Markov-Modulated Samples and Their Applications

Abstract

Samples which elements are dependent random variables are considered. This dependence arises due to total external random environment in which sample elements operate. The environment is described by continuous-time finite and ergodic Markov chain. Sample elements are positive random variables, which can be interpreted as lifetime till a failure. If this random variable is greater than value t > 0 and the environment has state i, then failure rate γ _i (t; β ⁽ⁱ⁾) is a known function of unknown coefficients $\beta ^{(i)} = (\beta _{1}^{(i)},\ldots,\beta _{m}^{(i)})$. Maximum likelihood equations are derived for estimators of {β ⁽ⁱ⁾}. A partial case when m = 1 and γ _i (t; β ⁽ⁱ⁾) = β _i is considered in detail.

Appendix

Lemma 3.1.

If elements of matrix G(t) are differentiable function of t, then

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial t}G(t)^{n}& =& \sum _{ i=0}^{n-1}G(t)^{i}{\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{n-1-i},\,n = 1,2,\ldots, {}\\ \frac{\partial } {\partial t}\exp (G(t))& =& \frac{\partial } {\partial t}\sum _{i=0}^{\infty }\frac{1} {i!}G(t)^{i} =\sum _{ i=1}^{\infty }\frac{1} {i!}\sum _{j=0}^{i-1}G(t)^{j}{\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{i-1-j}. {}\\ \end{array}$$

Proof.

Lemma 3.1 is true for n = 1 and 2. If one is true for n > 1, then

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial t}G(t)^{n+1}& =& {\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{n} + G(t) \frac{\partial } {\partial t}G(t)^{n} {}\\ & =& {\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{n} + G(t)\sum _{ i=0}^{n-1}G(t)^{i}{\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{n-1-i} {}\\ & =& \sum _{i=0}^{n}G(t)^{i}{\biggl [ \frac{\partial } {\partial t}G(t)\biggr ]}G(t)^{n-i}. {}\\ \end{array}$$

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