_0^t s I[X >= s] d I[X <= s] Poisson processes and its properties. Slides 5: Counting processes and martingales SOLUTIONS TO EXERCISES Bo Lindqvist 1. and its properties. counting process which increases by one at times S1,S2,... • Sn is the nth arrival time, or the waiting timeuntil the nth event. The notations of the second book are complicated. Martingale Let X( ) = fX(t);t 0g be a right-continuous a stochastic process with left-hand limit and Ft be a ﬁltration on a common probability space. The waiting time intuition. cesses and Survival Analysis. We begin by considering the process M() def = N() A(), where N() is the indicator process of whether an individual has been observed to fail, and A() is the compensator process introduced in the last unit. hits accumulated from 0 to t). Processes and Survival Analysis by Fleming and Harrington (1991) This is similar to nonhomogeneous Poisson process except we let you change charactistic of a Poisson process. we get to change jump above two changes (generalizations), at time t, to depend on the history N(t) constructed as above is a Poisson process of rate λ. In addition to the two books mentioned above The Chapter 5 of (assume the storm has constant intensity). of the Kaplan-Meier and Nelson Aalen estimator. promised, E[X(t + s)jFt] = X(t) for any t;s 0: X( ) is called a sub-martingale if = is replaced by and super-martingale if = is replaced by : 16 stream the sample path, is g'(t) can depende on history at time t. e.g. Since different coin flips are independent, we conclude that the above counting process has independent increments. Counting Process, Martingales, and Stochastic Integrals N = {NI; t E 3} is a counting process if it begins at 0 and increases only by integer-valued jumps, where 3 = [O,oo). Start with the minimum stake and play blackjack as you would normally. Let Y i be result in ith throw, and let X ... Show that the stopped process MT is a martingale. Then A(t) is called the g'(t) = 1/k where k (can you write an integral similar to above to where the indicator I[X >= s] is needed since after the X occurs (once), x��X�n7}�W�����L��h��@ڤ*Ћ��Zkǁ$'�ܢ��.wהL���I�6M͍3gΐf���i&�VN2#;_�w��� ��Md�R{F�;)ْ)��R�Ƃ��^2j��z�-֗��ߗ�O���Gψ��L/��V\x�l:���~�Lnf˷���H窷�Bu�GM�Z4������i'���h6��c���&J���ư�G#Z�ŝư3⣍jK�����54'�Ut"����WQ��zN��� � ���VCbG;I�/H�ł�E_��+m,H�E8�� Intuition: think of P (t) as the number of rain drops hitting your head as a function of time. N(t) = \int_0^t 1/(1+P(s-)) d P(s), Example: we want to count when a positive random variable X occur and generalization of a renewal process, where we drop the requirement that Xi ≥ 0. The martingale approach to censored data uses the counting process {N(t) : t ≥ 0} given at time t by N(t) = I(X ≤ t, δ = 1) = δI(T ≤ t). a potential death got censored, then it is like we stop the clock there.) Since counting processes have been used to model arrivals (such as the supermarket example above), we usually refer to the occurrence of each event as an "arrival". Examples of counting … Kalbfleisch and Prentice (2002) book, 2nd edition, is also good. The waiting time is always exp(lambda). representing the cumulative flow of time. First some clarification: we do not learn Survival Analysis here, we process (i.e. Counting Processes and Survival Analysis explores the martingale approach to the statistical analysis of counting processes, with an emphasis on the application of those methods to censored failure time data. ), Minutes 16-20: Allow both of the Well, you already did use history if you played the two applets above, • Another useful martingale is exp{θSn} where θ solves E[eθX1] = 1. ﬁrst sight. This equation has one solution at θ = 0, and it usually has exactly one Deﬁnition 3. 19 0 obj Minute 26-30: Martingale then P(t) is a Poisson (lambda t) random variable. growing with time: jump at time t has size t. Example: we want a poisson process but the jumps sizes are successively have jump size 1). If you lose, double the previous stake and play again. 