{"id":514,"date":"2015-01-06T17:32:57","date_gmt":"2015-01-06T17:32:57","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=514"},"modified":"2015-02-20T18:49:16","modified_gmt":"2015-02-21T00:49:16","slug":"6-building-mult-decrement-tables-from-associated-single-life-functions","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/6-building-mult-decrement-tables-from-associated-single-life-functions\/","title":{"rendered":"6. Building Mult-Decrement Tables from Associated Single Life Functions"},"content":{"rendered":"<p>To understand the relationship between multi-decrement and the associated single decrement tables, first note that<br \/>\n\\begin{eqnarray*}<br \/>\n~ _t p_x^j =\\exp \\left\\{ &#8211; \\int_0^t\\mu_{x+u}^{0j} ~ du \\right\\}<br \/>\n\\end{eqnarray*}<br \/>\nand so \\(0 \\le ~ _t p_x^j \\le 1\\). Thus<br \/>\n\\begin{eqnarray*}<br \/>\n~ _s p_x^{00} = \\prod_{k=1}^n ~ _s p_x^k \\leq ~<br \/>\n_s p_x^j ,<br \/>\n\\end{eqnarray*}<br \/>\nfor any (j). Thus, we have<br \/>\n\\begin{eqnarray*}<br \/>\n~ _t p_x^{0j} = \\int_0^t ~ _s p_x^{00} \\mu_{x+s}^{0j} ~ ds \\leq<br \/>\n\\int_0^t ~ _s p_x^j \\mu_{x+s}^{0j} ~ ds = ~ _t q_x^j .<br \/>\n\\end{eqnarray*}<\/p>\n<p>That is, the multi-decrement transition probability is less than or equal to the associated single decrement function.<\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/5-associated-single-decrement-models\/example\/\" title=\"Example\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/6-building-mult-decrement-tables-from-associated-single-life-functions\/1-constant-forces-of-transition\/\" title=\"1. Constant Forces of Transition\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>To understand the relationship between multi-decrement and the associated single decrement tables, first note that \\begin{eqnarray*} ~ _t p_x^j =\\exp \\left\\{ &#8211; \\int_0^t\\mu_{x+u}^{0j} ~ du \\right\\} \\end{eqnarray*} and so \\(0 \\le ~ _t p_x^j \\le &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":454,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-8i","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/514"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=514"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/514\/revisions"}],"predecessor-version":[{"id":1622,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/514\/revisions\/1622"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/454"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}