Certain Estimation Problems in Dependability using Percentiles

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 θ = (θ1, θ2,..., θk) , (k < n), which are labelling or indexing parameters. The parameters θ1, θ2,..., θk 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; θ1, θ2,..., θk). 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 θ1, θ2,..., θk. 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; θ),θ ∈ Θ ⊂ 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; θ). Alternatively, X1, X2,..., Xn are called independent and identically distributed (iid) random variables with pdf or pmf f (x; θ).