Scholarlink

<p>For a multinomial distribution, suppose we have prior knowledge about the sum of the probabilities of certain categories. This enables the construction of a submodel within the full (i.e., unrestricted) model. The maximum likelihood estimator (MLE) under such a submodel is generally expected to achieve higher estimation efficiency than the MLE under the full model. However, this article shows that this expectation does not always hold. We derive the asymptotic expansion of the risk of the MLE, with respect to Kullback–Leibler divergence, for both the full model and the “m-aggregation” submodel. The results indicate that the second-order term (the order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41884_2025_179_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(n^{-2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> term) of the submodel is larger than that of the full model unless the submodel is based on “solid” prior knowledge. From this theoretical finding, we conjecture that in some cases the submodel may yield a higher risk than the full model. We confirm this conjecture by presenting an explicit example in which the use of the submodel increases the risk.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Efficiency of maximum likelihood estimation for a multinomial distribution with known probability sums

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年10月18日

查看全文
  • Yo Sheena

摘要

For a multinomial distribution, suppose we have prior knowledge about the sum of the probabilities of certain categories. This enables the construction of a submodel within the full (i.e., unrestricted) model. The maximum likelihood estimator (MLE) under such a submodel is generally expected to achieve higher estimation efficiency than the MLE under the full model. However, this article shows that this expectation does not always hold. We derive the asymptotic expansion of the risk of the MLE, with respect to Kullback–Leibler divergence, for both the full model and the “m-aggregation” submodel. The results indicate that the second-order term (the order \(n^{-2}\) n - 2 term) of the submodel is larger than that of the full model unless the submodel is based on “solid” prior knowledge. From this theoretical finding, we conjecture that in some cases the submodel may yield a higher risk than the full model. We confirm this conjecture by presenting an explicit example in which the use of the submodel increases the risk.