# Estimation of Star-Shaped Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Continuous Star-Shaped Distributions

#### 2.1. The General Distribution Class

**Assumption**

**1.**

**Lemma**

**1.**

**Proof.**

#### 2.2. A Class of Two-Dimensional Distributions Whose Contour Defining Star Bodies Are Squared Sine Transformed Euclidean Circles

**Example**

**1.**

**Example**

**2.**

#### 2.3. Norm-Contoured Distributions

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

#### 2.4. Antinorm-Contoured Distributions

**Example**

**9.**

#### 2.5. Continuous Non-Concentric Elliptically Contoured Distributions

## 3. Estimation for Continuous Star-Shaped Distributions

#### 3.1. Parametric Estimators

- (1)
- Modified exponential model. $\theta =\tau \in (0,+\infty )$,$${f}_{\tau}(r)=\frac{1}{(d+1)(d-1)!}{\tau}^{-d}{r}^{d-1}\left(1+\frac{r}{\tau}\right){e}^{-r/\tau}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}r>0$$$${\int}_{0}^{\infty}r{f}_{\theta}(r)\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}r=\frac{d(d+2)\tau}{d+1}.$$
- (2)
- Weibull model. $\theta =(\tau ,a)\in (0,+\infty )\times (1,+\infty )$,$${f}_{\theta}(r)=\frac{a}{{\tau}^{d}\mathrm{\Gamma}(d/a)}{r}^{d-1}{e}^{-{(r/\tau )}^{a}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}r>0$$$${\int}_{0}^{\infty}r{f}_{\theta}(r)\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}r=\frac{\mathrm{\Gamma}(\frac{d+1}{a})\tau}{\mathrm{\Gamma}(\frac{d}{a})}.$$

#### 3.2. Nonparametric Estimators without Scale Fit

#### 3.2.1. Estimating μ and ${F}_{R}$

**Theorem**

**1.**

#### 3.2.2. Density Estimation

#### 3.2.3. Assumptions Ensuring Convergence Properties of Estimators

**Assumption**

**2.**

**Assumption**

**3.**

**Example**

**10.**

**Assumption**

**4.**

**Assumption**

**5.**

**Example**

**11.**

#### 3.2.4. Properties of the Density Estimator

**Theorem**

**2.**

**Theorem**

**3.**

- (i)
- Define$$\begin{array}{ccc}\hfill {\overline{\sigma}}^{2}(\tilde{x})& =& {\mathcal{O}}_{S}{(S)}^{-2}{\tilde{x}}^{1-d}{\psi}^{\prime}(\tilde{x})g(\tilde{x}){\int}_{-1}^{1}{k}^{2}(t)\phantom{\rule{0.166667em}{0ex}}dt,\hfill \\ \hfill \mathrm{\Lambda}(\tilde{x})& =& {\mathcal{O}}_{S}{(S)}^{-1}{\tilde{x}}^{1-d}{\psi}^{\prime}(\tilde{x})\frac{1}{p!}{\chi}^{(p)}(\psi (\tilde{x})){\int}_{-1}^{1}{t}^{p}k(t)\phantom{\rule{0.166667em}{0ex}}dt.\hfill \end{array}$$$${\widehat{\phi}}_{n}(x)-{\phi}_{g,K,\mu}(x)={Z}_{n}+{e}_{n},$$$$\sqrt{nb}{Z}_{n}\stackrel{d}{\u27f6}\mathcal{N}(0,{\overline{\sigma}}^{2}(\tilde{x}))\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}n\to \infty .$$
- (ii)
- If additionally ${lim}_{n\to \infty}{n}^{1/(2p+1)}b={C}_{4}$ holds true with a constant ${C}_{4}\ge 0$, then, for $n\to \infty $,$$\sqrt{nb}\left({\widehat{\phi}}_{n}(x)-{\phi}_{g,K,\mu}(x)\right)\stackrel{d}{\u27f6}\mathcal{N}({C}_{4}^{(2p+1)/2}\mathrm{\Lambda}(\tilde{x}),{\overline{\sigma}}^{2}(\tilde{x})).$$

#### 3.2.5. Reference Bandwidth

#### 3.3. Semiparametric Estimators Involving a Scale and a Parameter Fit

**Assumption**

**6.**

**Assumption**

**7.**

**Assumption**

**8.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

#### 3.4. Applications

**Example**

**12.**

**Example**

**13.**

## 4. Proofs

#### 4.1. Proof of Auxiliary Statements

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

#### 4.2. Proving Convergence of ${\widehat{F}}_{n}^{R}$

#### 4.3. Proving Strong Convergence of the Density Estimator

**Lemma**

**5.**

**Lemma**

**6.**

- where ${U}_{nil}:=\overline{k}(({u}_{l}-{\tilde{Y}}_{i})/b)\mathbf{1}\{|{\tilde{Y}}_{i}-{u}_{l}|\le b-{w}_{n}\}\lambda ({X}_{i})$,
- ${\overline{U}}_{nil}:=\overline{k}(({u}_{l}+{\tilde{Y}}_{i})/b)\mathbf{1}\{|{\tilde{Y}}_{i}-{u}_{l}|\le b-{w}_{n}\}\lambda ({X}_{i})$,
- ${V}_{inl}:=\mathbf{1}\{b-{w}_{n}<|{\tilde{Y}}_{i}-{u}_{l}|<b+{w}_{n}\}$.

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Proof**

**of Theorem 2:**

#### 4.4. Proving Asymptotic Normality of ${\widehat{\phi}}_{n}(x)$

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof**

**of Theorem 3.**

#### 4.5. Proofs When Additional Scale Fit Is Involved

**Lemma**

**11.**

**Proof.**

**Proof**

**of Theorem 4.**