{"id":497,"date":"2015-01-06T17:04:16","date_gmt":"2015-01-06T17:04:16","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=497"},"modified":"2015-02-20T18:42:29","modified_gmt":"2015-02-21T00:42:29","slug":"uniform-distributions-of-decrements-in-the-multiple-decrement-table","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/4-fractional-age-assumptions\/uniform-distributions-of-decrements-in-the-multiple-decrement-table\/","title":{"rendered":"Uniform Distributions of Decrements in the Multiple Decrement Table"},"content":{"rendered":"<p>We use the acronym (UDD) for &#8220;uniform distributions of decrements&#8221;. Note that this is <em>within the year<\/em>, not over the entire age range. Under this assumption, for integer age \\(x\\) and fraction \\(t\\) \\(( 0 \\leq t \\lt 1)\\), we have<br \/>\n\\begin{eqnarray*}<br \/>\n_t p_x^{0j} = t \\times p_x^{0j}<br \/>\n\\end{eqnarray*}<br \/>\nFor each decrement, exits are uniformly spaced over the year.<br \/>\nNow, if the (UDD) assumption holds for all decrements, then we may write<br \/>\n\\begin{eqnarray*}<br \/>\n~ _t p_x^{00} &#038;=&#038; 1 &#8211; sum_{j=1}^n ~ _t p_x^{0j} = 1 &#8211; sum_{j=1}^n t \\times p_x^{0j} \\\\<br \/>\n&#038;=&#038; 1 &#8211; t \\times p_x^{0\\bullet} .<br \/>\n\\end{eqnarray*}<br \/>\nWith Kolmogorov&#8217;s forward equation, we have<br \/>\n\\begin{eqnarray*}<br \/>\n\\frac{\\partial}{\\partial t} ~ _t p_x^{0j}= p_x^{0j}<br \/>\n= ~ _t p_x^{00} \\mu_{x+t}^{0j} .<br \/>\n\\end{eqnarray*}<br \/>\nThus, we may express the transition force as<br \/>\n\\begin{eqnarray*}<br \/>\n\\mu_{x+t}^{0j} = \\frac{p_x^{0j}}{1 &#8211; t \\times p_x^{0\\bullet}} .<br \/>\n\\end{eqnarray*}<\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/4-fractional-age-assumptions\/summary\/\" title=\"Summary\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/5-associated-single-decrement-models\/\" title=\"5. Associated Single Decrement Models\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We use the acronym (UDD) for &#8220;uniform distributions of decrements&#8221;. Note that this is within the year, not over the entire age range. Under this assumption, for integer age \\(x\\) and fraction \\(t\\) \\(( 0 &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":495,"menu_order":3,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-81","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/497"}],"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=497"}],"version-history":[{"count":4,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/497\/revisions"}],"predecessor-version":[{"id":1618,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/497\/revisions\/1618"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/495"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}