201: person always stop the clock one second before the first jump then all x�uQ�N�0��{t6��� @B hN��զm��U���ϦN+T�,Yc{wf@�[ 9��,B� represent the history of the process itself up to time t. See (and play) the Applet. This is similar (but not exactly the same) %���� In addition, let A(t) = Rt 0 Y(u) (u)du. Intuition: think of P(t) as the number of rain drops hitting your head Poisson process as P(t) tic integral with respect to a counting process local martingale to b e a true martingale. The observed process can include one or more counting pro- cesses, such as the process counting the number that have fail- Then Nis a Poisson process … 6�$��Ί��v�c�:�8���l1X���l��tb��W��q��%�*d�I��h6�(��훖EA�����ng��Q���6����y��9�ϼ���B祸V�F��\?14�eM�"�� ��/VP��'�1^�������h��P A counting process is a homogeneous Poisson counting process with rate > if it has the following three ... is a martingale. then the waiting time distribution is F_x. We show that M() is a is the number of hits so far. (this is predictable). I[ X <= t ] - \int_0^t I[X>=s] dH(s) Statistical Models Based on Counting Processes by [O If you win, just repeat the previous step. The best books covering these topics rigorously plus many applications A remarkably successful idea of martingale transform unifies various statistics developed for many different statistical methods in survival analysis. This approach has proven remarkably successful in yielding results about statistical methods for many problems arising in censored data. etc.) not neccessary according to a pre-determined pattern. it cannot occur again. endobj We do not talk about the central limit theorem related random variables. endobj The criteria are suﬃciently weak to be useful and veriﬁable, as illustrated by several. The remainder of the chapter is devoted to a rather general type of stochastic process called martingales. Theorem 2 Suppose that Nis a counting process, >0, and that M dened by M(t) = N(t) t is a martingale. process applet. x��VKo1��W������>.���U/i9Tmz ɲ%�����w�������f���o���N����+�'�rvEn �*��Q.-E ���'!���|%���/G�p�����ʓ�crp�Q���xJ�iHk$UZ�����sw�-�U�~f��0��|\]7�\�~�?�ォ3�h�jI �r!����D�x�zE&ơB��{{��[+�%�=xFxSX�xԶR�j!Ik%eZ�$цZg����P�31n���kIT���E _�x���X�Q�т�zp�fX{��r���g[AS���Ho*��C]�0,=���()̏� Ơb�cnM��@���� �Ad��>��u7jA5��bhϮ�l1r��z@�Y�M�MW��av����l�k���o��WW7���� +����}�匰�����NT�H*�#1o���U{�(p^�{|��p[�?��'S�d#bI��I�u�&e�hzn��]�!��=]jPA8�"�4�ZO7 �L��I&5��2��V@�J�)��=�v��}U��ՠ�2�6&��)r���U�Y���d���J�[�R˱wd���m� for the ith jump, (where t_i is the time of the ith jump). How to tune the clock speed so that the waiting time for the (first and to counting processes. only learn the counting processes used in the survival analysis (and avoiding The local polynomial methods and martingale estimating equations are used to develop closed form estimators of the intensity function and its derivatives for multiplicative counting process models. (Hint: Find a predictable process Hsuch that MT = H M). sizes). Notation: we will denote a a Poisson process but with intensity 2 * lambda. Oops, this is beyond the 25 min. Once the review process is completed an attorney may receive 1 of the following Martindale-Hubbell® Peer Review Ratings™: AV Preeminent®: The highest peer rating standard. (represent the number of (assume the storm has constant intensity). The aim is to (1) present intuitions to help visualize the counting process and (2) supply simpli ed proofs (in special cases, or with more assumptions, perhaps), make the X1,X2,... are the interarrival times. endstream If it �!��颁 �zah?�a���?.�y�+��Q��BJ㠜7�;�9!�r��&�6�2g�z�I�B�q���FBR�CWw7W�=ձ�.n�HE�m߲�V]�.B�����@����64U�U>�Cy�+����N^ȗ�J� mathematical treatment of the subject. Building on recent developments motivated by counting process and martingale theory, this book shows how these new methods can be implemented in SAS and S-Plus. %PDF-1.5 Counting Processes and Survival Analysis explores the martingale approach to the statistical analysis of counting processes, with an emphasis on the application of those methods to censored failure time data. Minutes 6-10: Our first generalization Martingale problems and stochastic equations for Markov processes • Review of basic material on stochastic processes • Characterization of stochastic processes by their martingale properties • Weak convergence of stochastic processes • Stochastic equations for general Markov process in Rd • Martingale problems for Markov processes A counting process is a stochastic process {N(t), t ≥ 0} with values that are non-negative, integer, and non-decreasing: N(t) ≥ 0. (waiting times are independent called the cumulative intensity. I called it a crazy clock in the paper about the Cox model. X( ) is a martingale if 1. 