# Inequalities for a distribution with monotone hazard rate

- 58 Downloads
- 4 Citations

## Summary

Let*X* be a positive random variable with the survival function\(\bar F\) and the density*f*. Let*X* have the moments μ=E(*X*) and μ_{2}=E(*X*^{2}) and put ε=|1-μ_{2}/2μ^{2}|. Put\(q(x) = f(x)/\bar F(x)\) and\(q_1 (x) = \bar F(x)/\int_x^\infty {\bar F(u)du} \). It is proved that the following inequalities hold:\(|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon e)\), for all*x*>0, if*q(x)* is monotone and that\(\int_0^\infty {|\bar F(x) - e^{ - x/\mu } |} dx \leqq 2\varepsilon \mu \), if*q*_{1}*(x)* is monotone. It is also shown that Brown's inequality\(|\bar F(x) - e^{ - x/\mu } | \leqq \varepsilon /(1 - \varepsilon )\) which holds whenever*q*_{1}*(x)* is increasing is not valid in general when*q*_{1} is decreasing.

## Key words

Characterization exponential distribution hazard rate mean residual life## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Azlarov, A. T. and Volodin, N. A. (1982). On proximity to exponential distribution with monotonic failure intensity,
*Theory Prob. Appl.*,**26**, 650. Summary of papers presented at Fifth Sem. on Problems of Stability of Stochastic Models on Nov., 1980.CrossRefGoogle Scholar - [2]Barlow, R. E., Marshall, A. W. and Proschan, F. (1963). Properties of probability distributions with monotone hazard rate,
*Ann. Math. Statist.*,**34**, 375–389.MathSciNetCrossRefGoogle Scholar - [3]Barlow, R. E. and Proschan, F. (1965).
*Mathematical Theory of Reliability*, John Wiley.Google Scholar - [4]Brown, M. (1983). Approximating IMRL distributions by exponential distributions, with applications to first passage times,
*Ann. Prob.*,**11**, 419–427.MathSciNetCrossRefGoogle Scholar - [5]Hall, P. (1979). On measures of distance of a mixture from its parent distributions,
*Stochastic Processes and their Appl.*,**8**, 357–365.MathSciNetCrossRefGoogle Scholar - [6]Heyde, C. C. and Leslie, J. R. (1976). On moment measures of departure from the normal and exponential laws,
*Stochastic Processes and Their Appl.*,**4**, 317–328.MathSciNetCrossRefGoogle Scholar - [7]Karlin, S., Proschan, F. and Barlow, R. E. (1961). Moment inequalities of Polya frequency functions,
*Pacific Jour. of Math.*,**11**, 1023–1033.CrossRefGoogle Scholar - [8]Marshall, A. W. and Proschan, F. (1972). Classes of distributions applicable in replacement with renewal theory implications,
*Proc. of Sixth Berkeley Symp. on Math. Statist. and Prob.*,**1**, 395–415.MathSciNetzbMATHGoogle Scholar - [9]Scoenberg, I. J. (1951). On Polya frequency functions,
*Jour. Analyse Math.*,**1**, 331–374.CrossRefGoogle Scholar