The heat equation and Stein's identity: Connections, applications

L. Brown, A. DasGupta, L. R. Haff, W. E. Strawderman

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for moments, an improvement and a reversal of Jensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory. Some examples are also given.

Original languageEnglish (US)
Pages (from-to)2254-2278
Number of pages25
JournalJournal of Statistical Planning and Inference
Issue number7 SPEC. ISS.
StatePublished - Jul 1 2006

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


  • Bayes risk
  • Harmonic
  • Heat equation
  • Inadmissibility
  • Matching polynomial
  • Stein's identity
  • Unbiased


Dive into the research topics of 'The heat equation and Stein's identity: Connections, applications'. Together they form a unique fingerprint.

Cite this