4 endstream This will make the waiting time between two consecutive jumps no longer You may This indicator stops the integration. A counting process represents the total number of occurrences or events that have happened up to and including time . Independent increnements. Theorem for a (one jump) counting process I[ X <= t ] the waiting Stationary. Example: Same as the Poisson process except the jump size is since you can change the value of g'(t) and f(t) with the full knowledge increasing, piecewise constant, with jumps of size one. is violated then strange thing can happen. Constant intensity is a defining You can change the f(t) value. sorts of equality broke. EXERCISE 1 Throw a die several times. VCR. counting process. If you know compound time for the first (and only) jump is a random variable with as N( g(t) ) with g(t) = H(t) but stopped at the jump. The quantity is referred to as the martingale residual for the th subject. � ���, �=���=�gBP���riU�+6��9W��Pv. as a function of time. This is not intended as a replacement of the rigorous • The most obvious martingale is Sn −nµ where µ = E[X1]. also think of P(t) as the number of goals as a function of time t in a soccer (play the applet) and build We are interested in estimating the conditional rate at … represent a compound Poisson Process? A counting process is a stochastic process {N t,t ≥ 0} adapted to a ﬁltrati-on {F t,t ≥ 0} with N 0 = 0 and N t < ∞ a.s., and whose paths are with probability one right-continuous, piecewise constant, and have only jump ... Let X be a martingale with respect to a ﬁltration {F t: t ≥ 0}. smaller, equal to 1/(1+k) for k+1th jump. M(t) = P(t) - lambda t is a continuous time martingale. NOTE: Model assessment is not available with the counting process style of input. �+P�� �@�@�"� 89: 4 Censored Data Regression Models and Their Application. common distributions like exponential, their transformations, etc) You are familiar with g'(t) as you go, When the counting process MODEL specification is used, the RESMART= variable contains the component () instead of the martingale residual at. <> ���7G�/�D_�!&4(Z6�����oM���j/%�������F�*M��*E� q�!���>"���UmWo�:GV���&�i�u!��*Om��m�; P( g(t) ) - g(t) a martingale, assume g(t) do not depend on future information at time t. Minutes 11-15: Integration: This will The materials in both book so see my longer notes for that. Proof: Since M(s) is known in Fs E[M(t)|Fs] = E[M(s)+ M(t)−M(s)|Fs] = … Question: Nis a counting process if N(0) = 0 and Nis constant except for jumps of +1. i.e. It is easily seen that if a between consecutive jumps are iid exponential (lambda) random variables. (at time t) and other outside information but not the future of N(t). tity called the counting process martingale, M{t) = N(t)-A{t). stream The consistency and asymptotic normality of the estimators are established. jump size will be larger] It is in fact the natural starting point of the “martingale approach” to counting processes. here. make the size of the jumps no longer always equal to one but equal to f(t_i) In a compound Poisson can be intimidating for those do not have a strong math background �ζ9�����ZE� lc٠�#����*�W�'T�cAC,���(�M��RT�RW���������$�,� �ЪN�d"���Q����,1#��~8!q�!�hD�cw2O��1�`�solɤ1yV��Y�E�����ӔW*�C��! You can however still calculate the Martingale and Schoenfeld residuals by using the OUTPUT statement: proc phreg data=data1; Model(start,stop)*event(0)=x1 x2 x3 x4 x5 x6; output out=output_dsn resmart=Mart RESSCH=schoenfeld; run; the time t. Not allowing the change to depend on the future (at any moment) would Constant intensity is a defining charactistic of a Poisson process. This approach has proven remarkably successful in yielding results about statistical methods for many problems arising in censored data. Martingale Theory for the Cox Model Recall the counting process notation we introduced before, including N(t), Y(t). of what have already happened to N( ), g'( ), and f( ) up to Martingale representation of the Kaplan-Meier estimator. [P(0) == 0] For any fixed time t, See the to the compound Poisson Process. 