summary. - ACT348H SUMMARY OF NOTES (Includes notes on...

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LC-1 ACT348H SUMMARY OF NOTES (Includes notes on Chapters 3-5 as well as 6-10)
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LC-2 SURVIVAL ANALYSIS Klein, Moeschberger Book, Chapters 2-3 Time-Until-Event Random Variable - Chapter 2 \ The random variable usually under consideration is a "time-until-death" or a random time-to-event variable is assumed to be , and may be continuous or discrete, with the followingrelated \ & \ ±! characteristics: survival or reliability function - W—B&±T—\²B & W—!&± " W—B&±! ²² lim B˜_ cumulative distribution function (cdf) J—B&±T—\³B&±"´W—B& ² J—!&± ! J—B&±" , B˜_ , probability density function (pdf) (if is continuous) \ 0—>&±J —> & >²! w - hazard function (if is continuous) \ for 2—B& ± ± ´ 68µW—B&¶· ! B²! 0—B& W—B& . .B cumulative hazard function (if is continuous) \ L—B&±´ 68µW—B&¶ ± 2—>&.> - ! B ² and W—B&±/B:µ´L—B&¶±/ ´ ! B mean remaining lifetime (if is continuous) \ - B³!² 7<6—B&±Iµ\´Bl\²B¶ ± ± ±/ ’’ BB __ —>´B&0—>&. > W—>&.> W—B & W—B& ° B , . ±I—\&±7<6—!& ± >¸0—>&.> ± !! ² Z+<—\&±I—\ &´µI—\& ¶ ± # >¸W—>&.>´ µ W—>&.>¶ # ## : "!! : \B -th quantile (or -th percentile) of for the number in the interval , : ´!²"µ : is the number such that ; J—B &· : W—B &·"´: :: the 50-th percentile is the of the distribution median I f then is continuous \ ± 0—>&.>± / ± / ± ¸/B:µ ´¶ B! _B ´L—B& ´ ! B 7<6—!& 7<6—B & 7<6—>& .> ² J—B& ± 0—>&. > 0—B&± ´ ± J—B ! B .. . B ²¶ and 2—B&±L —B & L—B& ± w ! B ²&
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LC-3 If , then there is a , with \ :—B & :—B &±T—\±B& is discret e probability mass function (pf) 44 for each point of probability (it is assumed that these points are indexed in order of increasing size - B 4 B &B &± "# ). In the discrete case, W—B& ± :—B & 2—B & ± ± ²"³2—B &´ !# B µB 4 B ¶B 4 4 , :—B& W— B& 4 4³" ²³ Example 3: The density function for a continuous survival distributionfor random variable is \ 0—B´µB / ²B¶ ! W—B´ ² J—B´ ² 2—B´ ² L—B´ ² I·\¸ ²Z +<·\ ¸ 7<6—B´ ¹B . Find and . Solution: W—B´ µ 0—>´.> µ > / .>µ —Bº"´ ’’ BB __ ¹ > ¹B J—B´µ"¹W—B´µ"¹—Bº"´ / ² 2—B´ µ µ µ² ¹B 0—B´ W—B´ B Bº" B/ —Bº"´/ ¹B ¹B L—B´ µ 2—>´.>µ¹ 68·W—B´¸µB¹68—Bº"´² ! B I·\¸ µ B»0—B´.B µ W—B´.B µ —Bº"´ / .Bµ¹ —Bº#´ / µ#² ! !! _ ¹ B ¹B Bµ! Bµ_ I·\ ¸ µ B »0—B´.B µ B »B / .B µ µ’² # # # ¹B $x " $ Z+<·\¸µI·\ ¸¹—I·\¸ ´ µ’¹ # µ#² # ## 7<6—B´ µ µ µ µ³ ¹> ¹ B ¹B ¹B W—>´. > —>º"´ / .> W—B ´ / —Bº#´/ Bº# Bº" ¤ Example 4: The survival function for a discrete survival distribution for random variable defined at \ non-negative integers is for , an integer. W—5´µ—³* ´ 5¼!5 5 Find . :—5´ ² J—5´ ² 2—5´ ² I·\ ¸ Z +<·\¸ an integer :—5´µW—5¹"´¹W—5´µ—³"´—³* ´ 5¼"² 5¹" J—5´µ"¹W—5´µ"¹—³* ´² 2—5´ µ µ µ³" 5 :—5 ´ —³"´—³*´ W—5¹"´ —³*´ 5¹" 5¹" .
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This note was uploaded on 09/27/2009 for the course ACT 348 taught by Professor Broverman during the Fall '04 term at University of Toronto- Toronto.

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summary. - ACT348H SUMMARY OF NOTES (Includes notes on...

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