Certain Estimation Problems in Dependability using Percentiles

<p>In statistical estimation problems we use random variables X1 X2... Xn that are independent and identically distributed as random variable X whose probability distribution is known but involves certain unknown parameters ? = (?1 ?2... ?k) (k < n) which are labelling or indexing parameters. The parameters ?1 ?2... ?k are not random variables. We will consider the following problem. The random variables X1 X2... Xn have a common distribution of continuous type defined by the probability density function f (x1 x2 . . . xn; ?1 ?2... ?k). We observe a point x = (x1 x2 . . . xn) ? X ? Rn of the variables where X is the sample space. It is required to use these observed values x1 x2 . . . xn to find estimates of unknown parameters ?1 ?2... ?k. We consider random sample defined as Definition 1.1.1 The random variables X1 X2... Xn are called a random sample of size n from the population f (x; ?)? ? ? ? Rk if X1 X2... Xn are mutually independent random variables and the marginal pdf or pmf of each Xi is the same function f (x; ?). Alternatively X1 X2... Xn are called independent and identically distributed (iid) random variables with pdf or pmf f (x; ?).</p><p> </p>