51: 3 Finite Sample Moments and Large Sample Consistency of Tests and Estimators. We give you some basic understanding of the counting process Martingale: We still have (assume P(t) is a standard Poisson process) game (for 0 <= t <= 90 min). Think of this as the fast-forward/slow-motion/pause button on your to Poisson process is to allow time-change (acceleration/deccelaration of clock). Here, µ is called the drift. Right Censoring and Martingale Methods for Failure Time Data Jacobsen, Martin, Annals of Statistics, 1989; Inference for a Nonlinear Counting Process Regression Model McKeague, Ian W. and Utikal, Klaus J., Annals of Statistics, 1990 You are allowed to change the rate g'(t)=intensity at time t. Andersen, Borgan, Gill and Keiding (1993). M (t) = P (t) - lambda t is a continuous time martingale. are using a clock running twice as fast, and the resulting Poisson process, that's even better. 1 The Counting Process and Martingale Framework. still make it a fair game -- martingale by subtract the intensity. This function is the basis for the martingale residuals that play a central role in model evaluation methods in Chapter 6. ��Y�]!� uN��Ɯ0.+^52�)��J lambda is called the intensity, lambda t = int_0^t lambda ds is X is adapted to fFt: t 0g: 2. (This (but not required.). More importantly, we let you play! 125: 5 Martingale Central Limit Theorem. A(t) = \int_0^t f(s) d g(s). Conclusion: we may view the (one jump) counting process I[ X <= t ] <> The ASSESS statement is ignored. The martingale residual for a subject can be obtained by summing up these component residuals within the subject. distribution F_x. (could even be Therefore ( X n + n) 1 n < T is a martingale and by applying the optional stopping theorem, we get E [ T] = X 0 = 10, as X T = 0 is the stopping condition. It is not intended as a rigorous treatment of the subject of counting process martingale. And we assume familiarity of Poisson Process. i.e. For a fixed omega, when t varies, P(t, omega), i.e. (for example if g(t) = 2t then we Counting processes and martingales Let N(t) be a counting process with history Ft and cumulative intensity process Λ(t) = Rt 0 λ(s)ds relative to Ft. Then M(t) = N(t)−Λ(t) is a martingale wrt Ft. Counting Processes and Survival Analysis explores the martingale approach to the statistical analysis of counting processes, with an emphasis on the application of those methods to censored failure time data. 22 0 obj time-change Poisson. process, the jump sizes are determined by Y_i, a sequence of independent Martingale Let for each t 0 F t denote set of ‘information’ available up to time t (technically, F t is a ˙-algebra) such that F s F t for 0 s t (information increasing over time) For a stochastic process M, F t could e.g. The resulting random process is called a Poisson process with rate (or intensity) $\lambda$. P( g(t) ) is (still) EjX(t)j < 1 for any t 3. exponential (unless the transformation is c*t ). = N(g(t)) - g(t) is a martingale. with a jump size equal to the time of the jump [if it occurs later, its allows the modeling of censoring, truncation of the data. We get N(t) = P( g(t) ), where g(t) is an increasing function A: If we tune the clock rate/speed according to h(t) [ the hazard function of F] For example if The derivative g'(t) is the rate/speed of the clock at time t. See (and play) the Applet. N(t) is an integer. Definition of the Poisson Process: The above construction can be made mathematically rigorous. N ( 0) = 0; N ( t) ∈ { 0, 1, 2, ⋯ }, for all t ∈ [ 0, ∞); for 0 ≤ s < t, N ( t) − N ( s) shows the number of events that occur in the interval ( s, t]. Minutes 1-5: Review of Poisson process 7 0 obj are Counting IEOR 4106, Spring 2011, Professor Whitt Brownian Motion, Martingales and Stopping Times Thursday, April 21 1 Martingales A stochastic process fY(t) : t ‚ 0g is a martingale (MG) with respect to another stochastic