Remark On Variance Bounds

TL;DR

This paper demonstrates that the variance of combined series can be used to derive simple proofs for well-known variance bounds, offering a new perspective on variance estimation.

Contribution

The paper introduces a novel approach using combined series variance formulas to simplify proofs of classical variance bounds.

Findings

Simplified proofs of variance bounds

New perspective on variance estimation

Potential for broader application in statistical analysis

Abstract

It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.

Equations

$$S_{n_1+n_2}^2=\frac{n_1}{n_1+n_2}{S}_{n_1}^2+\frac{n_2}{n_1+n_2}{S}_{n_2}^2+\frac{n_1{n}_2}{{\left(n_1+n_2\right)}^2}{\left(\overline{X}-\overline{Y}\right)}^2.$$

$$S_n^2=\frac{n-1}{n}{S}_{n-1}^2+\frac{1}{n-1}{\left(x_j-\overline{x}\right)}^2,$$

$$S_n^2\ge \frac{1}{n-1}{\left(x_j-\overline{x}\right)}^2,$$

$$S_n^2\ge \frac{n-1}{n}{S}_{n-1}^2.$$

$$S_n^2\ge \frac{m}{n}{S}_m^2.$$

$$S_n^2=\frac{n-2}{2}{S}_{n-2}^2+\frac{1}{2n}{\left(x_j-x_k\right)}^2+\frac{2}{n-2}{\left(\overline{x}-\frac{x_j+x_k}{2}\right)}^2,$$

$$S_n^2\ge \frac{1}{2n}{\left(x_j-x_k\right)}^2,$$

$$S_n^2\ge \frac{1}{2n}{\left(M-m\right)}^2.$$

$$S_n^2\ge \frac{1}{2n}{\left(M-m\right)}^2+\frac{2}{n-2}{\left(\overline{x}-\frac{m+M}{2}\right)}^2.$$

$$S_n^2\ge \frac{r}{n-r}{\left({\gamma }_r-\overline{x}\right)}^2,$$

$$S_{n_1+n_2}^2\ge \frac{n_1{n}_2}{{\left(n_1+n_2\right)}^2}{\left(\overline{X}-\overline{Y}\right)}^2.$$

$$\overline{x}-\sqrt{\frac{n-k}{k}}{S}_n\le x_k\le \overline{x}+\sqrt{\frac{k-1}{n-k+1}}{S}_n$$

Full text

Remark On Variance Bounds

R. Sharma

Department of Mathematics & Statistics

Himachal Pradesh University

Shimla -5,

India - 171005

email: rajesh  hpu  math@yahoo.co.in

**Abstract. **It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds. ** **

**AMS classification 60E15 **

Key words and phrases : Mean, Variance, Samuelson’s inequality.

1 Introduction

It is well known that if we have two sets of data $X$ and $Y$ containing $n_{1}$ and $n_{2}$ observations with means $\overline{X}$ and $\overline{Y}$ , and variances $S_{n_{1}}^{2}$ and $S_{n_{2}}^{2},$ respectively, then the combined variance of $n_{1}+n_{2}$ observations is given by

[TABLE]

Let $X=\left\{x_{j}\right\}$ be a sample of size one and let $Y$ be the sample of size $n-1$ drawn from the population $\left\{x_{1},x_{2},...,x_{n}\right\}$ such that $X\cap Y=\phi.$ Then $n_{1}=1,n_{2}=n-1,\overline{X}=x_{j},\overline{Y}=\frac{1}{n}\sum\limits_{i\neq j}x_{j},S_{1}^{2}=0,$ and it follows from (1.1) that

[TABLE]

where $\overline{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}$ and $S_{n}^{2}=\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}$are respectively the mean and variance of the data $\left\{x_{1},x_{2},...,x_{n}\right\}.$

Each summand in (1.2) is non-negative, so

[TABLE]

for all $j=1,2,...,n.$

The inequality (1.3) is known as Samuelson’s inequality (1968) in statistical literature. The inequality (1.3) was also established in mathematical literature by Laguerre (1880) in some different context and notations. Several alternative proofs of this inequality were given in literature, see Arnold and Balakrishna (1989), and Rassias and Srivastava (1999).

It may be noted here that the identity (1.2) also implies that

[TABLE]

Thus, if $S_{m}^{2}$ is the variance of a sample of size $m$ drawn from a population of size $n,$ then

[TABLE]

Let $X=\left\{x_{j},x_{k}\right\}$ be a sample of size $2$ and let $Y$ be the sample of size $n-2$ drawn from the population $\left\{x_{1},x_{2},...,x_{n}\right\}$ such that $X\cap Y=\phi.$ Then $n_{1}=2,$ $n_{2}=n-2,$ $\overline{X}=\frac{x_{j}+x_{k}}{2},$ $\overline{Y}=\frac{1}{n-2}\sum\limits_{i\neq j,k}x_{i},$ $S_{2}^{2}=\frac{\left(x_{j}-x_{k}\right)^{2}}{4},$ and it follows from (1.1) that

[TABLE]

for all $j=1,2,...,n$ and $n\geq 3.$

Each summand in (1.4) is non-negative, so

[TABLE]

for all $j=1,2,...,n.$ From (1.5), for $m\leq x_{i}\leq M,i=1,2,...,n,$ we have

[TABLE]

The inequality (1.6) is due to Nagy (1918). See also Nair (1948) and Thompson (1935).

Likewise, from (1.4), we have for $n\geq 3,$

[TABLE]

The inequality (1.7) provides a refinement of (1.6), see Sharma et al. (2008).

Mallows and Richter (1969) proves an extension of the Samuelson inequality (1.3). This says that if $\gamma_{r}$ is the mean of any subset of $r$ numbers chosen from the set $\left\{x_{1},x_{2},...,x_{n}\right\},$ then

[TABLE]

for $r=1,2,...,n-1.$

From (1.1), we have

[TABLE]

Let $X$ be a sample of size $r$ and let $Y$ be the sample of size $n-r$ drawn from the population $\left\{x_{1},x_{2},...,x_{n}\right\}$ such that $X\cap Y=\phi.$ Then $n_{1}=r,$ $n_{2}=n-r,$ $\overline{X}-\overline{Y}=\frac{n}{n-r}\left(\gamma_{r}-\overline{x}\right),$and so (1.8) follows from (1.9).

Likewise, we can deduce Boyd-Hawkins inequalities (1971) from (1.9). This says that if $x_{1}\leq x_{2}\leq...\leq x_{n},$ then

[TABLE]

for $k=2.3,...,n-1.$

The variance bounds have various extensions and applications in statistics, polynomials and matrix theory. We see that formula (1.1) provides further insight, and is very useful in the study of these inequalities. In this way we can study various further refinements, generalisations and extensions of the variance bounds.