Question

28 An experimenter knows that the distribution of the lifetime of a certain component is negative exponentially distributed with mean 1/0. On the basis of a random sample of size n of lifetimes he wants to estimate the median lifetime. Find both the maximum-likelihood and uniformly minimum-variance unbiased estimator of the median.

Answer 1

Solution:

Given Hati- the lifetime of a certain component it Negative exponentially distributed with Mean lo. Here, . Exponential distribution With Mean llo fcoe) - 02:03 .630 Median :- 6. ^f(x) dx = Ź » * *4x o => -emota - eno log O -logil2 loge Now, Loone likelihood function is, colai taking log on both sides logd a nlogo-of a value of o o in terms of median CM) Logdanlog (logo) log? Ticari dlogh Now Putting N M M an loga 1697. logo on the dm m2 क

+ loga [xi 0 313 31 m2 log2 nxi [ial m2 => n = log2 [na = loge. Maximum likelihood Estimator a of median is, sloga loge sono t m2 Noul, Ilogh dm again differentialing g Hogh ^-elog2 dne Σε α! me m3 , لماه € Co-logy). 2 loge m2 no Әм2 m3 2 loge me n loge m3 n m2 2n m2 m2 Thus -EL Iloge] 513 1M2 * since ña a logo is unbiased estimator & varma ► loge)varcs) 5) Cloge? - (hoge)? It follows that ñ = aloge is a minimum variance unbiased Ephematori