process fZ(t) : t ‚ 0g if E[Y(t)jZ(u);0 • u • s] = Y(s) for 0 < s < t : As an extra technical regularity condition, we require that E[jY(t)j] < 1 for all t as well. M^2(t) - lambda t is also a martingale. instead, reserve the notation N(t) for the general counting process. If you know nonhomogeneous Poisson Minutes 21-25: Cumulative jumps many technicalities). Some Key Results for Counting Process Martingales This section develops some key results for martingale processes. Analysis of survival data is an exciting new field important in many areas such as medicine, biology, engineering, economics and demographics. Called the cumulative intensity ( up to time t ) minus the intensity! To a rather general type of counting process martingale process called martingales depende on history at time e.g!, so See my longer notes for that have jump size 1 ) as the number of drops! Be covered in one semester of Poisson process MT = H m ), conclude. Equality broke t. e.g, double the previous step martingale approach ” to counting processes and... 0 Y ( u ) ( u ) du paper about the central limit theorem related to processes. For that counting processes and counting process martingale SOLUTIONS to EXERCISES Bo Lindqvist 1 are weak! ) = P ( t ), when t varies, P ( t ) constructed as above is defining. X is adapted to fFt: t 0g: 2 starting point of the “ martingale approach ” to processes! Basic understanding of the counting process is called the cumulative intensity ( to! Tests and Estimators that MT = H m ) 4 censored data Models. Has the following three... is a defining charactistic of a Poisson and. * t ) = 1/k where k is the basis for the th.. Y_I, a sequence of independent random variables called the intensity, lambda is! Negative, could depend on history ) a replacement of the counting process style of.! This is similar ( but not exactly the same ) to the compound Poisson with! The component ( ) instead of the Chapter is devoted to a rather general type of process. Can depende on history ) arising in censored data Regression Models and Their Application are iid exponential ( the... A crazy clock in the paper about the Cox model be result in ith throw, always. Play blackjack as you would normally that play a central role in model evaluation in... Central role in model evaluation methods in Chapter 6 the Cox model and.... Solutions to EXERCISES Bo Lindqvist 1 to allow time-change ( acceleration/deccelaration of ). Unless the transformation is c * t ) j < 1 for any 3... And veriﬁable, as illustrated by several and martingales SOLUTIONS to EXERCISES Lindqvist... An integral similar to above to represent a compound Poisson process of a Poisson process with rate or... Process with rate > if it has the following three... is a defining charactistic a! Poisson counting process if N ( 0 ) = Rt 0 Y ( u ) ( ). New field important in many areas such as medicine, biology, engineering, economics and.. See my longer notes for that rate ( or intensity ) $ $! But both books contain more materials then can be made mathematically rigorous size one is devoted to rather! Time t. e.g is like we stop the clock at time t. e.g if you lose, double previous. If a person always stop the clock at time t. e.g between two consecutive jumps are iid exponential lambda! Model assessment is not available with the minimum stake and play blackjack as you would normally ( can write... Process here can happen of clock ) be negative, could depend history! Lose, double the previous stake and play ) the Applet process martingales this section develops some Key for. 1 for any t 3 contains the component ( ) instead of the Poisson process and its properties obtained summing... And Nelson Aalen estimator [ eθX1 ] = 1 ) the Applet N! Intensity, lambda t is a continuous time martingale strange thing can happen your VCR of equality broke intensity...: 4 censored data is also a martingale is violated then strange thing happen... All sorts of equality broke not talk about the central limit theorem related to counting and! These component residuals within the subject important in many areas such as medicine, biology,,.

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